Keywords: Équations intégrales, Aéroacoustique, Équation d'Helmholtz convectée, Couplage éléments finis Integral equations, Aeroacoustics, Convected Helmholtz equation, Finite element and boundary element coupling, Combined fields integral equations, Reduced basis method, Certified approximation, A posteriori error bound, Empirical interpolation method, Nonintrusive approximation X

Méthodes de réduction de modèles appliquées à des problèmes d'aéroacoustique résolus par équations intégrales

Résumé : Cette thèse s'articule autour de deux thématiques : les méthodes numériques pour la propagation d'ondes acoustiques sous écoulement et les méthodes de réduction de modèles. Dans la première thématique, nous développons une méthode de couplage d'éléments finis et d'éléments de frontière pour résoudre l'équation d'Helmholtz convectée, lorsque l'écoulement est uniforme à l'extérieur d'un domaine borné. En particulier, nous proposons une formulation bien posée à toutes les fréquences de la source. Dans la deuxième thématique, nous proposons une solution au problème classique d'accumulation d'arrondis machine qui survient en calculant l'estimateur d'erreur a posteriori dans la méthode des bases réduites. Par ailleurs, nous proposons une méthode non intrusive pour calculer une approximation sous forme séparée des systèmes linéaires résultant de l'approximation en dimension finie de problèmes aux limites dépendant d'un ou plusieurs paramètres.

Mots-clés :

1

Introduction générale 1.1 Contexte industriel

La réduction du bruit en aéronautique constitue un enjeu industriel considérable. En effet, face à l'augmentation substantielle du trafic aérien, les normes de certification acoustique sont fortement renforcées afin de protéger les riverains des nuisances sonores essentiellement occasionnées par les avions aux abords des aéroports. Cette volonté de réduire le bruit d'origine aéroportuaire s'accompagne d'un besoin en outil numérique permettant de prévoir la propagation dans un fluide du son produit par un avion avant même sa construction. Nous nous intéressons en particulier au bruit généré par les turboréacteurs. Or, aujourd'hui, peu d'outils numériques et de méthodes sont disponibles pour la simulation de la propagation du son dans un fluide. Il existe des méthodes issues de la mécanique des fluides basées sur les équations de Navier-Stokes. Cependant, ces méthodes sont lourdes en terme de puissance de calcul et sont peu adaptées (difficulté pour extraire la contribution de l'acoustique). Une autre possibilité consiste à utiliser les équations d'Euler linéarisées. Ces équations se ramènent, dans le cas d'un écoulement nul, à la classique équation d'Helmholtz, qui peut être résolue par équation intégrale. C'est cette deuxième approche que nous choisissons d'exploiter et d'enrichir afin de pouvoir prendre en compte la convection de l'écoulement dans lequel se propagent les ondes.

Bruit généré par un avion

Le bruit généré par un avion est créé soit par la turbulence de l'écoulement autour de certaines parties de l'avion, soit directement par certaines pièces mécaniques, voir figure 1.1.

La génération du bruit par un turboréacteur est liée à son fonctionnement, en particulier aux quatre phases qui ont lieu simultanément : l'admission, la compression, la combustion et la détente. Chacune de ces quatre phases est à l'origine d'une contribution dans la perturbation acoustique totale créée par le turboréacteur. Lors de l'admission au niveau de l'entrée d'air, le bruit de soufflante est généré. Viennent ensuite le bruit des compresseurs et le bruit généré par la combustion. La poussée contribue au bruit de jet. Enfin, la turbine qui assure la rotation des arbres du turboréacteur est à l'origine du bruit de turbine. Des mesures expérimentales montrent que, lors de l'atterrissage et du décollage, pour les turboréacteurs actuels à double flux, la perturbation prépondérante est le bruit de soufflante, voir figure 1.2.

L'équation d'Helmholtz convectée

Pour cette introduction sur les phénomènes aéroacoustiques, nous nous référons à [START_REF] Hirschberg | An Introduction to Acoustics[END_REF]. Dans cette section, nous expliquons comment est obtenue l'équation utilisée pour la simulation d'ondes acoustiques dans les zones 2 et 3 de la figure 1.4. Nous partons des équations de Navier-Stokes pour l'écoulement total : masse :

∂ ∂t ρ + ∇ • (ρv) = 0, (1.1a) quantité de mouvement : ∂ ∂t (ρv) + ∇ • (ρv ⊗ v) = -∇p + ∇ • τ , (1.1b) énergie : ∂ ∂t (E) + ∇ • (Ev) = -∇ • q -∇ (pv) + ∇ (τ • v) , (1.1c)
où ρ est la densité, v est la vitesse, p est la pression, τ est le tenseur des contraintes, E est l'énergie totale et q est le flux de chaleur dû à la conduction thermique. L'énergie totale se décompose de la façon suivante : E = ρe+ 1 2 ρv 2 , où e est la densité d'énergie interne et 1 2 ρv 2 est la densité d'énergie cinétique. L'enthalpie est définie par h = e + p ρ . En notant T le température, l'entropie s peut être introduite via la loi fondamentale de la thermodynamique pour les processus réversibles :

T ds = de + pd ρ -1 = dh -ρ -1 dp.
(1. (1.5b)

Les capacités thermiques vérifient la relation de Mayer des gaz parfaits : C P -C V = R. Un gaz parfait est dit de Laplace si les capacités thermiques sont constantes, alors que pour des gaz réels loin des conditions normales de température et de pression, elles peuvent dépendre significativement de la température. Nous considérons que l'air est un gaz parfait. Le coefficient de Laplace d'un gaz parfait est donné par γ = C P C V . Pour l'air, R ≈ 286.73 J.kg -1 .K -1 et γ ≈ 1.402. D'après (1.5), un gaz parfait vérifie

ds = C V dp p -C P dρ ρ .
(1.6)

Les perturbations acoustiques dans l'air sont isentropiques. Elles se propagent à vitesse c, telle que

c 2 = ∂p ∂ρ s = γp ρ = γRT.
(1.7)

Dans le cas des gaz parfaits de Laplace, nous pouvons intégrer les relations (1.5) et (1. [START_REF] Barrault | An 'empirical interpolation' method: application to efficient reduced-basis discretization of partial differential equations[END_REF] pour obtenir e = C V T + e init , (1.8a)

h = C p T + h init , (1.8b) s = C V log p -C P log ρ + s init , (1.8c)
où e init , h init et s init sont les constantes d'intégration, correspondant à un état de référence du gaz.

Approximation acoustique

Dans le domaine acoustique, les termes visqueux et turbulent ne vont jouer un rôle que dans les zones de création de bruit aérodynamique, tandis que les perturbations acoustiques sont trop rapides pour être affectées par les effets de conduction thermique. Pour le calcul de la propagation acoustique, nous négligeons donc les termes dus au tenseur des contraintes τ et au flux de chaleur q. Pour mieux comprendre cette approximation, nous adimensionnons les équations en introduisant les quantités adimensionnées telles que : x := Lx, v := v 0 ṽ, t := L v 0 t, ρ := ρ 0 ρ, dp := ρ 0 v 2 0 dp, τ := µv 0 L τ , q := κ∆T L q, T := T 0 T , dT := ∆T d T et ds := C P ∆T T 0 ds avec les facteurs d'échelle L (longueur en m), v 0 (vitesse en m.s -1 ), ρ 0 (densité en kg.m -3 ), µ (viscosité dynamique en P a.s), κ (conductivité thermique en W.m -1 .K -1 ), T 0 (température en K), ∆T (variation de température en K), C p (capacité thermique à pression constante en J.kg -1 .K -1 ). Les équations de Navier-Stokes se réécrivent sous la forme masse : Maintenant que nous avons simplifié les équations de Navier-Stokes dans le régime de propagation acoustique qui nous intéresse, nous allons séparer la description de l'écoulement moyen et celle de la perturbation acoustique.

D D t ρ = -ρ ∇ • ṽ, (1.9a

Perturbation autour d'un écoulement moyen

Nous considérons que l'écoulement total est la somme d'un écoulement moyen stationnaire et d'une perturbation acoustique instationnaire, tels que

v = v 0 + v ′ , p = p 0 + p ′ , ρ = ρ 0 + ρ ′ , et s = s 0 + s ′ ,
(1.12) où l'indice 0 désigne l'écoulement moyen et l'apostrophe désigne la perturbation acoustique. Nous linéarisons dans le régime des petites perturbations et obtenons pour l'écoulement moyen masse : ∇ • (ρ 0 v 0 ) = 0, (1.13a) quantité de mouvement :

ρ 0 (v 0 • ∇) v 0 = -∇p 0 , (1.13b) entropie : (v 0 • ∇) s 0 =0, (1.13c)
et les relations (1.6) et (1.7) deviennent

ds 0 = C V dp 0 p 0 -C P dρ 0 ρ 0 , (1.14) et c 2 0 = γp 0 ρ 0 . (1.15)
Pour la perturbation acoustique, il vient au premier ordre masse :

∂ ∂t ρ ′ + ∇ • v 0 ρ ′ + v ′ ρ 0 = 0, (1.16a) quantité de mouvement : ρ 0 ∂ ∂t + v 0 • ∇ v ′ + ρ 0 v ′ • ∇ v 0 + ρ ′ (v 0 • ∇) v 0 = -∇p ′ ,
(1.16b) entropie :

∂ ∂t + v 0 • ∇ s ′ + v ′ • ∇s 0 = 0, (1.16c)
et avec la convention s init = 0,

s ′ = C V p 0 p ′ - C P ρ 0 ρ ′ .
(1.17)

Écoulement irrotationel et isentropique

Lorsque l'écoulement est irrotationel et que le domaine est simplement connexe, nous pouvons introduire un potentiel ϕ tel que v = ∇ϕ. La conservation de la quantité de mouvement (1.10b) s'écrit alors

ρ ∂ ∂t (∇ϕ) + (∇ϕ) • ∇ (∇ϕ) + ∇p = 0, (1.18)
ou encore

∇ ∂ ∂t ϕ + ∇ϕ 2 2 + 1 ρ ∇p = 0. (1.19)
Lorsque l'écoulement est isentropique partout (homentropique), nous pouvons utiliser (1.11) pour écrire 1

ρ ∇p = Kγρ γ-2 ∇ρ = ∇ Kγ γ -1 ρ γ-1 .
(1.20)

Nous pouvons alors intégrer (1. [START_REF] Boyaval | A reduced basis approach for variational problems with stochastic parameters: Application to heat conduction with variable Robin coefficient[END_REF]) en espace :

∂ ∂t ϕ + ∇ϕ 2 2 + Kγ γ -1 ρ γ-1 = f (t), (1.21)
où f (t) est une fonction du temps résultant de l'intégration en espace. Il est possible de prendre f (t) = 0, quitte à remplacer le potentiel ϕ par ϕ-t 0 f (t)dt, ce qui est possible dans la mesure où le potentiel n'est défini qu'à une fonction du temps près : ∇ ϕ -t 0 f (t)dt = ∇ϕ = v. Comme dans la Section 1.2.2, nous séparons la contribution stationnaire de la perturbation acoustique dans la définition du potentiel : ϕ = ϕ 0 + ϕ ′ (il suffit de supposer que l'écoulement stationnaire moyen est potentiel pour que la perturbation acoustique soit également potentielle). Le troisième terme du membre de gauche de (1.21) vérifie au premier ordre Nous supposons que les perturbations sont harmoniques. Cette hypothèse convient bien au profil de raies du bruit de soufflante du turboréacteur décrit dans la Section 1.1.1, où en pratique seules quelques pulsations décrivent la perturbation acoustique totale. Nous simplifions la description à une pulsation ω, telle que v ′ = Re ve -iωt , ρ ′ = Re ρe -iωt et ϕ ′ = Re φe -iωt .

Kγ γ -1 ρ γ-1 = Kγ γ -1 ρ γ-1 0 + Kγρ γ-2 0 ρ ′ . (1.
(1.24)

Les relations (1.16a) et (1. [START_REF] Buffa | Regularized combined field integral equations[END_REF]) deviennent alors masse :

-iω ρ + ∇ • (ρv 0 + ρ 0 ∇ φ) = 0, (1.25a) quantité de mouvement intégrée :

ρ 0 c 2 0 (iω φ -v 0 • ∇ φ) = ρ.
(1.25b) L'équation sur le potentiel acoustique est alors obtenue en injectant l'expression de ρ fournie par (1.25b) dans l'équation de conservation de la masse (1.25a) :

-iω

ρ 0 c 2 0 (iω φ -v 0 • ∇ φ) + ∇ • ρ 0 c 2 0 (iω φ -v 0 • ∇ φ) v 0 + ρ 0 ∇ φ = 0, (1.26)
ou encore, en posant k 0 := ω c 0 et M 0 := v 0 c 0 ,

ρ 0 k 2 0 φ + ik 0 M 0 • ∇ φ + ∇ • [ρ 0 (∇ φ -(M 0 • ∇ φ) M 0 + ik 0 φM 0 )] = 0. (1.27)
L'équation (1.27) est connue sous le nom d'équation d'Helmholtz convectée (voir [START_REF] Bécache | Perfectly matched layers for the convected helmholtz equation[END_REF]). Un terme source non nul peut être ajouté dans le membre de droite du bilan de masse (1.25a) conduisant à (1.27) avec ce même membre de droite. Dans le cas où l'écoulement moyen stationnaire est nul, ρ 0 et k 0 sont uniformes et M 0 = 0, conduisant à l'équation d'Helmholtz classique

∆ φ + k 2 0 φ = 0.
(1.28)

Dans la plupart des cas tests de cette thèse, nous considérons des objets plongés dans des écoulements potentiels et uniformes (correspondant aux zones 2 et 3 de la figure 1.4), et un terme source de type monopole acoustique. Un monopole acoustique localisé en x s ∈ R 3 , d'amplitude A s et de pulsation ω donne lieu à un terme source g(x, t) := A s δ xs (x) cos(ωt), où δ xs (x) est la distribution de Dirac centrée en x s . Pour simuler la propagation des perturbations acoustiques générées par une telle source dans un écoulement potentiel ou nul, il suffit d'ajouter A s δ xs au membre de droite des équations (1.27) et (1. [START_REF] Carstensen | On the adaptive coupling of FEM and BEM in 2-D-elasticity[END_REF]. Dans cette thèse, nous supposerons toujours que l'écoulement dans lequel est plongé l'objet est nonuniforme au plus dans un domaine borné de l'espace. A l'extérieur de ce domaine, l'écoulement est uniforme et par un changement de variable et de fonction inconnue, il est possible (nous y reviendrons au chapitre 3) de transformer l'équation d'Helmholtz convectée (1.27) en l'équation d'Helmholtz classique (1.28). Ainsi, nous nous ramènerons toujours au cas où la fonction cherchée satisfait (1.28) à l'extérieur d'un domaine borné.

De même, le potentiel de double couche est défini par D(λ)(x) := Γ ∇ y E(y -x)λ(y)ds(y), x ∈ R 3 \Γ . D'après [START_REF] Nédélec | Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems[END_REF]Theorem 3.

I + D [γ 0 u] Γ [γ 1 u] Γ = - γ - 0 u - γ - 1 u -.
(1.32) L'opérateur défini par bloc dans le membre de gauche de (1.32) est appelé projecteur de Calderón. Les deux relations dans (1.32) sont simplement des relations nécessaires vérifiées par une solution rayonnante par morceaux de l'équation d'Helmholtz u. Elles ne sont toutefois pas indépendantes (elles sont obtenues en pratique en prenant les traces de Neumann et Dirichlet de (1.30)). Dans un problème aux limites, un comportement peut être imposé à la surface Γ par le biais d'une condition aux limites qui, s'ajoutant à (1.32), peut conduire à un problème bien posé. Un exemple simple est celui où on suppose que u -≡ 0 et γ + 0 u = 0. Alors, par unicité du problème extérieur de Dirichlet, u + ≡ 0 et u ≡ 0. En revanche, si l'on suppose seulement γ + 0 u = γ - 0 u = 0, alors u + ≡ 0 est toujours vrai, mais u -n'est pas déterminée de façon unique si k correspond à une fréquence propre du Laplacien de Dirichlet dans Ω -. Deux autres exemples de condition aux limites sur Γ que nous étudierons en détail dans cette thèse sont les conditions de transmission et la condition de Robin. De façon générale, la méthode des équations intégrales consiste à utiliser les relations (1.32) et des conditions aux limites liées au problème physique pour obtenir un système d'équations bien posé d'inconnues [γ 0 u] Γ et [γ 1 u] Γ , puis à utiliser la formule de représentation (1.30) pour retrouver la fonction inconnue u à partir du saut de ses traces sur Γ .

Méthodes de réduction de modèle

L'équation d'Helmholtz convectée (1.27) est l'équation à la base de notre modèle d'aéroacoustique. Une fois que nous disposons d'une méthode numérique fiable, nous souhaitons être capables de résoudre ce problème pour de nombreuses valeurs de certains paramètres. Ces paramètres peuvent être la fréquence et la position de la source acoustique, ou encore le coefficient d'impédance des objets diffractants. Dans ce contexte, les méthodes de réduction de modèle prennent tout leur sens, car elles permettent d'obtenir très rapidement une approximation de la solution d'un problème. Ainsi, des études de propagation d'incertitude ou d'optimisation, requises en phase de conception d'un projet industriel et nécessitant d'évaluer la solution d'un problème pour de très nombreuses valeurs de certains paramètres, deviennent accessibles.

Dans cette section, nous présentons brièvement quelques méthodes de réduction de modèle.

Introduction : le fléau de la dimension

Le fléau de la dimension (curse of dimensionality) est une expression inventée par Bellman pour qualifier le comportement de l'augmentation de la complexité de description d'un espace avec l'ajout de dimensions supplémentaires, voir [START_REF] Bellman | Dynamic Programming[END_REF][START_REF] Bellman | Adaptive Control Processes[END_REF]. Ce fléau est rencontré dans de nombreuses disciplines. L'exemple le plus simple est peut-être combinatoire : supposons que nous disposons de d pièces de monnaie que nous lançons simultanément. Le nombre de résultats ordonnés possible est 2 d : la taille du problème augmente de façon exponentielle avec la dimension. Un autre exemple est le probème d'échantillonage d'un domaine. Prenons l'hypercube [0, 1] d que nous échantillonnons de façon uniforme dans chaque direction. Nous exigeons que la distance entre deux points soit inférieure à 0.1. En dimension 1, cela donne 11 points. En dimension quelconque, le nombre de points d'échantillonage vaut 11 d , et donc augmente de façon exponentielle avec la dimension.

Dans notre contexte d'aéroacoustique en aviation civile, nous disposons d'une méthode de simulation directe : à partir d'un modèle dépendant de certains paramètres, nous sommes capables de calculer le champ de pression acoustique diffracté. Maintenant, si nous sommes intéressés par l'exploration de l'ensemble des résultats engendrés par les solutions prises par un échantillonage de l'espace des paramètres, nous sommes victimes du fléau de la dimension : le problème aéroacoustique doit être résolu un nombre de fois qui augmente de façon exponentielle avec le nombre de paramètres. Une autre vision peut être de considérer les paramètres comme des variables du problème : notons u(x, µ 1 , µ 2 , • • • , µ d ) le champ de pression acoustique, calculé pour les valeurs µ 1 , µ 2 , • • • , µ d des paramètres. La recherche de la solution u consiste alors à déterminer le champ de pression en tout point de l'espace et pour toutes les valeurs possibles des paramètres. Supposons pour simplifier que x ∈ [0, 1] et que µ i ∈ [0, 1] pour tout 1 ≤ i ≤ d, et considérons u comme une fonction de y = (x, µ 1 , µ 2 , • • • , µ d ) ∈ [0, 1] d+1 . Dans le cadre d'une méthode numérique, il faudra reconstruire une approximation R(u) de u à partir d'un ensemble de N valeurs {u(y i )} 1≤i≤N , où y 1 , • • • , y N ∈ [0, 1] d+1 . Si (y i ) 1≤i≤N sont les noeuds d'une grille uniforme sur [0, 1] d+1 d'espacement h > 0 et si une reconstruction polynomiale est utilisée, il est connu que u -R(u) L ∞ ([0,1] d+1 ) ≤ Ch m , (1. [START_REF] Delnevo | Code acti3s harmonique, justification mathématique, Partie I[END_REF] où C > 0 est une constante indépendante de h et m l'ordre de la méthode. Comme le nombre de points d'échantillonnage N est en O h -(d+1) , l'erreur d'approximation vérifie

u -R(u) L ∞ ([0,1] d+1 ) ≤ CN -m d+1 , (1.34)
ce qui signifie que l'erreur de reconstruction converge d'autant plus lentement avec le nombre de points d'échantillonage que la dimension est grande. Il a été prouvé qu'il est impossible de construire des schémas de reconstruction tels que l'erreur converge plus rapidement [START_REF] Devore | Optimal nonlinear approximation[END_REF]. Dans le cas d'une approximation de Galerkin où les fonctions de base sont obtenues par tensorisation de fonctions de bases univariées, et où on suppose que l'on a N fonctions de base pour la dimension spatiale et N µ pour chaque dimension paramétrique, la taille du problème discrétisé est N N d µ et augmente donc de façon exponentielle avec le nombre de paramètres. Plus de détails peuvent être trouvés dans [START_REF] Erhlacher | Quelques modèles mathématiques en chimie quantique et propagation d'incertitudes[END_REF].

La dépendance exponentielle de la complexité du problème en fonction du nombre de paramètres rend impossible en pratique les approches brutales considérées ci-dessus. Dans la plupart des problèmes rencontrés en pratique, certains paramètres vont avoir une influence sur le résultat beaucoup plus faible que d'autres paramètres. Soit ǫ > 0, notons µ = (µ (1. [START_REF] Devore | Optimal nonlinear approximation[END_REF] Au prix d'une approximation sur le résultat, il est possible de réduire la dimension du problème de d à m. Le plus petit m pour lequel le cas d'égalité de (1.35) est valable pour ǫ = 0 est appelé dimension paramétrique intrinsèque du problème. Intuitivement, on veut fournir des efforts uniquement dans les directions qui influent le plus sur la qualité du résultat. Les méthodes de réduction de modèles ont été développées pour identifier les structures de dépendance principales du résultat en les paramètres et calculer rapidement des approximations de la solution.

1 , µ 2 , • • • , µ d ) ∈ [0,

Les représentations en produits tensoriels

Certaines méthodes de représentation en produits tensoriels cherchent à approcher une fonction u en grande dimension, par une somme de produits tensoriels de fonctions univariées :

u(x, µ 1 , • • • , µ d ) ≈ n k=1 r k (x)s (1) k (µ 1 ) • • • s (d) k (µ d ).
(1.36)

Le but de ces méthodes est de trouver la meilleure approximation possible, à nombre de termes de la somme n fixé.

Les algorithmes gloutons

Des présentations détaillées peuvent être trouvées dans [START_REF] Barron | Approximation and learning by greedy algorithms[END_REF][START_REF] Devore | Some remarks on greedy algorithms[END_REF][START_REF] Temlyakov | Greedy Approximation[END_REF]. Considérons un espace de Hilbert H muni du produit scalaire •, • H et de la norme associée • H . Un ensemble D de fonctions de H est appelé dictionnaire si pour tout élément g de D, g H = 1, et Span(D) = H.

Le problème générique consiste à chercher la meilleure approximation d'un élément u de H comme une combinaison linéaire d'au plus n éléments g 1 , • • • , g n ∈ D: (1. [START_REF] Duprey | Analyse Mathématique et Numérique du Rayonnement Acoustique des Turboréacteurs[END_REF] Avec des hypothèses techniques supplémentaires sur les fonctions de H, on peut montrer des taux de convergence polynomiaux pour PGA et OGA, voir [START_REF] Erhlacher | Quelques modèles mathématiques en chimie quantique et propagation d'incertitudes[END_REF] pour plus de détails.

(g 1 , • • • , g n ) ∈ argmin (d 1 ,••• ,dn)∈D u -P d 1 ,•

Proper Generalized Decomposition

Dans le contexte des équations aux dérivées partielles en grande dimension, la Proper Generalized Decomposition (PGD) est une méthode de réduction de modèle qui utilise les algorithmes gloutons présentés ci-dessus. Pour plus de détails, nous renvoyons à [START_REF] Ladevèze | Nonlinear computational structural mechanics: new approaches and nonincremental methods of calculation[END_REF][START_REF] Ammar | A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids[END_REF][START_REF] Nouy | A priori tensor approximations for the numerical solution of high dimensional problems: alternative definitions[END_REF]. Soit a : Ω × P → R, une fonction mesurable telle qu'il existe α, β > 0 tels que ∀(x, µ) ∈ Ω × P, α ≤ a(x, µ) ≤ β.

(1.41)

Nous considérons le problème de diffusion paramétrique suivant : Trouver u ∈ H := L 2 (P, H 1 0 (Ω)) tel que -∇ x • (a∇ x u) = f.

(1. [START_REF] Ern | Theory and Practice of Finite Elements[END_REF] Soit H un espace de Hilbert de fonctions multivariées u(x, µ 1 , • • • , µ d ) et H x , H 1 , ..., H d des espaces de Hilbert de fonctions univariées dépendant respectivement de x, µ 1 , ..., µ d . La PGD repose sur le choix de dictionnaire suivant : (1. [START_REF] Fares | The reduced basis method for the electric field integral equation[END_REF])

D := r ⊗ s (1) ⊗ • • • ⊗ s (d) | r ∈ H x , s (1) ∈ H 1 , • • • , s (d) ∈ H d , r ⊗ s (1) ⊗ • • • ⊗ s (d) H = 1 , (1.43) où r⊗s (1) ⊗• • •⊗s (d) (x, µ 1 , • • • , µ d ) = r(x)s (1) (µ 1 ) • • • s (d) (µ d )
où E(v) = v -u 2 H et v 2 H = Ω×P a(x, µ) |∇ x v(x, µ)| 2 dxdµ.
La PDG pour l'approximation de la solution de (1.42) consiste à déterminer itérativement r n , s d) .

(1) n , • • • , s (d) n ∈ H x × H 1 × • • • × H d tel que r n , s (1) n , • • • , s (d) n ∈ argmin (r,s (1) ,••• ,s (d) )∈Hx×H1ו••×Hd E n-1 k=1 r k ⊗ s (1) k ⊗ • • • ⊗ s (d) k + r ⊗ s (1) ⊗ • • • ⊗ s (
(1.45) En utilisant (1.39), nous remarquons que cela revient exactement à réaliser un algorithme PGA pour l'approximation de la solution de (1.42) dans H pour le choix de dictionnaire (1.43). En pratique, le calcul de r n , s

(1) n , • • • , s (d) n
à chaque itération se fait en résolvant les équations d'Euler associées au problème de minimisation (1.45) par une procédure de point fixe.

Le fléau de la dimension est contourné dans la mesure où le calcul d'une approximation de (1.42) à n termes nécessite de résoudre n problèmes de taille

N x + N 1 + • • • + N d , où N x et N i , 1 ≤ i ≤ d,
sont les dimensions des espaces vectoriels approchant respectivement H x et H i .

Une condition nécessaire pour que la PGD soit valable est que le problème soit symétrique, comme c'est le cas pour (1.42), pour que la forme faible soit équivalente à la condition d'Euler d'un problème de minimisation.

La méthode des bases réduites

La méthode des bases réduites est celle que nous avons choisie d'étudier dans cette thèse. Nous en rappelons ici le principe général et nous donnons quelques résultats théoriques connus à ce jour.

Introduction et problème réduit

Par la suite, nous notons P l'ensemble des valeurs considérées pour un paramètre µ, et P trial un échantillonage de P. Considérons la formulation variationnelle suivante : Trouver u µ ∈ V tel que a µ (u µ , u t ) = b µ (u t ), ∀u t ∈ V, (1. [START_REF] Giraud | When modified Gram-Schmidt generates a well-conditioned set of vectors[END_REF] où a µ est, dans le cas idéal, une forme bilinéaire, continue et coercive (uniformément en le paramètre µ ∈ P, c'est-à-dire avec des constantes de continuité M et de coercivité α coer indépendantes de µ), b µ est une forme linéaire et V un espace vectoriel de dimension N , approximation conforme d'un espace de Hilbert H. La méthode des bases réduites consiste dans un premier temps à précalculer des solutions u µ i de (1.46) pour des valeurs bien choisies (µ i ) 1≤i≤ N du paramètre au cours de la phase offline. Nous définissions l'espace vectoriel V N := Vect{u µ 1 , • • • , u µ N }. Puis, au cours de la phase online, pour des nouvelles valeurs de µ ∈ P, des approximations û N µ ∈ V N de la solution de (1.46) sont obtenues en résolvant

a µ (û N µ , u µ j ) = b µ (u µ j ), ∀j ∈ {1, ..., N }. (1.47) La décomposition de û N µ dans V N est notée û N µ = N i=1 γ i (µ)u µ i . (1.48) Estimateur d'erreur a posteriori Soit r N µ := G µ û N µ H , où G µ est l'application linéaire de H dans H telle que H ∋ u → G µ u := J (a µ (u, •) -b µ ) ∈ H, avec J l'isomorphisme de Riesz de H ′ dans H tel que pour tout l ∈ H ′ et tout v ∈ V, Jl, v H = l(v).
(1.49)

La quantité r N µ correspond à la norme duale du résidu associé à la formulation (1.46) calculé en la solution réduite û N µ . Dans le contexte de (1.46), on peut prouver que 1

M r N µ ≤ û N µ -u µ H ≤ 1 α coer r N µ .
(1.50)

Pour ce faire, considérons

a µ (û N µ -u µ , û N µ -u µ ) = a µ (û N µ , û N µ -u µ ) -b µ (û N µ -u µ ) = G µ û N µ , û N µ -u µ H , ( 1
û N µ -u µ 2 H ≤ G µ û N µ , û N µ -u µ H ≤ r N µ û N µ -u µ H , d'où l'inégalité de droite de (1.50) est directement obtenue. La formulation variationnelle (1.46) peut être éten- due sur H ′ pour obtenir a µ (û N µ -u µ , v) = a µ (u, •) -b µ , v H ′ ,H , où •, • H ′ ,H est le produit de dualité entre H ′ et H. Utilisons maintenant la continuité de a µ , il vient a µ (u, •) -b µ , v H ′ ,H ≤ M û N µ -u µ H v H . En particulier, r N µ = a µ (u, •) -b µ H ′ = sup 0 =v∈H aµ(u,•)-bµ,v H ′ ,H v H ≤ M û N µ - u µ H , l'inégalité de gauche de (1.50) est directement obtenue. Supposons que {u µ 1 , • • • , u µ N } ont été construites, on définit µ N +1 ∈ P trial tel que M û N µ N +1 -u µ N +1 H ≥ r N µ N +1 := max µ∈P trial r N µ ≥ α coer max µ∈P trial û N µ -u µ H , (1.52) et ainsi û N µ N +1 -u µ N +1 H ≥ γ max µ∈P trial û N µ -u µ H , (1.53)
où γ := αcoer M . Ainsi, le paramètre qui maximise r N µ sur P trial est celui qui maximise l'erreur û N µu µ H sur P trial à la constante γ près.

Efficacité online

On dit que a µ (u, v) et b µ (v) dépendent de µ de façon affine s'il existe µ → α k (µ) et a k (u, v), 1 ≤ k ≤ d a , et µ → β k (µ) et b k (v), 1 ≤ k ≤ d b , tels que a µ (u, v) = d a k=1 α k (µ)a k (u, v) et b µ (v) = d b k=1 β k (µ)b k (v). Sous l'hypothèse de dépendance affine, a µ (u µ i , u µ j ) = d a k=1 α k (µ)a k (u µ i , u µ j ) et b µ (u µ j ) = d b k=1 β k (µ)b k (u µ j ). Ainsi le problème réduit (1.47) peut être construit en complexité indépen- dante de N si les quantités a k (u µ i , u µ j ), 1 ≤ k ≤ d a , 1 ≤ i, j ≤ N , et b k (u µ j ), 1 ≤ k ≤ d b , 1 ≤ i, j ≤ N ,
sont calculées et stockées pendant la phase offline. Sous l'hypothèse de dépendance affine, on montre également que (on suppose ici pour simplifier que H est un espace vectoriel réel)

r N µ 2 = d b k=1 d b p=1 β k (µ)β p (µ) Jb k , Jb p H -2 d b k=1 d a l=1 N i=1 β k (µ)α l (µ)γ i (µ) Jb k , Ja l (u µ i , •) H + d a k=1 d a p=1 N i=1 N j=1 α k (µ)γ i (µ)α p (µ)γ j (µ) Ja k (u µ i , •), Ja p (u µ j , •) H ,
(1.54) où chaque évaluation de J est calculée en résolvant (1.49) dans la phase offline, et où les fonctions

α k (µ), 1 ≤ k ≤ d a , et β k (µ), 1 ≤ k ≤ d b , sont connues, et les coefficients γ i (µ), 1 ≤ i ≤ N

Algorithme de la phase offline

L'algorithme 1 détaille les étapes de la phase offline de la méthode des bases réduites. Le choix a priori de P trial est un problème difficile. Cependant, nous disposons d'un estimateur d'erreur a posteriori calculable en complexité indépendante de N . Pour tout µ ∈ P, il est possible, à chaque évaluation de la solution réduite û N µ , de quantifier l'erreur faite par l'approximation des base réduite. Dans la phase online, s'il arrive que, pour une certain µ * ∈ P, cette erreur soit supérieure à la tolérance fixée dans l'algorithme 1, il est possible d'incrémenter la base réduite en calculant u * µ et en ajoutant µ * à P select . Par ailleurs, l'efficacité du calcul de l'estimateur a posteriori permet d'exécuter la ligne 29 de l'algorithme 1 en complexité indépendante de N et de considérer des P trial de grande taille.

Algorithm 1 Phase offline de la méthode des bases réduites

1. for all k in {1, ..., d b } do 2.
Calculer Jb k 3. end for 4. for all k in {1, ..., d b } do 5. for all l in {1, ..., d b } do 6.

Calculer et sauvegarder (Jb

k , Jb l ) H [Terme de r N µ indépendant de N ] 7.
end for Calculer Ja k (uµ 1 , •)

19.
for all l in {1, ..., d b } do 20.

Calculer et sauvegarder (Jb

l , Ja k (uµ 1 , •)) H [Terme de r N µ linéaire en N ] 21.
end for 22. end for 23. for all k in {1, ..., d a } do 24. for all p in {1, ..., d a } do 25.

Calculer et sauvegarder (Ja

k (uµ 1 , •), Jap(uµ 1 , •)) H [Terme de r N µ quadratique en N ] 26.
end for 27. end for Incrémenter P select = P select ∪ {µ N +1 }

31.

Calculer uµ N +1 32.

Incrémenter V N +1 = Span{ V N , uµ N +1 } 33.
for all k in {1, ..., d b } do 34.

Calculer et sauvegarder b

k (uµ N +1 )
[Incrémenter le membre de droite de (1.47)]

35.

end for 36. for all k in {1, ..., d a } do 37.

Calculer Ja k (uµ N +1 , •)

38.

end for

39.

for all k in {1, ..., d a } do 40. for all i in {1, ..., N + 1} do 41.

Calculer et sauvegarder a

k (uµ i , uµ N +1 ) et a k (uµ N +1 , uµ i ) 42.
[Incrémenter le membre de gauche de (1.47)]

43.

for all l in {1, ..., d b } do 44.

Calculer et sauvegarder (Jb

l , Ja k (uµ N +1 , •))H [Terme de r N µ linéaire en N ] 45.
end for 46. for all p in {1, ..., d a } do 47.

Calculer et sauvegarder (Ja

k (uµ N +1 , •), Jap(uµ i , •))H [Terme de r N µ quadratique en N ]

Résultats de convergence

Les performances de la méthode des bases réduites sont quantifiées par la décroissance de l'erreur û N µu µ H avec N uniformément en µ. D'après le lemme de Cea,

û N µ -u µ H ≤ γ -1 inf v N ∈ V N u µ -v N H . (1.55)
Pour que l'erreur soit faible, il faut donc que V N approche bien l'ensemble F := {u µ , µ ∈ P}. L'épaisseur de Kolmogorov permet de quantifier cette notion :

d N (F ) := inf Y N ⊂H, dim(Y N )= N sup u∈F inf v N ∈Y N u -v N H . (1.56)
Si d N (F ) décroît rapidement avec N , la méthode des bases réduites est susceptible de donner une bonne approximation de u µ , pour tout µ ∈ P et pour N ≪ N . Des vitesses de convergence pour la méthode des bases réduites n'ont pas encore été démontrées dans le cas général, mais quelques résultats théoriques ont été proposés récemment dans [START_REF] Buffa | A priori convergence of the greedy algorithm for the parametrized reduced basis method[END_REF][START_REF] Binev | Convergence rates for greedy algorithms in reduced basis methods[END_REF][START_REF] Devore | Greedy algorithms for reduced bases in Banach spaces[END_REF]. Considérons Supposons que F est un sous-espace compact de H, alors pour tout N ≥ 1,

σ N (F ) := sup u∈F inf v N ∈ V N u -v N H , (1.57
σ N (F ) ≤ √ 2γ -1 min 1≤m≤ N {d m (F )} N -m N . (1.58) En particulier, σ 2 N (F ) ≤ √ 2γ -1 d N (F ).
Ce résultat contient et affine des résultats énoncés précédemment dans [START_REF] Buffa | A priori convergence of the greedy algorithm for the parametrized reduced basis method[END_REF][START_REF] Binev | Convergence rates for greedy algorithms in reduced basis methods[END_REF], en explicitant certaines constantes ([36, Corollary 3.3 (ii)-(iii)]) :

Supposons que F est un sous-espace compact de H. S'il existe des constantes C 0 , α > 0 telles que

d N (F ) ≤ C 0 N -α , (1.59) alors σ N (F ) ≤ 2 5α+1 γ -2 C 0 N -α . (1.60) S'il existe des constantes C 0 , c 0 , α > 0 telles que d N (F ) ≤ C 0 e -c 0 N α , (1.61) alors σ N (F ) ≤ 2C 0 γ -1 e -2 -1-2α c 0 N α . (1.62)
L'estimation (1.58) permet d'obtenir une vitesse de convergence dans le cas où les solutions de (1.46) sont bornées uniformément en µ dans un espace un peu plus régulier que H :

Supposons que H = H 1 (Ω) ou H = H 1 0 (Ω), qu'il existe une constante C indépendante de µ telle que pour tout µ ∈ P, u µ H 2 (Ω) ≤ C et que F est un sous-espace compact de H. Alors il existe une constante C ′ > 0 telle que sup µ∈P û2 N µ -u µ H 1 (Ω) ≤ C ′ N -1 2dx , (1.
sup µ∈P û2 N µ -u µ H 1 (Ω) ≤ γ -1 sup µ∈P inf v 2 N ∈V 2 N v 2 N -u µ H 1 (Ω) = γ -1 σ 2 N (F ). (1.64) D'après le théorème (1.58), sup µ∈P û2 N µ -u µ H 1 (Ω) ≤ √ 2γ -2 d N (F ). (1.65)
Considérons une base infinie e 1 , e

f ∈ H 2 (Ω), il existe g ∈ E N tel que f -g H 1 (Ω) ≤ C ′′ N -1 dx f H 2 (Ω) .
(1.66)

On peut par exemple prendre pour E N l'espace des polynômes continus affines par morceaux sur un maillage régulier de taille de maille

h ∼ N -1 d . En particulier, pour tout f ∈ H 2 (Ω), inf g∈E N f -g H 1 (Ω) ≤ C ′′ N -1 dx f H 2 (Ω) . (1.67) Comme F ⊂ H 2 (Ω) par hypothèse, sup f ∈F inf g∈E N f -g H 1 (Ω) ≤ C ′′ C N -1 dx , (1.68) si bien que d N (F ) ≤ C ′′ C N -1 dx . (1.69)
En utilisant cette borne dans (1.65), la relation (1.63) est obtenue avec

C ′ = √ 2CC ′′ γ -2 .
Nous remarquons que la vitesse de convergence est indépendante de la dimension du paramètre µ. Considérons les classes de problèmes elliptiques suivantes :

L µ u = f µ , dans Ω, u = 0, sur ∂Ω, (1.70) dans H 1 0 (Ω) et L µ u = f µ , dans Ω, ∇u • n = 0, sur ∂Ω, (1.71)
dans H 1 (Ω), où n est la normale sortante sur ∂Ω et

L µ u = - d i,j=1 ∂ ∂x j a µ ij ∂ ∂x i u , (1.72)
avec Ω ouvert borné de classe C 2 . On suppose de plus que a µ ij ∈ C 1 ( Ω) uniformément en µ et qu'il existe une constante c > 0 telle que pour tout µ ∈ P, f µ L 2 (Ω) ≤ c (ou simplement que f µ est indépendant de µ). D'après [START_REF] Brézis | Analyse fonctionnelle[END_REF]Théorèmes IX.25 et IX.26], il existe une constante c ′ > 0 qui ne dépend que de

Ω telle que u µ H 2 (Ω) ≤ c ′ f µ L 2 (Ω) . Par injection compacte de H 2 (Ω) dans H 1 (Ω), F est compact dans H 1 (Ω),

Contenu de la thèse

Cette thèse contient 2 parties et une annexe. Certains chapitres de ce manuscrit sont issus d'articles soumis ou publiés. Ils correspondent donc à l'origine à des travaux indépendants les uns des autres, écrits en langue anglaise. L'auteur s'excuse des quelques répétitions et changements de notation qui en résultent.

Production scientifique

Articles parus ou à paraître dans des revues à comité de lecture est l'équation d'Helmholtz classique (1.28) et la méthodologie présentée dans la section 1.3 est utilisée. En particulier, la condition de Robin modélisant le caractère impédant de l'objet complète les équations (1.32) pour donner un système d'équations bien posé sur les potentiels acoustiques. La preuve d'existence et unicité du problème obtenu est présentée et la preuve de la stabilité inf-sup de l'approximation en dimension finie par éléments de frontière est rappelée en suivant les travaux de [START_REF] Hsiao | Boundary Element Methods: Foundation and Error Analysis[END_REF]. La propriété de stabilité inf-sup est essentielle pour pouvoir utiliser la méthode des bases réduites : elle généralise l'hypothèse de coercivité faite dans la présentation des bases réduites dans la section 1.4.3. Enfin, les problèmes liés à l'existence de fréquences de résonance dans l'utilisation de certaines formulations intégrales sont rappelés et une solution à ce problème, inspirée des travaux de [START_REF] Buffa | Regularized combined field integral equations[END_REF], est présentée. Le contenu de ce chapitre n'est pas nouveau, mais sert à poser les bases des raisonnements présentés dans le chapitre suivant. Comme expliqué dans la section 1.4.3, le succès de la méthode des bases réduites repose sur la formule (1.54) pour l'estimateur a posteriori, car les termes de cette formule sont soit précalculables dans la phase offline de l'algorithme, soit calculables en complexité indépendante de la taille du problème. Cependant, comme illustré dans [Ar1] et [Ar3], cette formule est très sensible aux erreurs d'arrondis machine : au fur et à mesure que la méthode des bases réduites converge, les valeurs prises par l'estimateur d'erreur évalué selon la formule (1.54) sont typiquement de plusieurs ordres de grandeur plus grandes que l'erreur réellement commise. La formule (1.54) devient alors inopérante pour calculer l'estimateur d'erreur de manière précise. Le problème rencontré est similaire à celui qui intervient lorsque l'on évalue un polynôme près de ses racines : le résultat renvoyé par l'ordinateur peut être très différent du résultat exact. Dans [Ar1], nous avons proposé une nouvelle formule pour l'estimateur d'erreur a posteriori, sous la forme d'une combinaison linéaire d'évaluations de l'estimateur à des valeurs données du paramètre, qui sont calculées avec la formule G µ ûµ V pendant la phase offline. Cette dernière formule ne peut pas être utilisée dans la phase online, mais ne rencontre pas les problèmes de précision machine évoqués ci-dessus. Les coefficients de la combinaison linéaire dépendent de la valeur du paramètre en lequel nous souhaitons évaluer l'estimateur d'erreur et sont calculés en résolvant un système linéaire. La difficulté qui subsiste est que ce système linéaire peut être très mal conditionné dans certains cas. Dans [Ar3], nous résolvons ce problème en introduisant une nouvelle formule sur le même modèle que celle de [Ar1], mais dont le système linéaire à résoudre pour déterminer les coefficients de la combinaison linéaire est toujours bien conditionné. La stratégie utilisée est basée sur la méthode d'interpolation empirique. Un exemple numérique académique est proposé et la méthode est également testée sur le problème acoustique décrit dans le chapitre 2, avec comme paramètre le coefficient d'impédance des objets, et où la stabilité inf-sup a été prouvée. Dans ce problème, la constante inf-sup n'est pas indépendante du paramètre et une borne inférieure de cette constante n'est pas connue a priori, contrairement à ce qui a été supposé dans la section 1.4.3 où la constante de coercivité a été supposée indépendante du paramètre. Une possibilité pour calculer une borne inférieure de la constante inf-sup consiste à utiliser la méthode des contraintes successives [START_REF] Huynh | A successive constraint linear optimization method for lower bounds of parametric coercivity and inf-sup stability constants[END_REF], qui est rappelée à la fin de ce chapitre.

Le chapitre 6 reprend la prépublication [Pr1]. Dans la section 1.4.3, nous avons insisté sur le fait que la dépendance de l'opérateur et du second membre en les paramètres doit être affine pour pouvoir utiliser la méthode des bases réduites. Lorsque ce n'est pas le cas, une possibilité consiste à utiliser la méthode d'interpolation empirique pour rétablir l'hypothèse de dépendance affine de façon approchée. Cependant, cette méthode est intrusive, car elle nécessite de modifier les routines d'assemblage élémentaire du code considéré. Dans ce chapitre, nous proposons une méthode qui permet de rétablir l'hypothèse de dépendance affine de manière non intrusive, dans le sens où les routines d'assemblage élémentaires du code ne sont pas modifiées. Notre solution est également basée sur la méthode d'interpolation empirique et ne repose que sur l'hypothèse très raisonnable que l'utilisateur puisse récupérer les matrices complètes assemblées en certaines valeurs du paramètre qu'il aura sélectionnées de manière judicieuse. Un exemple numérique académique est proposé,et la méthode est testée sur un problème intégral résolu sur un maillage d'avion complet, ainsi que sur la deuxième formulation du problème aéroacoustique décrit dans le chapitre 3. Le chapitre 7 est basé sur la prépublication [Pr3]. Ce chapitre reprend l'idée de nonintrusivité introduite dans le chapitre 6. En particulier, de nombreuses variantes possibles pour obtenir une procédure non intrusive sont présentées, et les choix du chapitre 6 sont motivés. Enfin, la procédure est appliquée à la réduction de modèle par bases réduites aux problèmes aéroacoustiques présentés dans la partie I, et non simplement aux matrices du problème comme c'était le cas dans le chapitre 6. Les applications numériques de ce chapitre illustrent les contributions apportées par les deux parties de cette thèse.

Le chapitre 8 reprend la publication [Ar2]. Il décrit un travail réalisé à l'école d'été CEM-RACS 2011 à Luminy. Il est motivé par la problématique industrielle de simulation rapide du champ de température dans une cabine d'avion, en présence de sources thermiques situées dans la soute et produites par des composants électroniques. Dans un premier temps, nous écrivons un solveur 2D en FreeFem++ pour résoudre les équations de Navier-Stokes incompressibles puis l'approximation de Boussinesq. Ensuite, nous considérons l'équation de la chaleur sous convection constante et appliquons la méthode des bases réduites en prenant comme paramètres la conductivité thermique de différents éléments des composants électroniques.

Enfin, le chapitre B en annexe élargit la perspective industrielle et présente l'étude d'un modèle d'incertitudes non paramétriques, dans un contexte de vibration de plaques, pour résoudre un problème d'optimisation sous contraintes en probabilité. Dans le contexte d'aéroacoustique en aviation civile, une fois que nous disposons d'une méthode pour simuler la propagation du bruit généré par le turboréacteur, nous pouvons étudier la puissance acoustique transmise aux passagers à travers le fuselage. Nous déterminons une condition nécessaire à imposer au modèle des plaques du fuselage pour garantir que la probabilité que la puissance acoustique transmise ne dépasse pas un certain seuil soit inférieure à un seuil de tolérance donné.

In this chapter, we consider the acoustic scattering in the air at rest by an object whose surface is impedant, meaning that any incident wave will be partially absorbed and partially scattered. In what follows, such an object is called impedant. The goal here is to give the mathematical justification we need in Part II to apply the Reduced Basis Method for this problem. We point out why the variational formulation cannot be thought in the natural Sobolev spaces. We show that the Fredholm alternative framework can be recovered in a different functional setting, leading to a well-posed variational formulation at all frequencies when the surface of the object is Lipschitz. We show that the classical discrete approximation is inf-sup stable. Finally, we discuss invertibility issues caused by the presence of resonant frequencies of some integral formulation and present a remedy taken from the literature.

Physical setting

We consider a bounded object Ω -⊂ R 3 with boundary Γ and Ω + := R 3 \Ω -, where Ω - denotes the closure of Ω -. Any object such that Γ is Lipschitz can be considered, but we take a ball for simplicity of the presentation, see Figure 2.1. We consider a monopole source located in Ω + . The surface of the ball is impedant: the proportion of absorbed and scattered parts is quantified by the impedance coefficient µ, which is used in a Robin boundary condition at Γ . We suppose that µ > 0. We are interested in the computation of the scattered field p sc in Ω + . We denote p inc the known pressure field of wave number k created by the source in the absence of the sphere; the total acoustic field in Ω + is the sum of p inc and p sc . The complex conjugation is denoted by •.

Weak formulation

Preliminaries

We recall that a distribution u (u +,-:= u| Ω +,-) is a piecewise radiating Helmholtz solution, if

       ∆u +,-+ k 2 u +,-= 0 in Ω + ∪ Ω -, lim r→+∞ r ∂u + ∂r -iku + = 0, (2.1)
where the second equation is the Sommerfeld radiation condition. Let

M := N 1 2 I + D 1 2 I -D S , (2.2)
where N , D, D and S have been defined in (1.31). A piecewise radiating Helmholtz solution u verifies

M [γ 0 u] [γ 1 u] = - γ - 1 u - γ - 0 u -. (2.3)
The operator M is not injective: define w in such a manner that w| Ω -= 0 and w| Ω + is any non-zero field satisfying the Helmholtz equation and the Sommerfeld radiation condition at infinity. For instance, consider the problem

         ∆v + k 2 v = 0 in Ω + , γ + 0 v = g on Γ, lim r→+∞ r ∂v ∂r -ikv = 0,
where g ∈ H 12 (Γ ) is nonzero. This problem has a unique solution (see [START_REF] Mclean | Strongly elliptic systems and boundary integral equations[END_REF]Theorem 9.11 p.288]), which can be chosen for w| Ω + . Since w| Ω + is non-zero, [γ 0 w] and [γ 1 w] cannot be both zero owing to the representation formula (1.30). However,

M [γ 0 w] [γ 1 w] = 0 0 . (2.4)
Therefore, we have constructed a non-zero element of Ker(M ). We can understand that M is not injective by observing that both relations in the system (2.3) being obtained from the same relation (1.30), they are not independent. In other words, the system (2.

The Robin boundary condition

The unknown field p sc solves the following boundary value problem:

             ∆p sc + k 2 p sc = 0 in Ω + , γ + 1 p sc + i k µ γ + 0 p sc = -γ 1 p inc -i k µ γ 0 p inc on Γ, lim r→+∞ r ∂p sc ∂r -ikp sc = 0, (2.5)
where p inc is C 1 in a neighborhood of Γ (leading to γ 0 p inc := γ + 0 p inc = γ - 0 p inc and the same for γ 1 p inc ).

We define the distribution v :

Ω + ∪ Ω --→ C such that v |Ω -= -p inc , v |Ω + = p sc .
We introduce the jumps χ := [γ 0 v] and λ := [γ 1 v]. We have χ = γ + 0 p sc +γ 0 p inc and λ = γ + 1 p sc +γ 1 p inc , so that, using the Robin boundary condition on Γ in the system (2.5), there holds

λ + i k µ χ = 0. (2.6)
Many choices for v are possible (for instance, v |Ω -= 0). Our choice was motivated by the fact that the jumps χ and λ correspond to the exterior trace of the total acoustic potential. Other choices may have lead to jumps that have no physical meaning. Since v is a piecewise radiating Helmholtz solution, there holds

M χ λ = - γ 1 p inc γ 0 p inc . (2.7)
Injecting the Robin boundary condition, we obtain

N -ik 2µ I D D -S -iµ 2k I χ λ = γ 1 p inc -γ 0 p inc . (2.8)
The natural functional spaces of the involved integral operators are N :

H 1 2 (Γ ) → H -1 2 (Γ ), S : H -1 2 (Γ ) → H 1 2 (Γ ), D : H 1 2 (Γ ) → H 1 2 (Γ ) and D : H -1 2 (Γ ) → H -1 2 (
Γ ) (more details are given in Section 3.3.2). The variational form is as follows: find (χ, λ) ∈ H

1 2 (Γ ) × L 2 (Γ ) such that ∀ χ, λ ∈ H 1 2 (Γ ) × L 2 (Γ ),          N χ - ik 2µ χ, χ -1 2 , 1 2 + Dλ, χ -1 2 , 1 2 = γ 1 p inc , χ -1 2 , 1 2 , Dχ, λ -Sλ + iµ 2k λ, λ = -γ 0 p inc , λ , (2.9)
where (•, •) denotes the L 2 (Γ ) inner product: (λ, χ) := Γ λχ, and

•, • -1 2 , 1 2
denotes its extension to a duality pairing on

H -1 2 (Γ ) × H 1 2 (Γ ).
Remark 2.1 In the first equation of (2.9), ik 2µ χ makes sense as an element of H

1 2 (Γ ) ⊂ H -1 2 (Γ ).
In the second equation, we have taken λ, λ ∈ L 2 (Γ ). We cannot take λ, λ ∈ H -1 2 (Γ ) since the product λ, λ would not make sense. Furthermore, taking λ, λ ∈ H 1 2 (Γ ) would make the equivalence property between the variational formulation and the boundary value problem fail, since for instance, ∀ λ ∈ H 1 2 (Γ ), Dχ, λ = 0 does not imply Dχ = 0. It would be the case if λ, Dχ = 0 holds for all λ ∈ H -1 2 (Γ ).

Existence and uniqueness

Consider now

H := H 1 2 (Γ )×L 2 (Γ ) and H ′ := H -1 2 (Γ )×L 2 (Γ ). The norm • H is defined as (χ, λ) H := χ 2 H 1 2 (Γ ) + λ 2 L 2 (Γ ) 1 2
. We define a 0 and ã as the two sesquilinear forms such that

a 0 (χ, λ) , χ, λ = N 0 χ, χ -1 2 , 1 2 -i S 0 λ, λ -i k 2µ (χ, χ) + µ 2k λ, λ , ã (χ, λ) , χ, λ = (N -N 0 )χ, χ -1 2 , 1 2 -i (S -S 0 )λ, λ + Dλ, χ -1 2 , 1 2 + i Dχ, λ ,
(2.10) and b as the linear form such that

b χ, λ = γ 1 p inc , χ -1 2 , 1 2 -i γ 0 p inc , λ .
(2.11)

We have multiplied the second equation of (2.9) by i to obtain a required coercivity result, as we will see in what follows. The operators N 0 , S 0 , D 0 and D0 refer respectively to N , S, D and D taken at zero frequency. Let A 0 , H → H ′ , and Ã, H → H ′ , be the bounded operators defined respectively from a 0 and ã:

A 0 := N 0 -i k 2µ I 0 0 µ 2k -iS 0 , Ã := N -N 0 D iD -i(S -S 0 ) (2.

12)

We denote A := A 0 + Ã.

The weak form (2.9) can also be written as follows: find (χ, λ) ∈ H

1 2 (Γ ) × L 2 (Γ ) such that ∀ χ, λ ∈ H 1 2 (Γ ) × L 2 (Γ ), a (χ, λ) , χ, λ = b χ, λ , (2.13)
where a = a 0 + ã. Proof. We apply the Fredholm alternative.

i. Continuity:

We denote the continuity constants using C with a subscript. It is well-known (see [START_REF] Mclean | Strongly elliptic systems and boundary integral equations[END_REF]Theorem 6.11]) that N χ

H -1 2 (Γ ) ≤ C N χ H 1 2 (Γ )
, Dχ

H 1 2 (Γ ) ≤ C D χ H 1 2 (Γ )
, Sλ

H 1 2 (Γ ) ≤ C S λ H -1 2 (Γ )
, and Dλ

H -1 2 (Γ ) ≤ C D λ H -1 2 (Γ )
. These relations hold as well for at the zerofrequency particular case. Moreover,

Dχ L 2 ≤ Dχ H 1 2 (Γ ) ≤ C D χ H 1 2 (Γ )
, Dλ

H -1 2 (Γ ) ≤ C D λ H -1 2 (Γ ) ≤ C D λ L 2 (Γ ) , and
Sλ L 2 (Γ ) ≤ Sλ H 1 2 (Γ ) ≤ C S λ H -1 2 (Γ ) ≤ C S λ L 2 (Γ ) . Therefore, |a 0 (χ, λ) , χ, λ | ≤ C N 0 + C S 0 + k 2µ + µ 2k (χ, λ) H ( χ, λ) H , |ã (χ, λ) , χ, λ | ≤ C N 0 + C N + C S 0 + C S + C D + C D (χ, λ) H ( χ, λ) H , |b χ, λ | ≤ γ 1 p inc H -1 2 (Γ ) + γ 0 p inc L 2 (Γ ) ( χ, λ) H .
(2.14)

ii. Coercivity up to a compact perturbation:

We denote the coercivity constants using K with a subscript. From [76, Theorem 8.12 p.261], S 0 λ, λ 1 2 ,-

1 2 ∈ R, ∀λ ∈ H -1 2 (Γ ). Therefore, (S 0 λ, λ) ∈ R, ∀λ ∈ L 2 (Γ ). From [76, Theorem 8.21 p.267], N 0 χ, χ -1 2 , 1 2 ∈ R, ∀χ ∈ H 1 2 (Γ ) and N 0 is coercive on H 1 2 (Γ ). Then, Re (a 0 ((λ, χ) , (λ, χ))) = N 0 χ, χ -1 2 , 1 2 + µ 2k (λ, λ) ≥ min K N 0 , µ 2k (χ, λ) 2 H .
(2.15)

Hence, a 0 is H-coercive. Let us now examine the compactness properties of Ã. First, from [93, Lemma 3.9.8], N -N 0 is compact from H 1 2 (Γ ) into H -1 2 (Γ ). Then, we saw that S and S 0 are continuous from L 2 (Γ ) into H 1 2 (Γ ), and the injection of Lemma 3.3], the operator T := D -D * , where D * is the adjoint of D, is compact from

H 1 2 (Γ ) into L 2 (Γ ) is compact (we denote it as H 1 2 (Γ ) ֒→ L 2 (Γ )), therefore S -S 0 is compact from L 2 (Γ ) into L 2 (Γ ). The operator D is continuous from H 1 2 (Γ ) into H 1 2 (Γ ), and again H 1 2 (Γ ) ֒→ L 2 (Γ ) implies that D is compact from H 1 2 (Γ ) into L 2 (Γ ). Finally, we prove that D is compact from L 2 (Γ ) into H -1 2 (Γ ). From [54,
H -1 2 (Γ ) into H -1 2 (Γ ). In particular, T is compact from L 2 (Γ ) into H -1 2 (Γ ). We recall that since Γ is a closed surface, H -1 2 (Γ ) is the dual of H 1 2 (Γ ).
Then, by the Schauder theorem, D * is compact from L 2 (Γ ) into H -1 2 (Γ ) as the adjoint of a compact operator. Therefore, D is compact from L 2 (Γ ) into H -1 2 (Γ ) as the sum of two compact operators. Hence, Ã is compact. iii. Uniqueness:

The uniqueness property is shown in two steps. Suppose that (2.9) is verified for a zero right-hand side.

Step 1: Let us prove that λ = ik µ χ.

Let w(x) = -Sλ(x) + Dχ(x), ∀x ∈ R 3 \Γ . The distribution w is a piecewise radiating Helmholtz solution. Using Proposition 3.26, there holds

       γ + 1 w = -N χ + I 2 -D λ, γ + 0 w = I 2 + D χ -Sλ.
(2.16)

Using (2.9) with zero right-hand side, we obtain

         γ + 1 w = 1 2 λ - ik µ χ , γ + 0 w = iµ 2k λ - ik µ χ .
(2.17)

Let B be a ball containing Ω -. Using Green's formula,

B\Ω - |∇w| 2 -k 2 |w| 2 + γ + 1 w, γ + 0 w = ∂B w ∂ w ∂n . (2.18)
The normals on Γ and ∂B point "outward" (to infinity), see Figure 2.2. Using (2.17), γ 1 w + , γ 0 w + = Fig. 2.2. Definition of the surface ∂B.

iµ 4k λ -ik µ χ 2 L 2 (Γ )
. Therefore, since µ > 0, Im ∂B w

+ ∂w + ∂n = µ 4k λ -ik µ χ 2 L 2 (Γ )
≥ 0. From a corollary of the Rellich's Lemma (see [START_REF] Mclean | Strongly elliptic systems and boundary integral equations[END_REF]Lemma 9.9]), w| R 3 \ B ≡ 0. From the Cauchy-Kowalevski theorem (unique analytic continuation), there holds γ + 0 w = 0. Finally, from (2.17), λ = ik µ χ.

Step 2: We now prove that λ = -ik µ χ.

Using (2.3),          γ - 1 w = γ + 1 w -λ = - 1 2 λ + ik µ χ , γ - 0 w = γ + 0 w -χ = iµ 2k λ + ik µ χ . (2.19)
Applying the Green's formula on w in

Ω -yields R ∋ Ω - |∇w| 2 -k 2 |w| 2 = γ - 1 w, γ - 0 w . (2.20) From (2.19), γ - 1 w, γ - 0 w = - iµ 4k λ + ik µ χ L 2 (Γ ) ∈ iR. (2.21)
Using (2.20) and (2.21), we infer λ = -ik µ χ. Combining this result with that of Step 1, we obtain λ = χ = 0. This proves uniqueness and concludes the whole proof. ♦

Inf-sup stability of the discrete formulation

Let V 1 h and V 0 h be respectively the Lagrange finite element spaces P 1 and P 0 on Γ built using a shape regular mesh of Γ . The product space

H h := V 1 h ×V 0 h is a conforming approximation of H = H 1 2 (Γ ) × L 2 (Γ ). The numerical resolution is carried out with a Galerkin procedure on V 1 h × V 0 h using the boundary element method (BEM): Find (χ, λ) ∈ H h := V 1 h × V 0 h such that ∀ χ, λ ∈ H h ,          N χ - ik 2µ χ, χ -1 2 , 1 2 + Dλ, χ -1 2 , 1 2 = γ 1 p inc , χ -1 2 , 1 2 , Dχ, λ -Sλ + iµ 2k λ, λ = -γ 0 p inc , λ , (2.22)
where, for simplicity, we keep the same notation for the discretized unknowns.

Proposition 2.3

The following approximation properties of V 1 h and V 0 h (see [START_REF] Sauter | Boundary Element Methods[END_REF][START_REF] Ern | Theory and Practice of Finite Elements[END_REF]) hold: For all (χ, λ)

∈ H 1 (Γ ) × H 1 2 (Γ ), inf χ h ∈V 1 h χ -χ h H 1 2 (Γ ) ≤ C 1 h 1 2 χ H 1 (Γ ) , inf λ h ∈V 0 h λ -λ h L 2 (Γ ) ≤ C 2 h 1 2 λ H 1 2 (Γ ) , ( 2.23)
where C 1 and C 2 are constants independent of the mesh size h.

This leads to the following proposition: For all (χ, λ)

∈ H 1 (Γ ) × H 1 2 (Γ ) inf (χ h ,λ h )∈H h (χ, λ) -(χ h , λ h ) H ≤ Ch 1 2 χ H 1 (Γ ) + λ H 1 2 (Γ ) , (2.24)
where C is a constant independent of the mesh size h.

Proposition 2.4

Since the sesquilinear form a defined by (2.13) satisfies a Garding inequality (because it is coercive up to an additive compact perturbation), is injective and H h satisfies the approximation property (2.24), there exists ĥ > 0 and a constant γ such that for all mesh size h satisfying 0 < h ≤ ĥ, the following discrete inf-sup condition holds:

sup (0,0) =( χh , λh )∈H h a (χ h , λ h ), ( χh , λh ) ( χh , λh ) H ≥ γ (χ h , λ h ) H for all (χ h , λ h ) ∈ H h . (2.25)
Proof. We follow the proof of [55, Theorem 14] and detail it for completeness of the presentation.

We define H := H 1 (Γ ) × H 1 2 (Γ ). We define G hA 0 as the Galerkin projection corresponding to the operator A 0 by H ∋ x → G hA 0 x ∈ H h , i.e. the solution of: Find

G hA 0 x ∈ H h such that, ∀y h ∈ H h , A 0 G hA 0 x, y h H ′ ,H = A 0 x, y h H ′ ,H . (2.26)
We denote by C the continuity constant of A 0 and by α the coercivity constant of A 0 . We observe that for all y ∈ H,

G hA 0 y H ≤ C α y H since α G hA 0 y 2 H ≤ | A 0 G hA 0 y, G hA 0 y H ′ ,H | = | A 0 y, G hA 0 y H ′ ,H | ≤ C G hA 0 y H y H . Additionally, G hA 0 y -y H ≤ (1 + C α ) inf z h ∈H h y - z h H . Indeed, for all z h ∈ H h , α G hA 0 y -z h 2 H ≤ | A 0 (G hA 0 y -z h ), G hA 0 y -z h H ′ ,H | = | A 0 (y -z h ), G hA 0 y -z h H ′ ,H | ≤ C y -z h H G hA 0 y -z h H , so that G hA 0 y -z h H ≤ C α y -z h H ,
and the assertion follows from the triangle inequality.

Step 1. Let us show that ∀x ∈ H,

G hA 0 x -x H -→ h→0 0. (2.27)
Let x ∈ H and let ǫ > 0. Using the density of H in H, there exists y ∈ H such that x -

y H ≤ ǫ 2(1+ C α )
. Using the approximation property, there exists h 0 > 0 such that ∀h ≤ h 0 , inf

z h ∈H h y -z h H ≤ ǫ 2(1+ C α ) . Therefore, ∀h ≤ h 0 , G hA 0 y -y H ≤ ǫ 2 . Furthermore, G hA 0 (y - x) H ≤ C α x -y H ≤ C α ǫ 2(1+ C α )
. To sum up, ∀h ≤ h 0 , there holds

G hA 0 x -x H ≤ x -y H + y -G hA 0 y H + G hA 0 (y -x) H ≤ ǫ.
Step 2. Let L := I + A -1 0 à and L h := I + G hA 0 A -1 0 à mapping H to H. We have L -L h = (I -G hA 0 )A -1 0 Ã. We now prove that, for h small enough, L -1 h exists and there exists c 0 independent of h such that |||L -1 h ||| ≤ c 0 . We showed in Step 1 that I -G hA 0 converges pointwise to 0 as h → 0. Using a corollary of the Banach-Steinhaus theorem, I -G hA 0 converges to 0 in operator norm on compact sets. Moreover, since A -1 0 exists and is bounded, and à is compact, then A -1 0 à is compact. Therefore, we have

|||L -L h ||| → h→0 0. (2.28)
Hence, since L -1 exists (L -1 = A -1 A 0 with A invertible), there exists h 1 > 0 such that ∀0 < h ≤ h 1 , L -1 h exists. From (2.28), there exists 0

< h 2 ≤ h 1 such that ∀0 < h ≤ h 2 , |||L - L h ||| ≤ 1 2|||L -1 ||| . We have L h = [I -(L -L h )L -1 ]L. Then, using the Neumann series of L -L h , L -1 h = L -1 +∞ k=0 [(L-L h )L -1 ] k . Hence, ∀0 < h ≤ h 2 , |||L -1 h ||| ≤ |||L -1 ||| 1-|||L-L h ||||||L -1 ||| ≤ 2|||L -1 ||| =: c 0 .
Step 3. We now prove the discrete inf-sup stability of a. Since A = A 0 L, we obtain:

∀x h , xh ∈ H h , Re(a (x h , xh )) = Re( Ax h , xh H ′ ,H ) = Re( A 0 L h x h , xh H ′ ,H ) + Re( A 0 (L -L h )x h , xh H ′ ,H ). Hence, |a (x h , xh ) | + | A 0 (L -L h )x h , xh H ′ ,H | ≥ Re( A 0 L h x h , xh H ′ ,H ).
From the coercivity of A 0 , taking xh = L h x h ,

Re( A 0 L h x h , L h x h H ′ ,H ) ≥ α L h x h 2 H ≥ α c 0 x h H L h x h H ,
where we used the uniform boundedness of L -1 h in the last inequality. Moreover, still with

xh = L h x h , | A 0 (L -L h )x h , L h x h H ′ ,H | ≤ |||A 0 ||||||L -L h ||| x h H L h x h H .
Hence, for L h x h = 0,

|a (x h , L h x h ) | L h x h H ≥ α c 0 -|||A 0 ||||||L -L h ||| x h H . Since |||L -L h ||| → h→0 0, there exists 0 < h 3 ≤ h 2 such that, ∀0 < h ≤ h 3 , α c 0 -|||A 0 ||||||L -L h ||| ≥ α 2c 0 = α 4|||A -1 A 0 ||| =: ς. Therefore, ∀0 < h ≤ h 3 , sup xh H =0 |a (x h , xh ) | xh H ≥ |a (x h , L h x h ) | L h x h H ≥ ς x h H ,
* : Y * → X * is defined by (Au, v) Y,Y * = (u, A * v) X,X * , for all u ∈ X and v ∈ Y * . The boundary integral operators D : H 1 2 (Γ ) → H 1 2 (Γ ) and D : H -1 2 (Γ ) → H -1 2 (
Γ ) are transpose (or dual) but not adjoint, that is, they satisfy D = D * . In a complex-valued functions context, the convenient notion of symmetry is the Hermitian symmetry, naturally inherited from the chosen inner product: (u, v) = Γ ūv = Γ vu = (v, u). However, when considering finite dimensional approximations by means of finite elements and boundary elements, the basis functions are chosen to be real-valued. Consider the matrix M h obtained from (2.22), and the indices i, j such that M hi,j is in the lowerleft extradiagonal block. Then, M hi,j = (Dψ i , ϕ j ), where ψ i is a basis function of V 1 h and ϕ j is a basis function of V 0 h . There holds M hi,j = (ψ i , D * ϕ j ) = (ψ i , Dϕ j ) = (ψ i , Dϕ j ) = ( Dϕ j , ψ i ). Then, since ψ i and ϕ j are real-valued, there holds M hi,j = M hj,i , which means that the two extradiagonal blocks of M h are complex and symmetric (but not Hermitian symmetric). Actually, the whole matrix M h is symmetric.

-∆φ D n = λ D n φ D n in Ω -, γ - 0 φ D n = 0 on Γ, (2.29)
where the exponent D stands for the Dirichlet boundary condition enforced on Γ and where γ - 0 stands for the interior Dirichlet trace on Γ . We denote {λ D n } the set of eigenvalues and {φ D n } the set of eigenfunctions.

Likewise, there exist positive real numbers λ N n and nontrivial solutions φ N n of the following eigenvalue problem:

-∆φ N n = λ N n φ N n in Ω -, γ - 1 φ N n = 0 on Γ, ( 2

Proof. Let φ D

n be an eigenfunction of (2.29) and define the distribution w n ∈ D ′ (R 3 ) such that w - n := φ D n and w + n := 0. Using the representation formula (1.30),

w n = S(γ - 1 φ D n ). Since w n = S(γ - 1 φ D n ) = 0, we infer γ - 1 φ D n = 0. The interior Dirichlet trace of w n is γ - 0 S(γ - 1 φ D n ) = S(γ - 1 φ D n ) = γ - 0 w - n = γ - 0 φ D n = 0. Therefore, λ := γ - 1 φ D n is
I)) ⊥ = (Im( D -1 2 I) T ) ⊥ = Ker( D -1 2 I) = {0}, hence D -1 2 I is not surjective. Likewise, for all k 2 ∈ {λ N n }, D + 1 2 I
is noninvertible, as the transpose of the noninvertible operator D + 1 2 I.

CFIE for the exterior Helmholtz problem

Consider now the following exterior Helmholtz problem with a Dirichlet boundary condition:

         ∆u + k 2 u = 0 in Ω + γ + 0 u = -γ - 0 f inc on Γ lim r→+∞ r ∂u ∂r -iku = 0 (2.31)
where we suppose that f inc is such that γ - 0 f inc ∈ H 1 2 (Γ ). This exterior Helmholtz problem admits a unique solution in H(∆, Ω + ) [START_REF] Sauter | Boundary Element Methods[END_REF]Theorem 2.10.15], where H(∆, Ω + ) = {v ∈ H 1 (Ω + ), ∆v ∈ L 2 }. We define the distribution v such that v| Ω + := u and v| Ω -:= -f inc . Using the representation formula (1.30) and the boundary integral operators, there holds

     Sλ = -γ - 0 f inc , 1 2 I + D λ = -γ - 1 f inc , (2.32)
where

λ := [γ 1 v] Γ is
I) = {0}.
Only the physical solution γ + 1 u + γ - 1 f inc solves simultaneously both equations in (2.32). To avoid dealing with the overdetermined system (2.32), the idea of considering a linear combination of the equations in (2.32) was independently proposed in 1965 by Brakhage and Werner [START_REF] Brakhage | Über das Dirichletsche Außenraum Problem für die Helmholtzsche Schwingungsgleichung[END_REF], Leis [START_REF] Leis | Zur Dirichletschen Randwertaufgabe des Außenraumes der Schwingungsgleichung[END_REF] and Panich [START_REF] Panich | On the question of the solvability of the exterior boundary value problems for the wave equation and maxwell's equations[END_REF]: [START_REF] Björck | Loss and recapture of orthogonality in the modified Gram-Schmidt algorithm[END_REF] For all k > 0, for all η ∈ C such that Re(η) = 0, Ker S + iη 1 2 I + D = {0}.

Proposition 2.
Proof. Let λ ∈ Ker S + iη 1 2 I + D and set u = Sλ. The interior traces of u are γ - 0 u -= Sλ and γ -

1 u -= ( 1 2 I + D)λ, hence γ - 0 u -+ iηγ - 1 u -= 0. (2.33)
Since u is a radiating Helmholtz solution, there holds

Ω - |∇u| 2 -k 2 |u| 2 -γ - 1 u -, γ - 0 u - Γ = 0. (2.34)
Take the imaginary part of this relation. From Re(η) = 0 and (2.33), there holds γ - 1 u - L 2 (Γ ) = 0, which leads to γ - 1 u -= 0, and γ - 0 u -= 0 using again (2.33). Then, λ ∈ Ker(S) ∩ Ker 1 2 I + D = {0}. ♦

Likewise, there holds Proposition 2.17 For all k > 0, for all η ∈ C such that Re(η) = 0, Ker

1 2 I -D + iηN = {0}.
Consider the equation S + iη 1 2 I + D λ = g. This equation is set in H 1 2 (Γ ) and the unknown λ is sought in H -1 2 (Γ ). However, because of the term 1 2 I + D , the equation cannot be set in H 1 2 (Γ ) anymore. This means that, to write a variational formulation, we have to deal either with a pairing in H -1 2 (Γ ), or with a Petrov-Galerkin numerical approximation. A third possibility, which follows here, is the method introduced in [START_REF] Buffa | Regularized combined field integral equations[END_REF].

Consider the following Hermitian form:

δ Γ (p, q) := (∇ Γ p, ∇ Γ q) Γ + (p, q) Γ , ( 2.35)
for all p, q ∈ H 1 (Γ ). We define the regularizing operator M : H -1 (Γ ) → H 1 (Γ ) through the following implicit relation:

δ(M (p), q) Γ = (p, q) Γ , ( 2.36)
for all q ∈ H 1 (Γ ) (see [START_REF] Buffa | Regularized combined field integral equations[END_REF]Equation (3.10)]), where ∇ Γ denotes the surfacic gradient on Γ .

From [START_REF] Hiptmair | Stable FEM-BEM Coupling for Helmholtz Transmission Problems[END_REF], (λ, M λ)

Γ > 0 for all λ ∈ H -1 2 (Γ )\{0} and M is compact from H -1 2 (Γ ) into H 1 2 (
Γ ). Applying the regularizing operator to the second equation in (2.32), and then writing the CFIE linear combination leads to the following operator:

H -1 2 (Γ ) ∋ λ → S + iηM 1 2 I + D λ = -γ - 0 f inc -iηM γ - 1 f inc ∈ H 1 2 (Γ ). (2.37)
Theorem 2.18 For all k > 0, for all η ∈ C such that Re(η) = 0, Equation (2.37) has a unique solution λ.

Proof. Let k > 0 and η ∈ C such that Re(η) = 0. First, we write S +iηM 1 2 I + D = S 0 +(S -S 0 )+iηM 1 2 I + D , where S 0 is coercive, and (S -S 0 ) (see [START_REF] Sauter | Boundary Element Methods[END_REF]Lemma 3.9.8]) and M 1 2 I + D are compact in the natural trace spaces. Second, we prove that Ker S + iηM 1 2 I + D = {0}. The proof is close to that of Proposition 2.16. Let λ ∈ Ker S + iηM 1 2 I + D and set u = Sλ. Applying the interior trace operators to u yields γ - 0 u -= Sλ and γ - 1 u -= D + 1 2 I λ. Therefore,

γ - 0 u -+ iηM γ - 1 u -= 0. (2.38)
Since u is a radiating Helmholtz solution, there holds

Ω - |∇u| 2 -k 2 |u| 2 -γ - 1 u -, γ - 0 u - Γ = 0. (2.39)
Take the imaginary part of this relation. From Re(η) = 0 and (2.38), there holds γ - 1 u -, M γ - 1 u -= 0, which leads to γ - 1 u -= 0, and γ - 0 u -= 0 using again (2.38). Then, λ ∈ Ker(S) ∩ Ker 1 2 I + D = {0}. From the Fredholm alternative, Equation (2.37) has a unique solution λ.

♦

The operator M enables to recover the consistency in the functional spaces setting, with Γ Lipschitz, and well-posed Galerkin methods can then be derived.

Remark 2.19 Other regularizing operators M are possible. It is possible to regularize exterior

Helmholtz problems with Neumann boundary conditions in the same fashion (see [START_REF] Buffa | Regularized combined field integral equations[END_REF][START_REF] Hiptmair | Stable FEM-BEM Coupling for Helmholtz Transmission Problems[END_REF]). Remark 2.20 (Smooth boundary) When Γ is smooth, which is never the case in practice when considering a mesh of Γ , D is compact from L 2 (Γ ) into L 2 (Γ ) (see [START_REF] Buffa | Regularized combined field integral equations[END_REF]). The operator D is compact from L 2 (Γ ) into L 2 (Γ ) as the transpose of a compact operator. Besides, since for all λ ∈ H -1 2 (Γ ), Sλ andλ, the unknown in H -1 2 (Γ ). In particular, no regularization is required, because the Robin boundary condition (2.6) induces some regularity on λ. This enabled us to consider a functional space for λ which is more regular that H -1 2 (Γ ), namely L 2 (Γ ). Actually, the formulation (2.9) hides the principles of regularized CFIE. In Figure 2.4 we represent the condition numbers of matrices obtained after discretization of three integral equations, defined on very simple geometries. For the sphere, a mesh with 122 vertices has been used, and for the cube, a mesh with 221 vertices has been used. "Impedant" refers to the problem (2.9) of scattering by an impedant object. The two other problems deal with the inversion of the boundary integral operators N and S, for which noninversibility at respectively the Neumann and Dirichlet eigenvalues of the Laplacian defined in the interior domain is seen in Section 2.5.2. The formulation (2.9) does not suffer from any noninversibility problem at any frequency.

H 1 2 (Γ ) ≤ C λ H -1 2 (Γ ) ≤ C λ L 2 (Γ ) and H 1 2 (Γ ) ֒→ L 2 (Γ ) is compact, the operator S is also compact from L 2 (Γ ) into L 2 (Γ ). From the Fredholm alternative, S + iη 1 2 I + D is bijective from L 2 (Γ ) into L 2 (Γ )

Numerical illustration

A coupled FEM/BEM for the convected Helmholtz equation with non-uniform flow in a bounded domain

This chapter is an extended and detailed version of the article [Ar4].

Summary. We consider the convected Helmholtz equation modeling linear acoustic propagation at a fixed frequency in a subsonic flow around a scattering object. The flow is supposed to be uniform in the exterior domain far from the object, and potential in the interior domain close to the object. Our key idea is the reformulation of the original problem using the Prandtl-Glauert transformation on the whole flow domain, yielding (i) the classical Helmholtz equation in the exterior domain and (ii) an anisotropic diffusive PDE with skew-symmetric first-order perturbation in the interior domain such that its transmission condition at the coupling boundary naturally fits the Neumann condition from the classical Helmholtz equation. Then, efficient off-the-shelf tools can be used to perform the BEM-FEM coupling, leading to two novel variational formulations for the convected Helmholtz equation. The first formulation involves one surface unknown and can be affected by resonant frequencies, while the second formulation avoids resonant frequencies and involves two surface unknowns. Numerical simulations are presented to compare the two formulations.

Introduction

The scope of the present work is the computation of linear acoustic wave propagation at a fixed frequency in the presence of a flow. When the flow is at rest, the simplest model is the classical Helmholtz equation for the acoustic potential. This equation can be reduced to finding unknown functions defined on the surface of the scattering object and solving integral equations which can be effectively approximated by the Boundary Element Method (BEM) [START_REF] Sauter | Boundary Element Methods[END_REF]. When the medium of propagation is non-uniform, a volumic resolution has to be considered using, e.g., a Finite Element Method (FEM). If such non-uniformities occur only in a given bounded domain, it is possible to benefit from the advantages of both a volumic resolution and an integral equation formulation. Coupling BEM and FEM at the boundary of the given bounded domain allows this. Coupled BEM-FEM can be traced back to McDonald and Wexler [START_REF] Mcdonald | Finite-element solution of unbounded field problems[END_REF], Zienkiewicz, Kelly and Bettess [START_REF] Bettess | The coupling of the finite element method and boundary solution procedures[END_REF], Johnson and Nédélec [START_REF] Johnson | On the coupling of boundary integral and finite element methods[END_REF] and Jin and Liepa [START_REF] Jin | A note on hybrid finite element method for solving scattering problems[END_REF]. Over the last decade, such methods have been investigated, among others, for electromagnetic scattering [START_REF] Hiptmair | Coupling of finite elements and boundary elements in electromagnetic scattering[END_REF][START_REF] Levillain | Couplage éléments finis-équations intégrales pour la résolution des équations de Maxwell en milieu hétérogène[END_REF][START_REF] Leydecker | Adaptive FE-BE coupling for an electromagnetic problem in R3-A residual error estimator[END_REF], elasticity [START_REF] Carstensen | On the adaptive coupling of FEM and BEM in 2-D-elasticity[END_REF], and fluid-structure [START_REF] Domínguez | FE/BE coupling for an acoustic fluidstructure interaction problem. residual a posteriori error estimates[END_REF] or solid-solid interactions [START_REF] Maischak | A FEM-BEM coupling method for a nonlinear transmission problem modelling coulomb friction contact[END_REF][START_REF] Estorff | Coupled BEM/FEM approach for nonlinear soil/structure interaction[END_REF]. Coupled BEM-FEM for the classical Helmholtz equation can present resonant frequencies, leading to infinitely many solutions. All these solutions deliver the same acoustic potential in the exterior domain, but the numerical procedure becomes ill-conditioned. This problem has been solved in [START_REF] Buffa | Regularized combined field integral equations[END_REF][START_REF] Hiptmair | Stable FEM-BEM Coupling for Helmholtz Transmission Problems[END_REF], where a stabilization of the coupling, based on combined field integral equations (CFIE), has been proposed by introducing an additional unknown at the coupling surface.

When the medium of propagation is not at rest, the simplest governing equation is the convected Helmholtz equation resulting from the linearized harmonic Euler equations. Nonlinear interaction between acoustics and fluid mechanics is not considered herein; we refer to the early work of Lighthill for aerodynamically generated acoustic sources [START_REF] Lighthill | On sound generated aerodynamically. I. General theory[END_REF][START_REF] Lighthill | On sound generated aerodynamically. II. Turbulence as a source of sound[END_REF], to [START_REF] Hamilton | Nonlinear Acoustics: Theory and Applications[END_REF] for a review on nonlinear acoustics, and to [START_REF] Utzmann | Numerical Simulation of Turbulent Flows and Noise Generation[END_REF] for the coupling of Computational Aero Acoustic (CAA) and Computational Fluid Dynamics (CFD) solvers. Moreover, we assume that the flow is potential close to the scattering object and uniform far away from it. This geometric setup leads to a partition of the unbounded medium of propagation into two subdomains, the bounded interior domain near the scattering object where the flow is non-uniform and the unbounded exterior domain far away from the object where the flow is uniform. The main contribution of this work is the reformulation of the convected Helmholtz equation using the Prandtl-Glauert transformation on the whole flow domain, yielding (i) the classical Helmholtz equation in the exterior domain and (ii) an anisotropic diffusive PDE with skew-symmetric first-order perturbation in the interior domain such that its transmission condition at the coupling boundary naturally fits the Neumann condition from the classical Helmholtz equation. The Prandtl-Glauert transformation has been used in [START_REF] Dubois | Lorentz transform and staggered finite differences for advective acoustics[END_REF] for the uniformly convected Helmholtz equation. In the present case where the flow is non-uniform in the interior domain, this reformulation allows us to use efficient off-theshelf tools to perform a BEM-FEM coupling. Namely, a FEM is utilized in the interior domain to discretize the anisotropic second-order PDE, a BEM is utilized for the classical Helmholtz equation in the exterior domain, and Dirichlet-to-Neumann maps are used for the coupling. We emphasize that the key advantage of using the Prandtl-Glauert transformation is that the BEM part of the resolution only involves integral operators corresponding to the classical Helmholtz equation. We consider two approaches for the coupling, leading, to the authors' knowledge, to two novel coupled BEM-FEM formulations for the convected Helmholtz equation. The first formulation involves one surface unknown and can be affected by resonant frequencies, while the second one uses the stabilized CFIE technique from [START_REF] Buffa | Regularized combined field integral equations[END_REF][START_REF] Hiptmair | Stable FEM-BEM Coupling for Helmholtz Transmission Problems[END_REF], avoids resonant frequencies, and involves two surface unknowns. Our numerical results show that the first formulation yields results polluted by spurious oscillations in the close vicinity of resonant frequencies, whereas the second formulation yields consistent solutions at all frequencies. This advantage of the second formulation is particularly relevant in practice at high frequencies, where the density of resonant frequencies is higher.

We briefly discuss alternative methods from the literature to solve the convected Helmholtz equation in unbounded domains. In some cases with simple geometries, the far-field solution is analytically known [START_REF] Powles | Asymptotic and numerical solutions for shielding of noise sources by parallel coaxial jet flows[END_REF]. Boundary integral equations involving the Green kernel associated with the convected Helmholtz equation have been derived in [START_REF] Beldi | Some new results for the study of acoustic radiation within a uniform subsonic flow using boundary integral method[END_REF]. Other numerical methods include infinite finite elements [START_REF] Bettess | Infinite Elements[END_REF][START_REF] Zienkiewicz | Dynamic fuid-structure interaction. numerical modelling of the coupled problem[END_REF] and the method of fundamental solutions [START_REF] Fairweather | The method of fundamental solutions for scattering and radiation problems[END_REF]. An alternative approach to treat unbounded domains is to use Perfectly Matched Layers (PML), combined with a volumic resolution using, e.g., the FEM. Versions of PML for the convected Helmholtz equation are considered in [START_REF] Bécache | Perfectly matched layers for the convected helmholtz equation[END_REF][START_REF] Mitsoudis | Helmholtz equation with artificial boundary conditions in a two-dimensional waveguide[END_REF]. The use of PML allows one to consider unbounded domains of propagation, but the solution is only available within the domain of computation. This can be a drawback in the following situations: (i) when one is interested in the pressure field far away from the scattering object, or (ii) when scattering objects are located far away from each other so that the volumic resolution has to be carried out in a very large area. Instead, with coupled BEM-FEM, the volumic resolution only takes place in the areas where the flow is nonuniform, and the pressure can be retrieved at any point of the exterior domain using known representation formulae, regardless of the distance of this point to the scattering objects. However, coupled BEM-FEM exhibit matrices with dense blocks for the unknowns on the boundary, and an additional treatment is sometimes needed to avoid resonant frequencies. These two points are addressed in this work.

The material is organized as follows: the problem of interest is presented in Section 3.2, and coupling procedures are detailed in Section 3.3, where the main mathematical results are stated. The finite-dimensional approximation of the coupled formulations is addressed in Section 3.4, along with a discussion on the structure of the linear systems and the algorithms to solve them effectively. Finally, numerical results are presented in Section 3.5, and some conclusions are drawn in Section 3.6. The proofs of the mathematical results stated in the previous sections are presented in Section 3.7.

Aeroacoustic problem

This section describes the problem of acoustic scattering by a solid object in a non-uniform convective flow, together with the underlying physical assumptions.

Notation and preliminaries

Figure 3.1 describes the geometric setup. The interior domain, corresponding to the area near the scattering object where the convective flow is non-uniform, is denoted by Ω -. In the exterior domain, Ω + , the convective flow is assumed to be uniform. The complete medium of propagation, denoted by Ω ⊂ R 3 , is such that

Ω := Ω + ∪ Ω -∪ Γ ∞ = R 3 \{solid object}, where Γ ∞ := ∂Ω + ∩ ∂Ω -
is the boundary between the interior and exterior domains. The surface Γ ∞ is assumed to be Lipschitz. Such an assumption is sufficiently large to include for instance polyhedric surfaces resulting from the use of a finite element mesh in Ω -. The surface of the solid scattering object, ∂Ω -\Γ ∞ , is denoted by Γ and is assumed to be Lipschitz.

The speed of sound when the medium of propagation is at rest is denoted by c, the wave number by k, the density by ρ, and the acoustic velocity and pressure, respectively, by v and p. The rescaled velocity is defined as M := c -1 v, where M := |M | is the Mach number. The subscript ∞ refers to uniform flow quantities related to Ω + , whereas the subscript 0 refers to point-dependent flow quantities related to Ω -, that is, The source term g is time-harmonic with pulsation ω and is assumed to be located at Ω + for simplicity. This source term is an acoustic monopole located at x s ∈ Ω + of amplitude A s , so that g := A s δ xs cos(ωt), where δ denotes the Dirac mass distribution. The physical quantities are associated with complex quantities with the following convention on, for instance, the acoustic pressure: p ↔ Re (p exp (-iωt)). In what follows, we always refer to the complex

ρ |Ω -= ρ 0 (x), ρ |Ω + ≡ ρ ∞ , k |Ω -= k 0 (x), k |Ω + ≡ k ∞ , c |Ω -= c 0 (x), c |Ω + ≡ c ∞ , M |Ω -= M 0 (x), M |Ω + ≡ M ∞ .
∆ϕ + k 2 ∞ ϕ + 2ik ∞ M ∞ • ∇ϕ -M ∞ • ∇ (M ∞ • ∇ϕ) = g in Ω + . (3.3)
If there were no scattering object and if the convective flow were uniform in R 3 (and thus equal to the flow at infinity), the source term g would create an acoustic potential denoted by ϕ inc in R 3 . This potential, which solves (3.3) in R 3 , has an analytical expression, and ϕ inc and n • ∇ϕ inc are continuous across Γ ∞ . The acoustic potential scattered by the solid object is defined as ϕ sc := ϕϕ inc in Ω + . Eliminating the known acoustic potential ϕ inc created by the source yields

∆ϕ sc + k 2 ∞ ϕ sc + 2ik ∞ M ∞ • ∇ϕ sc -M ∞ • ∇ (M ∞ • ∇ϕ sc ) = 0 in Ω + . (3.4)

The Prandtl-Glauert transformation

The Prandtl-Glauert transformation was introduced by Glauert in 1928 [START_REF] Glauert | The effect of compressibility on the lift of an aerofoil[END_REF] to study the compressible effects of the air on the lift of an airfoil and was applied to subsonic aeroacoustic problems by Amiet and Sears in 1970 [START_REF] Amiet | The aerodynamic noise of small-perturbation subsonic flows[END_REF]. Herein, the Prandtl-Glauert transformation is applied in the complete medium of propagation and is based on the reduced velocity M ∞ . This transformation consists in changing the space and time variables as

       x ′ = γ ∞ M ∞ • x M ∞ + x -( M ∞ • x) M ∞ x ∈ Ω, t ′ = t - γ 2 ∞ c ∞ M ∞ • x t ∈ R, (3.5)
where

γ ∞ := 1 √ 1-M 2 ∞ and M ∞ := M -1 ∞ M ∞ with M ∞ := |M ∞ |.
The spatial transformation corresponds to a dilatation along M ∞ of magnitude γ ∞ , the component orthogonal to M ∞ being unchanged. In what follows, we suppose that M ∞ < 1, so that the Prandtl-Glauert transformation is a C ∞ -diffeomorphism from Ω × R to Ω ′ × R, where Ω ′ denotes the transformed medium of propagation.

The transformed problem

Let f be such that ϕ

(x) = f (x ′ ) exp (-ik ∞ γ ∞ (M ∞ • x ′ )), x ′ ∈ Ω ′ ; f inc
and f sc are defined from ϕ inc and ϕ sc in the same fashion, so that f inc is analytically known, and defined in

R 3 . Let ς(x ′ ) := ρ -1 ∞ exp (ik ∞ γ ∞ (M ∞ • x ′ )) g(x ′ ), x ′ ∈ Ω ′ .
In what follows, the transformed geometry, unknowns and operators are considered unless specified otherwise. For brevity, primes are omitted.

To apply the the Prandtl-Glauert transformation to a PDE in the frequency domain, one has first to change the differential operators as

∇u = N ∇ ′ u, ∇ • U = ∇ ′ • N U , ( 3.6)
for a scalar-valued function u and a vector-valued function U . Here,

N = I + C ∞ M ∞ M T ∞ with C ∞ = γ∞-1 M 2 ∞ and γ ∞ = 1 √ 1-M 2 ∞
, and ∇ ′ refers to derivatives with respect to the transformed variables x ′ . Moreover, it is readily verified that

N M = M + C ∞ P M ∞ , N M ∞ = γ ∞ M ∞ , N M • M ∞ = N M ∞ • M = γ ∞ P, ( 3.7)
where P = M • M ∞ . After changing the differential operators, one has to change the unknown function as

ϕ(x) = α(x ′ )f (x ′ ), where α(x ′ ) := exp (-ik ∞ γ ∞ (M ∞ • x ′ )).
We now show that, following the Prandtl-Glauert transformation, Equation (3.1) becomes

rk 2 βf + irkV • ∇ ′ f + ∇ ′ • irkf V + rΞ∇ ′ f = ς, ( 3.8)
where

r = ρ ρ∞ , β = (1 + qP ) 2 -q 2 M 2 ∞ , V = (1 + qP ) N M -qγ ∞ M ∞ , q = γ 2 ∞ k∞ k , Ξ = N (I -M M T )N , and ς = ρ -1 ∞ α -1 g. Dividing Equation (3.1) by ρ ∞ leads to ας = rk 2 ϕ + irkM • ∇ ′ ϕ + ∇ ′ • (r (∇ ′ ϕ -(M • ∇ ′ ϕ) M + ikϕM )). Applying (3.6), it is inferred ας = rk 2 ϕ + irkM • N ∇ ′ ϕ + ∇ ′ • rN N ∇ ′ ϕ -∇ ′ • r M • N ∇ ′ ϕ N M + ∇ ′ • (irkϕN M ) .
Substituting ϕ for αf and expanding the derivatives with respect to α yields

ας = αrk 2 f + αirkM • N ∇ ′ f + αrkk ∞ γ ∞ f (M • N M ∞ ) + ∇ ′ • αrN N ∇ ′ f -∇ ′ • (αirk ∞ γ ∞ f N N M ∞ ) -∇ ′ • αr M • N ∇ ′ f N M + ∇ ′ • (αirk ∞ γ ∞ f (M • N M ∞ ) N M ) + ∇ ′ • (αirkf N M ) , since ∇ ′ α = -αik ∞ γ ∞ M ∞ . Using (3.7
) and simplifying some terms leads to

ας = αrk 2 f + αirkM • N ∇ ′ f + αrk 2 qP f + ∇ ′ • αrN N ∇ ′ f -∇ ′ • (αirkγ ∞ qf M ∞ ) -∇ ′ • αr M • N ∇ ′ f N M + ∇ ′ • (αirk(1 + qP )f N M ) .
Expanding again the derivatives with respect to α yields

ς = rk 2 f + irkM • N ∇ ′ f + rk 2 qP f + ∇ ′ • rN N ∇ ′ f -irk ∞ γ ∞ N N ∇ ′ f • M ∞ -∇ ′ • (irkqγ ∞ f M ∞ ) -rkk ∞ qγ 2 ∞ f M 2 ∞ -∇ ′ • r M • N ∇ ′ f N M + irk ∞ γ ∞ M • N ∇ ′ f N M • M ∞ + ∇ ′ • (irk(1 + qP )f N M ) + rkk ∞ γ ∞ (1 + qP )f N M • M ∞ .
Using again (3.7) as well as the symmetry of N , it is inferred

ς = rk 2 f + irkN M • ∇ ′ f + rk 2 qP f + ∇ ′ • rN N ∇ ′ f -irkqγ ∞ M ∞ • ∇ ′ f -∇ ′ • (irkqγ ∞ f M ∞ ) -rk 2 q 2 M 2 ∞ f -∇ ′ • r N M M T N ∇ ′ f + irkqP N M • ∇ ′ f + ∇ ′ • (irk(1 + qP )f N M ) + rk 2 qP (1 + qP )f. (3.9)
The terms are now reorganized with respect to the orders of derivation of f to obtain

ς = rk 2 (1 + qP -q 2 M 2 ∞ + qP (1 + qP ))f + irk ((1 + qP )N M -qγ ∞ M ∞ ) • ∇ ′ f + ∇ ′ • (irkf ((1 + qP )N M -qγ ∞ M ∞ )) + ∇ ′ • rN N ∇ ′ f -∇ ′ • r N M M T N ∇ ′ f , (3.10) yielding (3.8).
We now show that, following the Prandtl-Glauert transformation, the boundary condition

(3.2) becomes irkf V + rΞ∇ ′ f • n ′ = 0 on Γ ′ , ( 3.11)
where Γ ′ denotes the transformed boundary Γ . The normals on the initial geometry are denoted by n, and the normals on the transformed geometry by n ′ . It is readily seen that

n = K ∞ N n ′ on Γ, (3.12)
where K ∞ is a normalization factor that is not needed in what follows. Owing to (3.6) and (3.12),

(3.2) becomes N ∇ ′ ϕ•N n ′ = 0. Hence, N ∇ ′ (αf )•N n ′ = 0, leading to (N ∇ ′ f -ik ∞ γ ∞ f N M ∞ )• N n ′ = 0. Since the flow is tangential on Γ , M • n = 0 on Γ . Hence, M • N n ′ = 0 on Γ , so that N ∇ ′ f -N M • ∇ ′ f M + ikf (1 + qP )M - k ∞ k γ ∞ N M ∞ • N n ′ = 0. (3.13)
Using the symmetry of N and (3.7), (3.13) leads to (3.11).

An important observation is that in

Ω + , β = γ 2 ∞ , V = 0 and Ξ = I, so that (3.18a) becomes ∆f + k2 ∞ f = ς in Ω + , ( 3.14)
where

k∞ := γ ∞ k ∞ . (3.15) Moreover, since supp(ς) ⊂ Ω + , f inc satisfies ∆f inc + k2 ∞ f inc = ς in Ω + , ∆f inc + k2 ∞ f inc = 0 in R 3 \Ω + . (3.16)
Eliminating f inc in (3.14) yields,

∆f sc + k2 ∞ f sc = 0 in Ω + . (3.17)
This is the classical Helmholtz equation with modified wave number k∞ . Another important property is that the matrix Ξ is symmetric positive definite in Ω -as well. To prove this, consider the matrices N , O and Ξ, that are are all symmetric.

If M 0 < 1, ∀U ∈ C 3 , U T ΞU = N U T O (N U ) ≥ 1 -M 2 0 N U 2 . Now consider N U 2 = M T ∞ N U M∞ 2 + N U - M T ∞ N U M∞ 2 , where M∞ := M ∞ M∞ . We have M T ∞ N U M∞ = 1 √ 1-M 2 ∞ M T ∞ U M∞ and N U - M T ∞ N U M∞ = U - M T ∞ U M∞ , therefore, for all x ∈ Ω - U T Ξ(x)U ≥ 1 -M 2 0 (x) 1 1-M 2 ∞ M T ∞ U M∞ 2 + U - M T ∞ U M∞ 2 ≥ 1 -M 2 0 (x) U 2 . Moreover, for all U , W ∈ C 3 , there holds U • Ξ(x)W ≤ 1+M 2 0 (x) 1-M 2 ∞ U W , ∀x ∈ Ω -.
In summary, the boundary value problem we consider is

rk 2 βf + irkV • ∇f + ∇ • (irkf V + rΞ∇f ) = ς in Ω, (3.18a) (irkf V + rΞ∇f ) • n = 0 on Γ, (3.18b) lim r→+∞ r ∂(f -f inc ) ∂r -i k∞ (f -f inc ) = 0, (3.18c) where f is searched in H 1 loc (Ω) := {u ∈ H 1 (K), ∀K ⊂ Ω compact}. Equation (3.18a
) is the transformed convected Helmholtz equation, (3.18b) the transformed boundary condition, and the condition at infinity (3.18c) the classical Sommerfeld radiation condition that guarantees existence and uniqueness for Helmholtz exterior problems [START_REF] Mclean | Strongly elliptic systems and boundary integral equations[END_REF]Theorem 9.10]. In the general case, the Sommerfeld radiation condition is written for the scattered potential, since some incident acoustic potentials, e.g., plane waves, do not verify it.

Coupling procedure

Problem (3.18) is separated into an interior problem posed in Ω -and an exterior problem posed in Ω + in view of using different numerical methods in each subdomain. Specifically, the problem in the interior domain is solved by means of finite elements, whereas the problem in the exterior domain is solved by means of boundary elements. The main purpose of this section is to derive two coupling procedures between the interior and exterior problems.

The transmission problem

The one-sided Dirichlet traces on Γ ∞ of a smooth function u in Ω + ∪ Ω -are defined as γ ± 0 u ± := u ± | Γ∞ , and the one-sided Neumann traces as γ ± 1 u ± := (∇u ± ) | Γ∞ • n, where u ± := u| Ω ± and where n is the unit normal vector to Γ ∞ conventionally pointing towards Ω + (see Figure 3.1). The one-sided normal traces on Γ ∞ of a smooth vector field σ in Ω + ∪ Ω -are defined as γ ± n σ ± := σ ± | Γ∞ • n, where σ ± := σ| Ω ± . These trace operators are extended to bounded linear operators γ ± 0 :

H 1 (Ω ± ) → H 1 2 (Γ ∞ ), γ ± 1 : H(∆, Ω ± ) → H -1 2 (Γ ∞ ) and γ ± n : H(div, Ω ± ) → H -1 2 (Γ ∞ ), where H 1 (Ω ± ), H 1 2 (Γ ∞ ), and
H -1 2 (Γ ∞ ) are the usual Sobolev spaces on Ω ± and Γ ∞ , H(div, Ω ± ) = {σ ∈ L 2 (Ω ± ), ∇ • σ ∈ L 2 (Ω ± )}, with L 2 (Ω ±
) the Lebesgue space of square integrable functions on Ω ± , and Lemma 20.2]). It is actually sufficient to consider functional spaces on compact subsets of Ω + to define exterior traces on

H(∆, Ω ± ) = {u ± ∈ H 1 (Ω ± ), ∆u ± ∈ L 2 (Ω ± )} (see [100,
Γ ∞ . Let X denote the surface Γ or Γ ∞ . The L 2 (X)-inner product •, • L 2 (X),L 2 (X) : L 2 (X) × L 2 (X) → C is defined as λ, µ L 2 (X),L 2 (X) := X λ(y)µ(y)ds(y). (3.19)
This inner product can be extended to a duality pairing on H

-1 2 (X) × H 1 2 (X) denoted by •, • H -1 2 (X),H 1 2 (X)
. Define now the product

(λ, µ) X :=      λ, µ H -1 2 (X),H 1 2 (X) if λ ∈ H -1 2 (X), µ ∈ H 1 2 (X), µ, λ H -1 2 (X),H 1 2 (X) if λ ∈ H 1 2 (X) , µ ∈ H -1 2 (X).
(3.20)

We consider the following transmission problem where the one-sided normal trace γ - n,Γ on Γ from Ω -is used to formulate the boundary condition (3.18b): Proof. See Section 3.7. ♦

rk 2 βf -+ irkV • ∇f -+ ∇ • (irkf V + rΞ∇f ) -= 0 in Ω -, (3.21a) ∆f sc + k2 ∞ f sc = 0 in Ω + , (3.21b) γ - n,Γ (irkf V + rΞ∇f ) -= 0 on Γ, (3.21c) γ + 0 f + -γ - 0 f -= 0 on Γ ∞ , (3.21d) γ + 1 f + -γ - 1 f -= 0 on Γ ∞ , (3.21e) lim r→+∞ r ∂(f + -f + inc ) ∂r -i k∞ (f + -f + inc ) = 0. ( 3
+ 1 f + -γ - n (irkf V + rΞ∇f ) -= 0,

Basic ingredients of the coupling procedure

The coupling procedure hinges on a weak formulation in the interior domain Ω -and a Dirichlet-to-Neumann map (DtN ) associated with the classical Helmholtz equation (3.21b) in the exterior domain Ω + .

Derivation of Dirichlet-to-Neumann maps

For u ∈ H 1 (Ω + ∪ Ω -), the jump and average of its Dirichlet traces across Γ ∞ are defined respectively as

[γ 0 u] Γ∞ := γ + 0 u + -γ - 0 u -and {γ 0 u} Γ∞ := 1 2 γ + 0 u + + γ - 0 u -. For u ∈ H(∆, Ω + ∪ Ω -),

the jump and average of its Neumann traces across Γ

∞ are defined re- spectively as [γ 1 u] Γ∞ := γ + 1 u + -γ - 1 u -and {γ 1 u} Γ∞ := 1 2 γ + 1 u + + γ - 1 u -. When a trace is single-valued at Γ ∞ , we omit the superscripts ±.
In what follows, Helmholtz equations, as well as corresponding boundary integral operators, are written for the transformed wave number k∞ γ ∞ k ∞ , cf. (3.15) . A function u defined over R 3 is said to be a piecewise Helmholtz solution if u| Ω + and u| R 3 \Ω + solve the classical Helmholtz equation (3.21b) respectively in Ω + and R 3 \Ω + . A radiating piecewise Helmholtz solution is a piecewise Helmholtz solution that satisfies the Sommerfeld radiation condition (3.21f). For all λ ∈ C 0 (Γ ∞ ), the single-layer potential is defined as S(λ)(x) := Γ∞ E(yx)λ(y)ds(y),

x ∈ R 3 \Γ ∞ , where E(x) := exp(i k∞|x|) 4π|x|
is the fundamental solution of the classical Helmholtz equation (3.21b) with wave number k∞ satisfying the Sommerfeld radiation condition (3.21f). For all µ ∈ C 0 (Γ ∞ ), the double-layer potential is defined as [START_REF] Sauter | Boundary Element Methods[END_REF]Theorem 3.1.16], these operators can be extended to bounded linear operators S :

D(µ)(x) := Γ∞ ∇ y E(y -x)µ(y)ds(y), x ∈ R 3 \Γ ∞ . From
H -1 2 (Γ ∞ ) → H 1 loc (R 3 ) and D : H 1 2 (Γ ∞ ) → H 1 loc (R 3 \Γ ∞ )
. Moreover, both map onto radiating piecewise Helmholtz solutions. Recalling [80, Theorem 3.1.1], a radiating piecewise Helmholtz solution u can be represented from its Dirichlet and Neumann jumps across

Γ ∞ in the form u = -S([γ 1 u] Γ∞ ) + D([γ 0 u] Γ∞ ) in Ω + ∪ (R 3 \Ω + ). (3.22)
The single-layer and double-layer potentials satisfy the following jump relations across Γ ∞ [80, Theorem 3.1.2]:

[γ 0 (Sλ)] Γ∞ = 0, [γ 1 (Sλ)] Γ∞ = -λ, ∀λ ∈ H -1 2 (Γ ∞ ), [γ 0 (Dµ)] Γ∞ = µ, [γ 1 (Dµ)] Γ∞ = 0, ∀µ ∈ H 1 2 (Γ ∞ ). (3.23)
The operators

S : H -1 2 (Γ ∞ ) → H 1 2 (Γ ∞ ), Sλ := γ 0 (Sλ) , D : H 1 2 (Γ ∞ ) → H 1 2 (Γ ∞ ), Dµ := {γ 0 (Dµ)} Γ∞ , D : H -1 2 (Γ ∞ ) → H -1 2 (Γ ∞ ), Dλ := {γ 1 (Sλ)} Γ∞ , N : H 1 2 (Γ ∞ ) → H -1 2 (Γ ∞ ), N µ := -γ 1 (Dµ) , (3.24)
are respectively the single-layer, double-layer, transpose (or dual) of the double-layer, and hypersingular boundary integral operators. The Dirichlet and Neumann traces are well-defined, and the mapping properties can be found in [START_REF] Mclean | Strongly elliptic systems and boundary integral equations[END_REF]Theorem 7.1]. The following trace identities are directly derived from (3.23):

γ 0 S = S, γ ± 1 S = D ∓ 1 2 I, γ ± 0 D = D ± 1 2 I, γ 1 D = -N, (3.25)
Moreover, if u is a radiating piecewise Helmholtz solution, taking the interior traces of (3.22) and using (3.25) leads to

1 2 I -D S N 1 2 I + D [γ 0 u] Γ∞ [γ 1 u] Γ∞ = - γ - 0 u - γ - 1 u -. ( 3
R 3 \Ω + from (3.16)). Since f inc is continuous across Γ ∞ , [γ 0 v] Γ∞ = γ + 0 f sc + γ - 0 f - inc = γ + 0 f sc + γ + 0 f + inc = γ + 0 f + . (3.27) Likewise, [γ 1 v] Γ∞ = γ + 1 f + .
Other choices can be made for v (in particular v| R 3 \Ω + = 0). Here, the jumps correspond to the traces of the total transformed acoustic potential, and a direct coupling with the equation in Ω -(having f as unknown), is then possible. In what follows, we drop ± superscripts for the Dirichlet and Neumann traces of f and f inc since the traces are single-valued. Then, (3.26) applied to v yields

1 2 I -D S N 1 2 I + D γ 0 f γ 1 f = γ 0 f inc γ 1 f inc . (3.28)
Various identities relating γ 0 f and γ 1 f can be derived from (3.28), and these identities can be used to define DtN maps. For example, using the first line of (3.28) yields formally

γ 1 f = DtN 0 (γ 0 f ) := S -1 γ 0 f inc + D -1 2 I γ 0 f ,

Remark 3.3

The block operator defined on the left-hand side of (3.28) is not injective.

Weak formulation in the interior domain Ω

- Let Φ := f | Ω -where f solves (3.21). Multiplying (3.21a) by a test function Φ t ∈ H 1 (Ω -)
and using a Green formula together with the boundary condition (3.21c) at Γ yields

V(Φ, Φ t ) -γ - 1 Φ, γ - 0 Φ t Γ∞ = 0, (3.29)
with the sesquilinear form

V(Φ, Φ t ) := Ω - rΞ∇Φ • ∇Φ t - Ω - rk 2 βΦΦ t + i Ω - rkV • Φ∇Φ t -Φ t ∇Φ . (3.30)
Using the transmission conditions (3.21d)-(3.21e), γ - 0 Φ = γ 0 f and γ - 1 Φ = γ 1 f , so that the coupling with the exterior problem can be written as γ - 1 Φ = DtN (γ - 0 Φ). This yields the following coupled formulation:

Find Φ ∈ H 1 (Ω -) such that ∀Φ t ∈ H 1 (Ω -), V(Φ, Φ t ) -DtN (γ - 0 Φ), γ - 0 Φ t Γ∞ = 0. (3.31)

Unstable coupled formulation

To carry out the coupling, a first classical DtN map is considered. Since this DtN map is not well-defined at some frequencies, the resulting coupled formulation is not well-posed at these frequencies, and is therefore called unstable. From (3.28), recalling

γ - 0 Φ = γ 0 f and γ - 1 Φ = γ 1 f , there holds 1 2 I -D S N 1 2 I + D γ - 0 Φ γ - 1 Φ = γ 0 f inc γ 1 f inc . (3.32)
Using the first line of (3.32), γ -

1 Φ = S -1 D -1 2 I γ - 0 Φ + γ 0 f inc .
At this point, the inverse of S is written formally. Conditions of inversibility are discussed below. From the second line of (3.32), γ -

1 Φ = -N (γ - 0 Φ) + 1 2 I -D γ - 1 Φ + γ 1 f inc .
Injecting into the right-hand side of this relation the expression of γ - 1 Φ derived above yields the DtN affine map:

DtN unstab : H 1 2 (Γ ∞ ) → H -1 2 (Γ ∞ ) such that γ - 1 Φ = DtN unstab (γ - 0 Φ) := -N (γ - 0 Φ) + 1 2 I -D S -1 D - 1 2 I (γ - 0 Φ) + γ 0 f inc + γ 1 f inc . (3.33)
The operator inversion requires to introduce the auxiliary field

λ ∈ H -1 2 (Γ ∞ ) such that D - 1 2 I (γ - 0 Φ) -Sλ = -γ 0 f inc , ( 3.34)
yielding

DtN unstab (γ - 0 Φ) = -N (γ - 0 Φ) + 1 2 I -D (λ) + γ 1 f inc . (3.35)
Injecting DtN unstab (γ - 0 Φ) from (3.35) into the formulation (3.31) yields, using (3.34), the following coupled variational formulation:

Find (Φ, λ) ∈ H such that, ∀ Φ t , λ t ∈ H, V(Φ, Φ t ) + N (γ - 0 Φ), γ - 0 Φ t Γ∞ + D - 1 2 I (λ), γ - 0 Φ t Γ∞ = γ 1 f inc , γ - 0 Φ t Γ∞ , (3.36a) D - 1 2 I (γ - 0 Φ), λ t Γ∞ -S(λ), λ t Γ∞ = -γ 0 f inc , λ t Γ∞ , ( 3.36b)
with product space

H := H 1 (Ω -) × H -1 2 (Γ ∞ ) and inner product (Φ, λ) , Φ t , λ t H := Φ, Φ t H 1 (Ω -) + λ, λ t H -1 2 (Γ∞)
. The formulation (3.36) is called unstable since it admits infinitely many solutions at some frequencies of the source, leading to incorrect numerical results.

Remark 3.4

The DtN unstab affine map was proposed by Costabel to obtain a symmetric coupling in the case of self-adjoint operators [START_REF] Costabel | Symmetric methods for the coupling of finite elements and boundary elements[END_REF]. The DtN unstab map can be well-defined for certain operators: for instance, for transmission problems for the Laplace equation, this map leads to a well-defined symmetric system. In the system (3.36), the only non-symmetric contribution results from the vector V in the sesquilinear form V, cf. (3.30). The system becomes symmetric when the flow is uniform everywhere. However, since D and D are dual but not adjoint operators, there is no Hermitian symmetry.

Proposition 3.5 If f solves (3.21), then (f -, γ 1 f ) solves (3.36). Conversely, if (Φ, λ) solves (3.36), then R(Φ, λ) solves (3.21), where R : H → H 1 loc (Ω\Γ ∞ ) is such that R(Φ, λ)| Ω -:= Φ and R(Φ, λ)| Ω + := (-S(λ) + D(γ - 0 Φ) + f inc )| Ω + .
Proof. See Section 3.7. ♦

The main difficulty with the unstable coupled formulation (3.36) stems from the fact that Ker(S) depends on whether -k2 ∞ belongs to the set Λ of Dirichlet eigenvalues for the Laplacian on the bounded domain R

3 \Ω + . Specifically, Ker(S) = {0} if -k2 ∞ / ∈ Λ, while Ker(S) contains nontrivial elements if -k2 ∞ ∈ Λ. Proposition 3.6 If f solves (3.21), then for all λ * ∈ Ker(S), (f -, γ 1 f + λ * ) solves (3.36).
Proof. This is a direct consequence of Ker(S) = Ker( D -1 2 I).

♦ Theorem 3.7 If -k2 ∞ / ∈ Λ, then problem (3.36) is well-posed. If -k2 ∞ ∈ Λ, then (3.36
) admits infinitely many solutions of the form (f -, γ 1 f + λ * ), where f is the solution to (3.21) and λ * is any element in Ker(S).

Proof. See Section 3.7.

♦ Remark 3.8 Let -k2 ∞ ∈ Λ.
Owing to Proposition 3.5, for any couple (Φ, λ) solving (3.36), R(Φ, λ) solves (3.21). However, even if our goal is to solve (3.21), we will see in Section 3.5 that the numerical procedure to approximate (3.36) fails to the point that R(Φ, λ) is dominated by numerical errors.

The formulation (3.36) is written on a geometry and for unknown functions that has been transformed by the Prandtl-Glauert transformation. The physical acoustic potential in the canonical system of coordinates is obtained by applying the inverse Prandtl-Glauert transformation to R(Φ, λ). The formulation (3.36) looks similar to the classical symmetrical coupled formulation proposed by Costabel [START_REF] Costabel | Symmetric methods for the coupling of finite elements and boundary elements[END_REF], and recalled in the case of the Helmholtz transmission problem in [54, equation (15)]. In the formulation (3.36), the integral operators are written at the transformed wave number k∞ , which differ from the wavenumber k ∞ of the source. The local convection of the acoustic potential by the mean flow is taken into account through the Prandtl-Glauert transformation and the volumic term V, which differ from classical coupled formulations written for the classical (nonconvected) Helmholtz equation.

Stable coupled formulation

The idea of considering a linear combination of S and 1 2 I + D to derive well-posed boundary integral equations was independently proposed in 1965 by Brakhage and Werner [START_REF] Brakhage | Über das Dirichletsche Außenraum Problem für die Helmholtzsche Schwingungsgleichung[END_REF], Leis [START_REF] Leis | Zur Dirichletschen Randwertaufgabe des Außenraumes der Schwingungsgleichung[END_REF] and Panich [START_REF] Panich | On the question of the solvability of the exterior boundary value problems for the wave equation and maxwell's equations[END_REF]. This is the so-called Combined Field Integral Equation (CFIE). However, S and

D map H -1 2 (Γ ∞ ) into different spaces (H 1 2 (Γ ∞ ) and H -1 2 (Γ ∞ ) respectively).
This inconsistency in the functional setting can be solved by considering a regularizing compact operator from

H -1 2 (Γ ∞ ) into H 1 2 (Γ ∞ )
, as observed by Buffa and Hiptmair [START_REF] Buffa | Regularized combined field integral equations[END_REF]. We briefly recall the approach of [START_REF] Buffa | Regularized combined field integral equations[END_REF] and apply it to the present setting. Let ∇ Γ∞ denote the surfacic gradient on Γ ∞ . Consider the following Hermitian sesquilinear form: For all p, q ∈ H 1 (Γ ∞ ),

δ Γ∞ (p, q) := (∇ Γ∞ p, ∇ Γ∞ q) Γ∞ + (p, q) Γ∞ , ( 3.37)
and the regularizing operator M :

H -1 (Γ ∞ ) → H 1 (Γ ∞ ) is defined through the following implicit relation: For all p ∈ H 1 (Γ ∞ ), δ Γ∞ (M p, q) = (p, q) Γ∞ , ∀q ∈ H 1 (Γ ∞ ). (3.38)
It is readily seen that M = (-∆ Γ∞ + I) -1 , where ∆ Γ∞ is the Laplace-Beltrami operator on Γ ∞ .

Many choices of DtN maps based on CFIE strategies with the regularizing operator M lead to well-posed systems whatever the value of k∞ . The present choice hinges on the inversion of the operator S + iηM [START_REF] Ammar | A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids[END_REF].1], this operator is bijective as long as the coupling parameter η is such that Re(η) = 0. To do so, the first line of (3.32) and the application of M to the second line of (3.32) are used to obtain

1 2 I + D mapping H -1 2 (Γ ∞ ) into H 1 2 (Γ ∞ ) since, from [23, Lemma
1 2 I -D + iηM N S + iηM 1 2 I + D N 1 2 I + D γ - 0 Φ γ - 1 Φ = γ 0 f inc + iηM γ 1 f inc γ 1 f inc . (3.39)
Then, using both equations in (3.39) in the same fashion as in Section 3.3.3 leads to DtN stab :

H 1 2 (Γ ∞ ) → H -1 2 (Γ ∞ ) such that γ - 1 Φ =DtN stab (γ - 0 Φ) := -N (γ - 0 Φ) + 1 2 I -D S + iηM 1 2 I + D -1 - 1 2 I -D + iηM N (γ - 0 Φ) + γ 0 f inc + iηM γ 1 f inc + γ 1 f inc . (3.40)
The operator inversion requires to introduce the auxiliary field

λ ∈ H -1 2 (Γ ∞ ) such that S + iηM 1 2 I + D (λ) + 1 2 I -D + iηM N (γ - 0 Φ) = γ 0 f inc + iηM (γ 1 f inc ), (3.41)
so that

DtN stab (γ - 0 Φ) = -N (γ - 0 Φ) + 1 2 I -D (λ) + γ 1 f inc . (3.42)
The evaluation of M involving an operator inversion as well, it requires to introduce another

auxiliary field p ∈ H 1 (Γ ∞ ) such that, for all q ∈ H 1 (Γ ∞ ), δ Γ∞ (p, q) = N (γ - 0 Φ), q Γ∞ + 1 2 I + D (λ), q Γ∞ -(γ 1 f inc , q) Γ∞ , ( 3.43)
so that equation (3.41) can be rewritten

S(λ) + 1 2 I -D (γ - 0 Φ) + iηp = γ 0 f inc . (3.44)
Injecting DtN stab (γ - 0 Φ) from (3.42) into the formulation (3.31) yields, using (3.43) and (3.44), the following stable coupled variational formulation:

Find (Φ, λ, p) ∈ H such that ∀ Φ t , λ t , p t ∈ H, V(Φ, Φ t ) + N (γ - 0 Φ), γ - 0 Φ t Γ∞ + D - 1 2 I (λ), γ - 0 Φ t Γ∞ = γ 1 f inc , γ - 0 Φ t Γ∞ , (3.45a) D - 1 2 I (γ - 0 Φ), λ t Γ∞ -S(λ), λ t Γ∞ + iη p, λ t Γ∞ = -γ 0 f inc , λ t Γ∞ , (3.45b) N (γ - 0 Φ), p t Γ∞ + D + 1 2 I (λ), p t Γ∞ -δ Γ∞ (p, p t ) = γ 1 f inc , p t Γ∞ , ( 3.45c)
with product space

H := H 1 (Ω -)×H -1 2 (Γ ∞ )×H 1 (Γ ∞ ) and inner product (Φ, λ, p) , Φ t , λ t , p t H := Φ, Φ t H 1 (Ω -) + λ, λ t H -1 2 (Γ∞) + p, p

Finite-dimensional approximation

The coupled formulations (3.36) and (3.45) are approximated by finite element and boundary element methods. The underlying results are well-known from both theories and can be directly applied to the present setting.

Discrete finite element spaces

Let M be a shape-regular tetrahedral mesh of Ω -. The mesh F ∞ of Γ ∞ is composed of the boundary faces of M. Let h M > 0 denote the mesh size, V 1 M the space of continuous piecewise affine polynomials on M, S 0 M the space of piecewise constant polynomials on F ∞ , and S 1 M the space of continuous piecewise affine polynomials on

F ∞ . Let H M := V 1 M × S 0 M ,

and

H M := V 1 M × S 0 M × S 1 M . The discretization of (3.36) reads: Find (Φ M , λ M ) ∈ H M such that, ∀ Φ t M , λ t M ∈ H M , a unstab (Φ M , λ M ) , Φ t M , λ t M = b unstab Φ t M , λ t M , ( 3.46)
with a unstab and b unstab readily deduced from (3.36), while the discretization of (3.45) reads:

Find (Φ M , λ M , p M ) ∈ H M such that, ∀ Φ t M , λ t M , p t M ∈ H M , a stab (Φ M , λ M , p M ) , Φ t M , λ t M , p t M = b stab Φ t M , λ t M , p t M , (3.47)
with a stab and b stab readily deduced from (3.45). Since H M ⊂ H and H M ⊂ H, both approximations are conforming.

In what follows, A B denotes the inequality A ≤ cB with positive constant c independent of the mesh size and of the discrete and exact solutions. The following classical approximation properties are available (see [START_REF] Brenner | The Mathematical Theory of Finite Element Methods[END_REF][START_REF] Ern | Theory and Practice of Finite Elements[END_REF][START_REF] Sauter | Boundary Element Methods[END_REF]):

inf Φ M ∈V 1 M Φ -Φ M H 1 (Ω -) h M Φ H 2 (Ω -) , inf λ M ∈S 0 M λ -λ M H -1 2 (Γ∞) h M λ H 1 2 (Γ∞) , inf p M ∈S 1 M p -p M H 1 (Γ∞) h M p H 2 (Γ∞) . (3.48)
Hence, the following approximation properties hold:

∀(Φ, λ) ∈ H 2 (Ω -) × H 1 2 (Γ ∞ ), inf (Φ M ,λ M )∈H M (Φ, λ) -(Φ M , λ M ) H h M Φ H 2 (Ω -) + λ H 1 2 (Γ∞) , ( 3.49)
and

∀(Φ, λ, p) ∈ H 2 (Ω -) × H 1 2 (Γ ∞ ) × H 2 (Γ ∞ ), inf (Φ M ,λ M ,p M )∈H M (Φ, λ, p) -(Φ M , λ M , p M ) H h M Φ H 2 (Ω -) + λ H 1 2 (Γ∞)
+ p H 2 (Γ∞) .

(3.50) Remark 3.12 Taking a polynomial approximation with one order less for H -1 2 (Γ ∞ ) than for H 1 (Ω -) and H 1 (Γ ∞ ) enables all the approximations to be at the same order in h M .

Discretization of the coupled formulations

Unstable formulation

Let (θ i ) 1≤i≤p and (ψ i ) 1≤i≤q denote finite element bases for V 1 M and S 0 M respectively. These basis functions are real-valued. The decompositions of

Φ M ∈ V 1
M and λ M ∈ S 0 M on these bases are written in the form

Φ M = p i=1 Φ Mi θ i and λ M = q i=1 λ Mi ψ i . Let u unstab M = (Φ Mi ) 1≤i≤p (λ Mi ) 1≤i≤q , B unstab =   γ 1 f inc , γ - 0 θ i Γ∞ 1≤i≤p -(γ 0 f inc , ψ i ) Γ∞ 1≤i≤q   ,
(3.51)

A unstab =   V(θ j , θ i ) + N (γ - 0 θ j ), γ - 0 θ i Γ∞ D -1 2 I (ψ j ), γ - 0 θ i Γ∞ D -1 2 I (γ - 0 θ j ), ψ i Γ∞ -(S(ψ j ), ψ i ) Γ∞   , (3.52)
where in A unstab the index i refers to the rows and the index j to the columns. The linear system resulting from (3.46) is

A unstab u unstab M = B unstab . (3.53)
To better understand the structure of the linear system (3.53), the basis functions (θ i ) 1≤i≤p of V 1 M are separated into two sets: the basis function (θ F∞ i ) 1≤i≤p F∞ associated to the vertices of F ∞ , and (θ M i ) 1≤i≤p M , such that p = p F∞ + p M. The matrix A unstab is written

A unstab =      V(θ M j , θ M i ) V(θ F∞ j , θ M i ) 0 V(θ M j , θ F∞ i ) V(θ F∞ j , θ F∞ i ) + N (γ - 0 θ F∞ j ), γ - 0 θ F∞ i Γ∞ D -1 2 I (ψ j ), γ - 0 θ F∞ i Γ∞ 0 D -1 2 I (γ - 0 θ F∞ j ), ψ i Γ∞ -(S(ψ j ), ψ i ) Γ∞      .
(3.54) The blocks of the matrix in (3.54) are denoted

A unstab =    A unstab 1,1 A unstab 1,2 0 A unstab 2,1 A unstab 2,2 A unstab 2,3 0 A unstab 3,2 A unstab 3,3    . ( 3

Stable formulation

Let (ξ i ) 1≤i≤r denote a finite element basis for S 1 M . The decomposition of p M ∈ S 1 M on this basis is written in the form p M = r i=1 p Mi ξ i . Let

u stab M =    (Φ Mi ) 1≤i≤p (λ Mi ) 1≤i≤q (p Mi ) 1≤i≤r    , B stab =     γ 1 f inc , γ - 0 θ i Γ∞ 1≤i≤p -(γ 0 f inc , ψ i ) Γ∞ 1≤i≤q (γ 1 f inc , ξ i ) Γ∞ 1≤i≤r     ,
(3.56)

A stab =       V(θ j , θ i ) + N (γ - 0 θ j ), γ - 0 θ i Γ∞ D -1 2 I (ψ j ), γ - 0 θ i Γ∞ 0 D -1 2 I (γ - 0 θ j ), ψ i Γ∞ -(S(ψ j ), ψ i ) Γ∞ iη(ξ j , ψ i ) Γ∞ N (γ - 0 θ j ), ξ i Γ∞ D -1 2 I (ψ j ), ξ i Γ∞ -δ Γ∞ (ξ j , ξ i ) Γ∞       , (3.57)
with the same convention on the indices i and j of A stab . The linear system resulting from (3.47) is

A stab u stab M = B stab . (3.58)
Like the previous section, the matrix of the linear system (3.58) is written

A stab =         V(θ M j , θ M i ) V(θ F∞ j , θ M i ) 0 0 V(θ M j , θ F∞ i ) V(θ F∞ j , θ F∞ i ) + N (γ - 0 θ F∞ j ), γ - 0 θ F∞ i Γ∞ D -1 2 I (ψ j ), γ - 0 θ F∞ i Γ∞ 0 0 D -1 2 I (γ - 0 θ F∞ j ), ψ i Γ∞ -(S(ψ j ), ψ i ) Γ∞ iη(ξ j , ψ i ) Γ∞ 0 N (γ - 0 θ F∞ j ), ξ i Γ∞ D -1 2 I (ψ j ), ξ i Γ∞ -δ Γ∞ (ξ j , ξ i ) Γ∞         . (3.59)
The blocks of the matrix in (3.59) are denoted

A stab =      A stab 1,1 A stab 1,2 0 0 A stab 2,1 A stab 2,2 A stab 2,3 0 0 A stab 3,2 A stab 3,3 A stab 3,4 0 A stab 4,2 A stab 4,3 A stab 4,4      . (3.60)
All the blocks are complex-valued. The blocks

A stab 1,1 , A stab 1,2 , A stab 1,3 , A stab 2,1 , A stab 2,2 , A stab 2,3 , A stab 3,

Inf-sup stability of the discretized formulations

From the Fredholm setting and the approximation properties (3.49) and (3.50), the following discrete inf-sup conditions can be derived following [55, Theorem 14].

Proposition 3.13 If -k2 ∞ / ∈ Λ and h M is small enough, there holds, for all (Φ M , λ M ) ∈ H M , sup (0,0) =(Φ t M ,λ t M )∈HM a unstab (Φ M , λ M ) , Φ t M , λ t M (Φ t M , λ t M ) H (Φ M , λ M ) H . (3.61)
At all frequencies and if h M is small enough, there holds, for all

(Φ M , λ M , p M ) ∈ H M , sup (0,0,0) =(Φ t M ,λ t M ,p t M )∈HM a stab (Φ M , λ M , p M ) , Φ t M , λ t M , p t M (Φ t M , λ t M , p t M ) H (Φ M , λ M , p M ) H . (3.62)

Convergence

From the inf-sup stability of the discrete problems, the following error estimates easily follow from [55, Theorem 13].

Proposition 3.14 If -k2

∞ / ∈ Λ and h M is small enough, the discrete problem (3.46) has a unique solution (Φ M , λ M ) ∈ H M , and the following optimal error estimate holds:

(Φ, λ) -(Φ M , λ M ) H inf (Φ t M ,λ t M )∈HM (Φ, λ) -Φ t M , λ t M H , (3.63)
where (Φ, λ) is the unique solution of (3.36). At all frequencies and if h M is small enough, the discrete problem (3.47) has a unique solution (Φ M , λ M , p M ) ∈ H M , and the following optimal error estimate holds:

(Φ, λ, p) -(Φ M , λ M , p M ) H inf (Φ t M ,λ t M ,p t M )∈HM (Φ, λ, p) -Φ t M , λ t M , p t M H , (3.64)
where (Φ, λ, p) is the unique solution of (3.45).

Remark 3.15

The constant in (3.63) depends on k∞ , and its value explodes as -k2 ∞ tends to an element of Λ. The constant in (3.64) depends on k∞ as well, but remains bounded on any bounded set of frequencies.

Numerical resolution

Both the unstable and stable formulations have been implemented in the EADS in-house boundary element software called ACTIPOLE [START_REF] Delnevo | Code acti3s harmonique, justification mathématique, Partie I[END_REF][START_REF] Delnevo | Code acti3s, justifications mathématiques, Partie II : presence d'un écoulement uniforme[END_REF]. This software can treat general threedimensional geometries. The iterative solver is a block-GMRES [START_REF] Langou | Solving large linear systems with multiple right-hand sides[END_REF][START_REF] O'leary | The block conjugate gradient algorithm and related methods[END_REF][START_REF] Saad | GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems[END_REF] with no restart, suitable for non-symmetric linear systems. The "block" means that the solver treats all the right-hand sides simultaneously. The restart is an option that enables to save memory, but generally degrades the convergence. Since the solver stores on hard drive created during the iterations on hard drive, the memory usage is not an issue. In the iterative solver, the specificity of each block is taken into account. Matrix-vector products involving sparse blocks are optimized accordingly, and matrix-vector products involving boundary integral terms can be accelerated using a fast multipole method and out-of-core parallelization techniques. The preconditioner uses a combination of a sparse approximate inverse (SPAI) preconditioner [START_REF] Carpentieri | Sparse preconditioners for dense linear systems from electromagnetic applications[END_REF][START_REF] Carpentieri | Combining fast multipole techniques and an approximate inverse preconditioner for large electromagnetism calculations[END_REF] and the sparse direct solver MUMPS [START_REF] Amestoy | Multifrontal parallel distributed symmetric and unsymmetric solvers[END_REF]. More precisely, for each of the dense diagonal blocks

A unstab 2,2 , A unstab 3,3
and A stab 2,2 , A stab 3,3 , the SPAI preconditioner searches for an approximation of the inverse of these blocks. Consider any these blocks, denoted A in the following. A is made sparse by keeping, in each column, the interaction terms between the corresponding basis function and the ones in its vicinity (in the sense that the corresponding vertices or faces are nearby). The result of this operation is denoted by A sp , and define

S A sp = {M ∈ C n,n | ∀ 1 ≤ i, j ≤ n such that A sp i,j = 0, M i,j = 0}.
The SPAI preconditioner of A is given by P := argmin

S A sp A sp M -I F , where • F
denotes the Frobenius norm, and n the number of rows of A. Notice that for the blocks A unstab 2,2 and A stab 2,2 , the SPAI preconditioner is computed ignoring the volumic contributions. For the sparse diagonal blocks, the preconditioner is taken as the inverse of each blocks. The inverse is not actually computed: since MUMPS provides a factorization of each of these blocks, each time a product preconditioner-vector is needed when constructing the Krylov vectors of the iterative method, two triangular systems are efficiently solved using this factorization. The preconditioner for the whole system is block diagonal, each bloc being a SPAI or MUMPS preconditioner.

Numerical results

The purpose of this section is the comparison between the unstable formulation (3.36) and the stable formulation (3.45) with the coupling parameter η = 1. Both formulations have been implemented in the EADS in-house boundary element software called ACTIPOLE [START_REF] Delnevo | Code acti3s harmonique, justification mathématique, Partie I[END_REF][START_REF] Delnevo | Code acti3s, justifications mathématiques, Partie II : presence d'un écoulement uniforme[END_REF]. This software can treat general three-dimensional geometries. The iterative solver is a block generalized conjugate residual method [START_REF] O'leary | The block conjugate gradient algorithm and related methods[END_REF][START_REF] Langou | Solving large linear systems with multiple right-hand sides[END_REF] based on a generalized minimal residual method [START_REF] Saad | GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems[END_REF], suitable for non-symmetric systems. A sparse approximate inverse preconditioner [START_REF] Carpentieri | Sparse preconditioners for dense linear systems from electromagnetic applications[END_REF][START_REF] Carpentieri | Combining fast multipole techniques and an approximate inverse preconditioner for large electromagnetism calculations[END_REF] is used. Consider an ellipsoid with major axis directed along the z-axis. This object is included inside a larger ball. The external border of the ball after discretization is the surface Γ ∞ . A potential flow is computed around the ellipsoid and inside the ball, such that the flow is uniform outside the ball, of Mach number 0.3 and directed along the z-axis. An acoustic monopole source lies upstream of the ball, on the z-axis as well. Four different meshes are considered, see Table 3.1. For accuracy reasons, a rule of thumb in boundary elements method for the classical Helmholtz equation consists in imposing that the mean edge is at least eight to ten times smaller than the wavelength of the source. In our software, we first generate the mesh and then apply the Prandtl-Glauert transformation. Therefore, in this test case, the mesh is at most extended by a factor γ ∞ ≈ 1.048. Moreover, the integral operators are computed at the transformed wavenumber k∞ = γ ∞ k ∞ , resulting on a wavelength of approximately 0.21 m. We then verify that, for Mesh 1, the mean edge of the transformed mesh is eight times smaller than the wavelength. As a consequence, the three other meshes do not satisfy the rule of thumb, but will be used as comparison supports and in numerical experiments requiring a large number of resolutions. In this test case, the extension of the edges of the mesh and the modification of the wavenumber induced by the Prandtl-Glauert transformation are mild, but can become very large as M ∞ is close to 1. The first effect can be canceled by computing the mesh on the geometry already modified by the Prandtl-Glauert transformation.

From Table 3.1, for fine meshes, the number of basis functions used to discretize the unknown p for the variational formulation (3.47) takes a smaller part in the total number of basis functions than for coarse meshes. Therefore, the relative complexity added to (3.36) by the third equation of (3.45) decreases with the total number of unknowns, which is an interesting property when it comes to industrial test cases. In what follows, a frequency

f is called resonant if -k2 ∞ = -4π 2 f 2 γ 2 ∞ ∈ Λ,
where Λ is the set of Dirichlet eigenvalues for the Laplacian on R 3 \Ω + . The set Λ depends on the geometric shape of the coupling surface Γ ∞ , which slightly changes after each discretization.

Comparison of pressure fields

As seen in Theorem 3.7, the unstable formulation (3.36) is not well-posed at resonant frequencies. First, a prospective study to identify a resonant frequency for each of the four meshes is carried out by monitoring the condition number of the matrices produced by the discretized version of the unstable formulation (3.36). A resonant frequency for Mesh 1, Mesh 2, Mesh 3, and Mesh 4 is identified around 1509.849 Hz, 1513.431 Hz, 1521.015 Hz, and 1535.704 Hz respectively. The convergence of the iterative solver is monitored by requiring that the Euclidian norm of the relative residual is smaller than 10 -6 . Additional tests indicate that the discretized solution to the stable formulation does not change much below this value of the relative residual. For Mesh 1, away from a resonance, say at 1500 Hz, the scattered pressure fields computed with the unstable and stable formulations are very similar. This holds as well for the total pressure fields, see Figure 3.3. At the resonant frequency 1509.849 Hz, the unstable formulation (3.36) yields pressure maps quite different from the ones at 1500 Hz, whereas the stable formulation (3.45) yields pressure maps very similar to the ones at 1500 Hz, see Figure 3.4. From Proposition 3.5, at a resonant frequency, the solutions to (3.36) that differ from the solution to (3.45) should not affect the scattered pressure field in the exterior domain. Actually, the distortion of the scattered field with the unstable formulation (3.36) is the result of the significant magnification of discretization and numerical errors by the ill-conditioning of the linear system approximating (3.36).

Auxiliary variable p

In Figure 3.5, the left plot indicates that with Mesh 1, the magnitude of p is around 0.5% of the scattered pressure. The right plot shows the behaviour of the magnitude of p (measured as p L ∞ (Γ∞) ) with respect to the stopping criterion of the iterative solver for the four meshes. The finer the mesh, the smaller the auxiliary variable p, which is consistent with the fact that the p-component of the solution to (3.45) vanishes (see Proposition 3.9). of the peak at the resonance does not appear to depend on the mesh. In the right plot, a larger bandwidth is considered with Mesh 2. Owing to the frequency sampling (every 5 Hz), some resonances may be missed and the local maxima may not be accurately reached (in particular, from the left plot, the local maximum of 7.2 for log(cond(M )) at 1513.431 Hz is very underestimated). The stable formulation (3.45) produces somewhat larger condition numbers for the large majority of the frequencies, but, unlike the unstable formulation (3.36), it presents no resonance. Moreover, from the Weyl formula, the number of resonant frequencies smaller than f increases as f 3 2 , making the need for a stable formulation even more important for simulations at higher frequencies.

Comparison of condition numbers

Convergence

To further study the impact of the ill-conditioning of the unstable formulation (3.36) on the computed solution, the preconditioning is not used in what follows. First, the value of the acoustic pressure on a network of 10000 points located further than 0.5 m from the center of the sphere (therefore in Ω + ) is computed using the stable formulation (3.45) with Mesh 1 at the resonant frequency 1509.849 Hz. This computed acoustic pressure is called the accurate pressure. Next, the acoustic pressure on the same network of points is computed for different values of the number of iterations of the solver, using the unstable formulation (3.36) and the stable formulation (3.45) with Mesh 1 at the resonance 1509.849 Hz. The relative difference between the computed pressure and the accurate pressure in Euclidian norm is called the relative error. Figure 3.7 presents the relative residual and the relative error with respect to the number of iterations. With the unstable formulation (3.36), the relative residual decreases irregularly. In particular, it stays constant during around 200 iterations. The relative error decreases, stays constant, rises after 400 iterations, and finally stabilizes at a large value, whereas the relative residual keeps converging to zero. As for every ill-conditioned problem, the relative residual cannot be used to ascertain convergence towards the correct solution. In particular, after 600 iterations, the relative residual is extremely small, while the error is of order one. With the are decoupled from (3.45c), and (3.45a)-(3.45b) become (3.36), so that the curve for η = 0.001 is similar to the curve of the unstable formulation for Mesh 4 in Figure 3.6. The condition number appears to be the smallest for η in the range 1 to 10, and worsens for lower and higher values of η. This motivates the choice η = 1 made in the above simulations.

Conclusion

In this work, we derived two coupled boundary element / finite element methods for the convected Helmholtz equation with non-uniform flow in a bounded domain. The first one leads to an unstable formulation, while the second one leads to a stable formulation. The unstable formulation involves two equations and is well-posed except at some resonant frequencies of the source, while the stable one is unconditionally well-posed, but involves three equations. Even if the unstable formulation admits infinitely many solutions at resonant frequencies, the pressure field resulting from any of these solutions equals the one resulting from the stable formulation. However, our numerical results show that at resonant frequencies, the discretization of the unstable formulation is so ill-conditioned that the pressure field is very different from the one produced by the stable formulation. Moreover, the stable formulation remains tractable within large industrial problems since the relative complexity added by its third equation decreases with the size of the mesh. Its interest is also enhanced by the fact that, at higher frequencies, the density of resonant frequencies is more important.

As long as the uniform flow assumption in the exterior domain is reasonable, more complex flows in the interior domain can be considered, as well as more complex boundary conditions at the surface of the scattering object. These extensions only require to modify the finite element part of the present methodology. An important development for practical simulations is the introduction of modal sources in the interior domain, which is the purpose of [Pr2].

Another interesting extension of this work is the resolution of parametrized aeroacoustic problems, with the frequency of the source as a parameter, using reduced-order models, for instance by means of Proper Generalized Decomposition or Reduced Basis methods. Using the unstable formulation may involve ill-conditioned numerical resolutions if the frequency range of interest contains resonant frequencies, whereas the stable formulation guarantees well-posedness of the procedure. Moreover, the complexity of the online stage of the reduced-order model is not increased by the third equation of the stable formulation.

Annex: Proof of the mathematical results

As preliminary results, we derive Propositions 3.16 and 3.17 Notice that we do no start with the well-posedness of (3.21) since, to our knowledge, a result directly applicable to it is not available in the literature. Herein, we prove the well-posedness of (3.45) which is equivalent to (3.21).

Proposition 3.16 Problem (3.21) has at most one solution in H 1 loc (Ω).

Proof. Let f ∈ H 1 loc (Ω) solve (3.21) with ς = 0 (so that f inc = 0). From Proposition 3.1, f solves (3.18). Let B be an open ball containing Ω -. Let f t ∈ H(∆, B). Using Green's first identity, 0 = Ω∩B -rk 2 βf -ikrV • ∇f -∇ • (irkf V + (rΞ∇f ) f t = Ω∩B rΞ∇f • ∇f t -rk 2 βf f t -ikrV • (∇f f t -∇f t f ) -γ - 1,∂B f, γ - 0,∂B f t ∂B , ( 3.65)
where γ - 0,∂B and γ - 1,∂B are the Dirichlet and Neumann traces on ∂B from B. Taking

f t = f yields γ - 1,∂B f, γ - 0,∂B f ∂B = Ω∩B rΞ∇f • ∇f -rk 2 βf f -2krV • Im∇f f , (3.66) so that Im γ - 1,∂B f, γ - 0,∂B f ∂B = 0. Using Rellich Lemma (see [76, Lemma 9.9]), since f ∈ H 1 loc (R 3
\B) solves the classical Helmholtz equation in R 3 \B and satisfies the Sommerfeld radiation condition, as well as Im γ -

1,∂B f, γ - 0,∂B f ∂B ≥ 0, it is inferred that f | R 3 \B ≡ 0. Equation (3.18a) with ς = 0 can be written L(f ) := rk 2 β + ∇ • (irkV ) f + 2irkV • ∇f + ∇ • (rΞ∇f ) = 0 in Ω. (3.67)
From [45, Theorem 1.1], since rΞ is uniformly elliptic with Lipschitz continuous coefficients, and rk 2 β + ∇ • (irkV ) and 2irkV have bounded coefficients, the differential operator L satisfies the strong unique continuation property in Ω. Hence, f | R 3 \B ≡ 0 implies that the only H 1 loc (Ω) solution of (3.18) with ς = 0 is f ≡ 0 in Ω. The assertion follows from Proposition 3.1. ♦ Proposition 3.17 Let (Φ, λ, p) solve (3.45). There holds p = 0, λ = γ - 1 Φ, and

rk 2 βΦ -+ irkV • ∇Φ -+ ∇ • irkΦ -V + rΞ∇Φ -= 0 in L 2 (Ω -), (3.68a) γ - n,Γ (irkΦV + rΞ∇Φ) = 0 in H -1 2 (Γ ), (3.68b
)

N (γ - 0 Φ) + D + 1 2 I (λ) = γ 1 f inc in H -1 2 (Γ ∞ ), (3.68c) D - 1 2 I (γ - 0 Φ) -S(λ) = -γ 0 f inc in H 1 2 (Γ ∞ ). (3.68d)
Proof. Set σ := irkΦV + rΞ∇Φ. Owing to (3.45a) and recalling the definition (3.30) of the sesquilinear form V, there holds,

∀Φ t ∈ H 1 (Ω -), Ω - σ•∇Φ t = Ω - rk 2 βΦΦ t +i Ω - rkV •∇ΦΦ t -N (γ - 0 Φ) + D - 1 2 I (λ) -γ 1 f inc , γ - 0 Φ t Γ∞ . (3.69)
Restricting the test function Φ t to C ∞ c (Ω -) shows that (3.68a) holds in L 2 (Ω -). Moreover, since

Ω - σ • ∇Φ t = - Ω - (∇ • σ) Φ t + γ - n σ, γ - 0 Φ t Γ∞ + γ - n,Γ σ, γ - 0,Γ Φ t Γ , ( 3.70)
and recalling that γ - n σ = γ - 1 Φ on Γ ∞ , owing to the property of the convective flow at Γ ∞ ,

N (γ - 0 Φ) + D - 1 2 I (λ) -γ 1 f inc , γ - 0 Φ t Γ∞ + γ - 1 Φ, γ - 0 Φ t Γ∞ + γ - n,Γ σ, γ - 0,Γ Φ t Γ = 0.
(3.71) Then, by the surjectivity of the trace operators γ - [

0 from H 1 (Ω -) onto H 1 2 (Γ ∞ ) and γ - 0,Γ from H 1 (Ω -) onto H 1 2 (Γ ) (see
N (γ - 0 Φ) + D - 1 2 I (λ) + γ - 1 Φ = γ 1 f inc , (3.72a) D - 1 2 I (γ - 0 Φ) -S(λ) -iηp = -γ 0 f inc . (3.72b) Owing to (3.45c) and (3.72a), it is inferred that δ Γ∞ (p, p t ) = (λ-γ - 1 Φ, p t ) Γ∞ for all p t ∈ H 1 (Ω -). From the definition (3.38) of M , there holds p = M (λ -γ - 1 Φ). Let x := λ -γ - 1 Φ. Then, from (3.26), (3.72) can be written    1 2 I -D S N 1 2 I + D       γ - 0 Φ λ    =    -iηM x + γ 0 f inc x + γ 1 f inc    , ( 3.73)
so that (-iηM x + γ 0 f inc , x + γ 1 f inc ) belongs to the range of the block operator defined on the left-hand side of (3.73). Under this condition, from [54, Theorem 4.1], citing [START_REF] Petersdorff | Boundary integral equations for mixed Dirichlet, Neumann and transmission problems[END_REF], a radiating piecewise Helmholtz solution u such that γ - 0 u = -iηM x + γ 0 f inc and γ - 1 u = x + γ 1 f inc can be constructed. Consider v such that v| Ω + = 0 and v| R 3 \Ω + = u| Ω -, and w such that w| Ω + = 0 and w| R 3 \Ω + = f inc . Since v and w are radiating piecewise Helmholtz solutions, û := vw is also a radiating piecewise Helmholtz solution. Since [γ 0 û] = iηM x and [γ 1 û] = -x, (3.26) implies where a := a 1 + a 2 . We show the well-posedness of (3.78) by proving successively that (i) a 1 and a 2 are bounded on H × H and b is bounded on H, (ii) a 1 is H-coercive, (iii) the linear map associated with a 2 is compact from H into H. With the uniqueness of the solution (Proposition 3.18), the assertion follows from the Fredholm alternative.

   1 2 I -D S N 1 2 I + D       iηM x -x    =    iηM x -x    . ( 3
equation in R 3 \Ω + , therefore γ - 1 ũ-, γ - 0 ũ- Γ∞ = R 3 \Ω + {|∇ũ| 2 -k2 ∞ |ũ| 2 } ∈ R. ( 3
of R(Φ, λ) is γ + 0 R(Φ, λ) = γ + 0 (-S(λ) + D(γ - 0 Φ) + f inc ) = -S(λ) + (D + 1 2 I)(γ - 0 Φ) + γ 0 f inc . Using (3.68d), there holds γ + 0 R(Φ, λ) = γ - 0 Φ, which is γ - 0 R(Φ,
a 1 (Φ, λ, p) , Φ t , λ t , p t := Ω - rΞ∇Φ • ∇Φ t + N 0 (γ - 0 Φ), γ - 0 Φ t Γ∞ + S 0 (λ), λ t Γ∞ + δ Γ∞ (p, p t ) + D0 - 1 2 I (λ), γ - 0 Φ t Γ∞ - D 0 - 1 2 I (γ - 0 Φ), λ t Γ∞ , a 2 (Φ, λ, p) , Φ t , λ t , p t := - Ω - rk 2 βΦΦ t + i Ω - rkV • Φ∇Φ t -Φ t ∇Φ + N -N 0 (γ - 0 Φ), γ - 0 Φ t Γ∞ + S -S 0 , λ t (λ) Γ∞ + D -D0 (λ), γ - 0 Φ t Γ∞ -D -D 0 , λ t (γ - 0 Φ) Γ∞ -iη p, λ t Γ∞ -N (γ - 0 Φ), p t Γ∞ - D + 1 2 I (λ), p t Γ∞ , ( 3
Φ t , λ t , p t := γ 1 f inc , γ - 0 Φ t Γ∞ + γ 0 f inc , λ t Γ∞ -γ 1 f inc , p t Γ∞ . ( 3
(i) From Section 3.2.4, Ξ∇Φ

• ∇Φ t ≤ 1+M 2 0 1-M 2 ∞ ∇Φ ∇Φ t . Then, Ω - rΞ∇Φ • ∇Φ t ≤ 1 1 -M 2 ∞ r L ∞ (Ω -) 1 + M 2 0 L ∞ (Ω -) Φ H 1 (Ω -) ||Φ t || H 1 (Ω -) . (3.79)
The other volumic integrals are simply controlled by

Ω - rkV • Φ∇Φ t -Φ t ∇Φ ≤ 2 rk L ∞ (Ω -) V L ∞ (Ω -) 3 Φ H 1 (Ω -) Φ t H 1 (Ω -) , Ω rk 2 βΦΦ t ≤ rk 2 β L ∞ (Ω -) Φ H 1 (Ω -) Φ t H 1 (Ω -) .
(3.80)

From [START_REF] Mclean | Strongly elliptic systems and boundary integral equations[END_REF]Theorem 6.11], all the involved integral operators are bounded in their natural trace spaces. The boundedness constant of an operator A is denoted by C A , and the continuity constant of the interior Dirichlet trace operator is denoted by

C γ - 0 : γ - 0 Φ H 1 2 (Γ∞) ≤ C γ - 0 Φ H 1 (Ω -) . Moreover, since H 1 (Γ ∞ ) ⊂ H 1 2 (Γ ∞ ), there exists a constant C Γ∞ > 0 such that p H 1 2 (Γ∞) ≤ C Γ∞ p H 1 (Γ∞)
. These inequalities lead to

a 1 (Φ, λ, p) , Φ t , λ t , p t ≤ 2 1 + 1 1 -M 2 ∞ r L ∞ (Ω -) 1 + M 2 0 L ∞ (Ω -) + C S 0 +C γ - 0 1 + C D 0 + C D0 + C 2 γ - 0 C N 0 (Φ, λ, p) H Φ t , λ t , p t H , a 2 (Φ, λ, p) , Φ t , λ t , p t ≤ 2 rk 2 β L ∞ (Ω -) + 2 rk L ∞ (Ω -) V L ∞ (Ω -) 3 + C Γ∞ 1 2 + |η| + C D + C S 0 +C S + C γ - 0 C Γ∞ C N + C D 0 + C D + C D0 + C D + C 2 γ - 0 (C N 0 + C N ) (Φ, λ, p) H Φ t , λ t , p t H , b Φ t , λ t , p t ≤ √ 2 (C γ - 0 + C Γ∞ ) γ 1 f inc H -1 2 (Γ∞) + γ 0 f inc H 1 2 (Γ∞) Φ t , λ t , p t H .
(ii) Unlike D and D, the operators D 0 and D0 are real-valued. They are therefore adjoint, so that D0 -

1 2 I (λ), γ - 0 Φ Γ∞ - D 0 - 1 2 I (γ - 0 Φ), λ t Γ∞ ∈ iR. (3.81)
From [32, Theorem 2], the operators N 0 and S 0 are strongly elliptic in their natural trace spaces. Moreover, from Section 3.2.4, for all

U ∈ C 3 , U • ΞU ≥ 1 -M 2 0 U 2 . Then, there holds Re (a 1 ((λ, Φ, p) , (λ, Φ, p))) = Ω rΞ∇Φ • ∇Φ + N 0 (γ - 0 Φ), γ - 0 Φ Γ∞ + S 0 (λ), λ Γ∞ + δ Γ∞ (p, p) ≥ inf Ω - r 1 -M 2 0 ∇Φ 2 L 2 (Ω -) + K N 0 γ - 0 Φ 2 H 1 2 (Γ∞) + K S 0 λ 2 H -1 2 (Γ∞) + p 2 H 1 (Γ∞) , ( 3
.82) where the coercivity constant of an operator A is denoted by K A . From the Petree-Tartar Lemma [START_REF] Ern | Theory and Practice of Finite Elements[END_REF]Lemma A.38], it can be shown that there exists a constant C Ω -> 0 such that

Φ H 1 (Ω -) ≤ C Ω -∇Φ L 2 (Ω -) 3 + γ - 0 Φ H 1 2 (Γ∞)
. Therefore,

Re (a 1 ((λ, Φ, p) , (λ, Φ, p))) ≥ 1 2C 2 Ω - min inf Ω - r 1 -M 2 0 , K N 0 Φ 2 H 1 (Ω -) + K S 0 λ 2 H -1 2 (Γ∞) + p 2 H 1 (Γ∞) ≥ min inf Ω -r 1 -M 2 0 2C 2 Ω - , K N 0 2C 2 Ω - , K S 0 , 1 (Φ, λ, p) 2 H .
(3.83)

(iii) Let V be a Hilbert space, let A be an operator from V to V , and let a be a sesquilinear form such that, for all u, v ∈ V , a(u, v) = (Au, v) V . A classical result states that A is compact if and only if, for all weakly convergent sequences (u n ), (v n ) ∈ V N such that u n ⇀ u and v n ⇀ v, there holds, up to subsequences, a(u n , v n ) → a(u, v). Let (Φ 1n , λ 1n , p 1n )⇀(Φ 1 , λ 1 , p 1 ) and (Φ 2n , λ 2n , p 1n )⇀(Φ 2 , λ 2 , p 1 ) be two weakly convergent sequences in H. Since the injection of

H 1 (Ω -) into L 2 (Ω -) is compact, then, up to subsequences, Φ in →Φ i and ∇Φ in ⇀∇Φ i , i = 1, 2, in L 2 (Ω -). Therefore, up to subsequences, - Ω - rk 2 βΦ 1n Φ 2n + i Ω - rkV • Φ 1n ∇Φ 2n -Φ 2n ∇Φ 1n → - Ω - rk 2 βΦ 1 Φ 2 + i Ω - rkV • Φ 1 ∇Φ 2 -Φ 2 ∇Φ 1 . (3.84)
Moreover, from [93, Lemma 3.9.8], S -S 0 , N -N 0 , D -D 0 and D -D0 are compact operators in their natural trace spaces. Hence, up to subsequences, S -S 0 (λ 1n )→ S -S 0 (λ 1 ) in

H 1 2 (Γ ∞ ),
   1 2 I + D -S -N 1 2 I - D       γ - 0 Φ λ    =    γ - 0 Φ γ - 1 Φ    , ( 3.85)
so that γ - 0 Φ, γ - 1 Φ belongs to the range of the block operator defined on the left-hand side of (3.85). Under this condition, from [54, Theorem 4.1], citing [START_REF] Petersdorff | Boundary integral equations for mixed Dirichlet, Neumann and transmission problems[END_REF], a radiating piecewise Helmholtz solution u such that γ + 0 u = γ - 0 Φ and γ + 1 u = γ - 1 Φ can be constructed. Consider the function v defined as v| Ω + = u| Ω + and v| R 3 \Ω + = 0. The function v is still a radiating piecewise Helmholtz solution, and its jumps of traces are

[γ 0 v] Γ∞ = γ - 0 Φ and [γ 1 v] Γ∞ = γ - 1 Φ. From (3.26), 1 2 I -D γ - 0 Φ + Sγ - 1 Φ = -γ - 0 v -= 0.
Together with the first line of (3.85), it is deduced that λγ - 1 Φ ∈ Ker(S) = Ker( D -1 2 I). Therefore, (3.68c) and (3.68d) hold. Then, since single-layer and double-layer potentials are radiating piecewise Helmholtz solutions, R(Φ, λ) satisfies (3.21b) The scope of this chapter is to validate the BEM/FEM coupled formulation (3.36), that uses the new transformed convected Helmholtz equation (3.18a). We suppose that the frequency of the source is a nonresonant frequency for the considered problem (more precisely, -k2 ∞ / ∈ Λ). This can be assessed by verifying that the formula (1.30) does not produce absurd values in R + \Ω + , as it is the case on the left plots of Figure 3.4. All the simulations on this chapter are three-dimensional.

Validation campaign

The formulation (3.36) is compared to other numerical methods in the following cases:

-flow at rest and uniform properties, comparison with ACTIPOLE with only BEM in Section 4.1.1, -flow at rest and different properties, comparison with an analytic solution computed by means of Mie series in Section 4.1.2, -nonuniform flow and nonuniform properties, qualitative comparison with ACTI-HF and ISVR in Section 4.1.4. Then, industrial test cases are presented in Section 4.2.

Flow at rest and uniform properties:

M = M ∞ = 0, ρ = ρ ∞ , c = c ∞ ,

comparison with ACTIPOLE with only BEM

The code we used to implement the formulations (3.36), ACTIPOLE [START_REF] Delnevo | Code acti3s harmonique, justification mathématique, Partie I[END_REF][START_REF] Delnevo | Code acti3s, justifications mathématiques, Partie II : presence d'un écoulement uniforme[END_REF], is a boundary element method code designed by EADS-IW and Airbus to solve Helmholtz exterior problems with uniform (and zero) flows. In its original form, ACTIPOLE computes only surface potentials, and then cannot be used with an inhomogeneous Ω -. This original state of the code can be used as a reference when the fluid characteristics in Ω -are equal to the one in Ω + .

We first consider the geometry presented in Figure 4.1.In this figure, the plain cube is solid object perfectly reflecting the incoming acoustic waves (with boundary condition (3.2) on the acoustic potential), whereas the hatched areas correspond to the interior domain Ω -meshed with 11510 tetrahedra. The boundary of Ω -and the boundary of the solid object are each composed of 2418 triangles. A quantitative comparison is presented in Table 4.1, where we consider a network of 100 points uniformly distributed on a part of a hyperplane located in Ω + . The difference comes from the fact that with the formulation (3.36), the pressure in the interior domain is computed on a mesh, introducing approximation errors, while with the reference version of ACTIPOLE, the pressure in this domain is computed using the representation formula (1.30). We notice with additional computation that the difference decreases when the mesh size decreases, providing us with the first validation argument.

Number of measure points 10x10

(-0.5, -0.5, 0.7)

Points defining the part of hyperplane (-0.5, 0.5, 0.7) (0.5, -0.5, 0.7)

Relative difference on the scattered pressure 1.08 %

Flow at rest and uniform properties:

M = M ∞ = 0, ρ = ρ ∞ , c = 2c ∞ ,

comparison with an analytic solution computed by means of Mie series

Suppose that there is no diffracting object, and that M = M ∞ = 0, ρ = ρ ∞ , c = 2c ∞ . In this case, the pressure field is perturbed when crossing Γ ∞ by the difference of the speed of sound between Ω -et Ω + . It is possible to get, for a simple geometry, an approximation of the pressure field in Ω + in the form of a Mie series.

The test case is composed of a ball of radius 1, meshed with tetrahedra and centered at the origin, such that (ρ 0 , c 0 , M 0 ) = (1.2 kg.m -3 , 680 m.s -1 , 0), embedded in the exterior domain where The mesh we used respects the convention of the length of the longest edge being of the order of one tenth of the wavelength of the source for frequencies up to 85 Hz. The solution using Mie series was provided by the company IMACS. We also consider the pressure field in the case c = c ∞ (therefore in the absence of the interior domain Ω -), whose real part is given by cos (kr) 4πr , where r is the distance between the source, are refer to it as pressure "in the absence of ball".

(ρ ∞ , c ∞ , M ∞ ) = (1.2 kg.m -3 , 340 m.s -1 , 0), see
In Figure 4.4, the curve of the pressure computed using formulation (3.36) and the one provided by IMACS are very similar, and these two curves are significantly different from the one correspond to the case in the absence of ball. In Figure 4.5 are represented the difference between the formulation (3.36) and the case in the absence of ball on the one hand, and the difference between the solution computed by IMACS and the case in the absence of ball on the other hand. The relative difference is less than 0.25% for f < 85 Hz and less than 5% up to 500 Hz.

The difference between the solution using the formulation (3.36) and the solution provided by IMACS is much smaller than the difference of each of these solutions with the reference. Therefore, the perturbation induced by the change in the speed of sound computed by our code is validated by comparison with the solution by IMACS. We have verified that this conclusion also holds when the visualization point is located in the interior domain Ω -.

Uniform flow and properties: M

= M ∞ = 0.5, ρ = ρ ∞ , c = c ∞ ,

comparison with ACTIPOLE with only BEM

The test case is the same as in Section 4.1.1, except that there is now an equal nonzero uniform flow in Ω -and Ω + , such that M = M ∞ = 0.5. The Mach number being nonzero, the Prandtl-Glauert transformation is used for both the formulation (3.36) and the reference version of ACTIPOLE.

-the source is a monopole of magnitude 1 and frequency 4054 Hz located at r S = 292.5 mm, -the hot jet stream is modeled by a cylinder of length 2 m and radius r J = 117 mm (theoretically, its length is infinite), the thickness of the shear layer is δ = 0.14 mm.

Introduction

In many problems, such as optimization, uncertainty propagation or real-time simulation, one has to evaluate an objective function for a large number of values of some parameters. Evaluating this objective function often implies solving a parametrized partial differential equation for a given parameter value. In an industrial context, one evaluation of the objective function can already be a challenging numerical problem. To keep reasonable computational costs, various model reduction techniques have been developed to speed up computations. We focus on the Reduced Basis (RB) method [START_REF] Machiels | Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems[END_REF][START_REF] Machiels | Reliable real-time solution of parametrized partial differential equations: Reduced-basis output bound methods[END_REF]. This method has been applied to many kinds of problems, including nonlinear problems such as the viscous Burgers equation [START_REF] Veroy | Reduced-basis approximation of the viscous Burgers equation: rigorous a posteriori error bounds[END_REF] or the steady incompressible Navier-Stokes equations [START_REF] Veroy | Certified real-time solution of the parametrized steady incompressible navier-tokes equations: rigorous reduced-basis a posteriori error bounds[END_REF].

As described in Section 5.2, the RB method consists in replacing the sequence

P ∋ µ Eµ → u µ → Q(u µ ) by the sequence P ∋ µ ʵ → ûµ → Q(û µ ).
Here, P denotes the parameter set, E µ : µ → u µ the model problem, ʵ : µ → ûµ its lower-dimensional approximation, Q(u µ ) the quantity of interest, and Q(û µ ) its RB approximation. More specifically, the RB method consists in two steps: (i) A so-called offline stage, where solutions to E µ for well-chosen values of the parameter µ are computed. During this stage, N problems of size N are solved (with N ≪ N ), and some quantities related to the N solutions are stored, and (ii) a so-called online stage, where the precomputed quantities are used to solve ʵ for many values of µ. In this stage, a certification of the approximation is possible by means of an a posteriori error bound. An important feature in the RB method is the use of an online-efficient a posteriori error bound. The notion of online-efficiency is defined in Section 5.2.4. Moreover, the a posteriori error bound must be as sharp as possible to faithfully represent the error. However, as noticed for example in [86, pp.148-149], standard a posteriori error bounds are subject to round-off errors, especially for the computation of accurate solutions. This difficulty can be encountered in complex industrial applications in the following two cases. First and most importantly, when the stability constant of the underlying bilinear (or sesquilinear) form is very small, the classical formula for the error bound fails to certify the approximation, even at a relatively crude error level, as illustrated in Section 5.5 where the stability constant is about 10 -6 and the classical error bound stagnates at about 10 -4 . Second, in some industrial codes, the single-precision format is used to speed up computations, when high precision is not needed. In this case, the classical formula for the error bound fails to deliver values below 10 -4 for a stability constant of order 1. The purpose of this work is an explanation of these facts and the derivation of a new method to compute the error bound in an accurate and online-efficient way. Additionally, the new formula uses potentially less precomputed quantities than the classical formula.

In Section 5.2, we briefly recall the main ingredients of the RB method, namely (i) the construction of the reduced problem, (ii) the a posterior error bound, (iii) the notion of onlineefficiency, and (iv) the offline stage during which the vectors of the reduced basis are constructed. We then explain in Section 5.3 why the classical formula for computing the error bound is illconditioned in regard of round-off errors. In Section 5.4, we present our new procedure based on the Empirical Interpolation Method (EIM). A version of the EIM stabilized with respect to round-off errors is also derived, and the various procedures to compute the error bound are compared on a simple one-dimensional diffusion problem. In Section 5.5, we apply this new procedure to a three-dimensional acoustic scattering problem.

The reduced basis method

The model problem

We suppose that the problem of interest has the following discrete variational form, depending on a parameter µ in a parameter set P: for a finite-dimensional space V of dimension N (with N ≫ 1 resulting, e.g., from discretization), find u µ ∈ V such that

E µ : a µ (u µ , v) = b(v), ∀v ∈ V, (5.1)
where a µ is an inf-sup stable bounded sesquilinear form on V × V and b is a continuous linear form on V. We work in complex vector spaces in view of our application to acoustic scattering. In what follows, the complex conjugate of z ∈ C is denoted z * . We define the Riesz isomorphism J from V ′ to V such that for all l ∈ V ′ and all u ∈ V, (Jl, u) V = l(u), where (•, •) V denotes the inner product of V (antilinear with respect to its second argument) with associated norm

• V . We denote β µ := inf u∈V sup v∈V |a µ (u, v)| u V v V
> 0 the inf-sup constant of a µ and βµ a computable positive lower bound of β µ . For simplicity, we consider that the linear form b is independent of the parameter µ. The extension to µ-dependent b is straightforward. We refer to the discrete solution u µ as the "truth solution".

The reduced problem

Suppose that a reduced basis, consisting of N solutions u µ i of E µ i , i ∈ {1, ..., N }, has already been constructed. To alleviate the notation, we denote u i the function u µ i . How the parameters µ i are chosen is briefly outlined in Section 5.2.5. Given a parameter value µ ∈ P, the reduced problem is then a Galerkin procedure written on the linear space V = Span{u 1 , ..., u N } ⊂ V: find ûµ ∈ V such that ʵ : a µ (û µ , u j ) = b(u j ), ∀j ∈ {1, ..., N }.

(5.

2)

The approximate solution on the reduced basis is written as

ûµ = N i=1 γ i (µ)u i . ( 5.3)
Recalling the exact and approximate quantities of interest Q(u µ ) and Q(û µ ), respectively, the quality of the approximation for a given µ ∈ P is quantified by the error measure Q(u µ ) -Q(û µ ) . When we obtain a satisfying error measure with N ≪ N , the RB strategy is successful. Two main cases are generally considered: (i) the so-called general-purpose case, where one is interested in the whole solution: Q = Q = Id and • = • V , and (ii) the so-called goaloriented case, where Q is a linear form on V and

• = | • |. The operator Q is consistently built so that Q(u µ ) -Q(û µ ) vanishes for µ = µ i , i ∈ {1, ..., N }.

A posteriori error bound

In the standard RB method, the a posteriori error bound is a residual-based bound. In what follows, we refer to it simply as error bound. Since this error bound is an upper bound, it provides a way to certify the approximation made by the reduced basis.

Proposition 5.1 (General-purpose case)

The following error bound holds: For all µ ∈ P,

u µ -ûµ V ≤ E 1 (µ) := β-1 µ G µ ûµ V , (5.4) with G µ the linear map from V to V such that V ∋ u → G µ u := J (a µ (u, •) -b) ∈ V.
Proof. See [START_REF] Patera | Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations[END_REF]Section 4.3.2]. ♦

In the goal-oriented case, one possible approach is to introduce the following dual problem:

Find v µ ∈ V such that E d µ : a µ (w, v µ ) = Q(w), ∀w ∈ V. (5.5)
We wrote the dual problem on the same discrete space V, but another space can be considered.

A reduced basis procedure is also carried out for the problem E d µ , resulting in an approximation vµ of v µ . The approximate quantity of interest is then defined as

Q(û µ ) := Q(û µ ) -(G µ ûµ , vµ ) V ,
where the second term is the so-called dual-based correction.

Proposition 5.2 (Goal-oriented case)

The following error bound holds: For all µ ∈ P,

Q(u) -Q(û µ ) ≤ E go 1 (µ) := βd µ -1 G µ ûµ V G d µ vµ V , (5.6)
where

G d µ is the linear map from V to V such that V ∋ v → G d µ u := J (a µ (•, v) -Q) ∈ V and βd µ is a computable lower bound of β d µ = inf u∈V sup v∈V |a µ (v, u)| u V v V . Obviously, β d µ = β µ if a µ is Hermitian.
Proof. See [START_REF]Mathematical modelling and numerical simulation in materials science[END_REF]Proposition 23].

♦
In what follows, we mainly focus on the general-purpose case. Extensions to the goaloriented case are straightforward.

Online-efficiency of the RB method

The notion of online-efficiency is central to the RB method. Definition 5. [START_REF] Amiet | The aerodynamic noise of small-perturbation subsonic flows[END_REF] The RB method is said to be online-efficient if in the online stage, (i) the reduced problems can be constructed in complexity independent of N , and (ii) the error bound can be computed in complexity independent of N . Definition 5. [START_REF] Ammar | A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids[END_REF] The sesquilinear form a µ is said to depend on µ in an affine way if there exist d functions α k (µ) : P → C and d µ-independent sesquilinear forms a k bounded on V × V such that

a µ (u, v) = d k=1 α k (µ)a k (u, v), ∀u, v ∈ V. (5.7)
In what follows, we always assume that the affine decomposition (5.7) holds. This decomposition is instrumental to achieve online-efficiency.

Property 5.5 If a µ depends on µ in an affine way, then the RB method is online-efficient.

Proof. (i) The reduced matrix writes ( µ ) j,i = a µ (u i , u j ) and the reduced right-hand side ( B) j = b(u j ), for all 1 ≤ i, j ≤ N . There holds µ = d k=1 α k (µ) Âk , where ( Âk ) ij := a k (u i , u j ). Therefore, provided the d matrices Âk and the vector B are precomputed during the offline stage, the reduced problems are constructed in complexity independent of N .

(ii) The operator G µ inherits the affine dependence of a µ on µ since, for all u ∈ V,

G µ u = -Jb + d k=1 α k (µ)Ja k (u, •) = G 00 + d k=1 α k (µ)G k u, ( 5.8)
where G 00 := -Jb ∈ V and G k u := Ja k (u, •) ∈ V for all k ∈ {1, ..., d}. Using this affine decomposition and recalling (5.3), we infer

E 1 (µ) = β-1 µ G 00 + N i=1 d k=1 α k (µ)γ i (µ)G k u i V .
(5.9)

The scalar product on which the norm in (5.9) hinges can be expanded to provide another formula for the error bound (see [86, eq.(4.61)]):

E 2 (µ) = β-1 µ   (G 00 , G 00 ) V + 2Re N i=1 d k=1 γ i (µ)α k (µ)(G k u i , G 00 ) V + N i,j=1 d k,l=1 γ i (µ)α k (µ)γ * j (µ)α * l (µ)(G k u i , G l u j ) V   1 2 , ( 5.10)
which is computed in complexity independent of N in the online stage provided that (G 00 , G 00 ) V , (G k u i , G 00 ) V and (G k u i , G l u j ) V are precomputed during the offline stage, and provided that a lower bound βµ of the stability constant of a µ is also computed in complexity independent of N (which is possible, for example, by the Successive Constraint Method, see [START_REF] Huynh | A successive constraint linear optimization method for lower bounds of parametric coercivity and inf-sup stability constants[END_REF][START_REF] Chen | Improved successive constraint method based a posteriori error estimate for reduced basis approximation of 2d Maxwell's problem[END_REF]). ♦

An important observation made in [Ar1], and that will be useful below, is that the formula (5.10) defining E 2 can be rewritten in an equivalent way as

E 2 (µ) := β-1 µ δ 2 + 2Re(s t xµ ) + x * t µ S xµ 1 2 , ( 5.11)
where δ := G 00 V , s and xµ are vectors in C d N with components s I := (G k u i , G 00 ) V and (x µ ) I := α k (µ)γ i (µ), and S is a matrix in C d N ,d N with coefficients S I,J := (G k u i , G l u j ) V (with I and J re-indexing respectively (k, i) and (l, j), for all 1 ≤ k, l ≤ d and all 1 ≤ i, j ≤ N ). The t superscript denotes the transposition. The vector s and the matrix S depend on the reduced basis functions {u i } 1≤i≤ N but are independent of µ, and the vector xµ depends on the RB approximation ûµ via the coefficients γ i (µ). Notice that the term between parenthesis on the right-hand side of (5.11) is a multivariate polynomial in xµ of total degree 2. We would like to stress that E 1 (µ) = E 2 (µ) (in infinite precision arithmetic): the indices 1 and 2 are used to denote two different ways to compute the same quantity. In particular, E 1 (µ) is not online efficient, while E 2 (µ) is.

The offline stage

Fix a discrete subset of parameters P trial ⊂ P. In the offline stage, the parameters µ i (from which the reduced basis is constructed) are chosen by a greedy algorithm as elements of P trial . We denote P select the set of these selected parameters; see [START_REF] Patera | Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations[END_REF]Section 3.3] for a presentation of the greedy algorithm. At each step of the algorithm, the new quantities a k (u i , u j ) and b(u j ) are computed and stored, as well as the new components of the vector s and of the matrix S to be used in the formula (5.11) for E 2 . This task, as that of evaluating G 00 , typically requires inverting the stiffness matrix in V by solving, for all k ∈ {1, ..., d} and all i ∈ {1, ..., N }, the variational problem: find w i,k ∈ V such that

E Gi,k : (w i,k , v) V = a k (u i , v), ∀v ∈ V.
(5.12)

Then, G k u i = w i,k can be computed. The computation of (G k u i , G l u j ) V follows from the solutions of E Gi,k and E Gj,l . Since the error bounds are evaluated using the formula E 2 (µ), for all µ ∈ P trial , with the current state of the reduced basis, finding the maximum of the error bound on P trial is of complexity independent of N . This allows one to consider very large sets P trial without increasing too much the complexity of the whole offline procedure.

Round-off errors and online certification

In this section, we explain why the online-efficient error bound (5.11) may be sensitive to round-off errors.

Elements of floating-point arithmetic

In a computer, real numbers are represented by a finite number of bits, called floatingpoint representation. Current architectures are optimized for a format used by a large majority of softwares: IEEE 754 double-precision binary floating-point format. Let x be a real number. The floating point representation of x is denoted by f l(x). When a (nonzero) real number is rounded to the closest floating-point number, the relative error on its floating-point representation is bounded by a number, ǫ, called the machine precision. In double precision, ǫ = 5 × 10 -16 (see [START_REF] Goldberg | What every computer scientist should know about floating point arithmetic[END_REF]Section 1.2]). Let x and y be real numbers. When computing the operation x + y, the result returned by the computer can be different from its theoretical value. Whenever the difference is substantial, a loss of significance occurs. A well-known case of loss of significance is when x and y are almost opposite numbers. Suppose that x = -y. We denote by maxfl(x + y) the result that the computer returns when the maximal accumulation of round-off errors occurs when computing the summation. There holds

|maxfl(x + y)| ≈ 2ǫ|x|.
(5.13)

When implementing an algorithm, one should ensure that each step is free of such a loss of significance. In some cases, simply changing the order of the operations can prevent these situations. As an illustration, consider x = 1, y = 1 + 10 -7 , and the operation x 2 -2xy + y 2 . This is a sum of terms where the first intermediate result in the sum is 14 orders larger than the result. Therefore, a loss of significance is expected. The relative error of this computation is about 8 × 10 -4 . Computing (xy) 2 , which is the factorization of the considered operation, leads to a relative error of about 10 -9 . Thus, the terms of the sum are only 7 orders larger than the results, leading to a less catastrophic loss of significance. In this specific case, the remedy consists in carrying out the sum before the multiplication. In the RB context, the evaluation of the formula E 2 suffers from such a loss of significance, as we now explain.

Validity of the formulae E 1 and E 2 for computing the error bound

Consider the two formulae E 1 , see (5.9), and E 2 , see (5.11), for computing the error bound. Definition 5. [START_REF] Barrault | An 'empirical interpolation' method: application to efficient reduced-basis discretization of partial differential equations[END_REF] The formula E k , k = 1, 2, is said to be valid for computing the error bound with tolerance tol if max

µ∈P select (E k (µ)) ≤ tol. (5.14)
From a theoretical viewpoint, the error u µ -ûµ V and the residual G µ u µ vanish for all µ ∈ P select . Hence, any formula for computing the residual-based error bound vanishes as well and therefore is valid with any tolerance. However, the validity of a formula for computing the error bound is to be considered in the presence of some adverse phenomenon introducing errors in the computation, see Figure 5.1. The greedy algorithm in the offline stage stops when max µ∈P trial (E k (µ)) < tol RB , where tol RB denotes the maximum acceptable error made by the RB approximation. Therefore, if the minimum tolerance for which an error bound E k is valid is larger than tol RB , then the greedy algorithm cannot converge and will keep increasing the set P select although the error can be actually very small. We examine the validity of the formulae E 1 and E 2 for computing the error bound in the presence of two independent phenomena: round-off errors and approximate reduced basis functions u i (in the context of inexact linear algebra solvers for E µ i ).

Round-off errors

We investigate the influence of round-off errors when computing the error bounds E 1 (µ) and E 2 (µ). As observed at the end of Section 5.3.1, the computation of a polynomial using a factorized form is more accurate than using the developed form, in particular at points close to its roots. Here, βµ E 2 (µ)

2 is a multivariate polynomial of degree 2 in xµ computed in a developed form, whereas the scalar product (G µ u µ , G µ u µ ) V used in the computation of E 1 (µ) is not developed.

In this section, we neglect the round-off errors introduced when solving E µ and ʵ , so that the reduced basis functions u i and the reduced solutions ûµ are considered free of round-off errors. We also suppose that the computable positive lower bound βµ of the inf-sup constant is computed free of round-off errors, see Remark 5.9. Proposition 5.7 Let µ ∈ P select and let maxfl( βµ E k (µ)), k = 1, 2, denote the evaluation of βµ E k (µ) when the maximum accumulation of round-off errors occurs. There holds

maxfl( βµ E 1 (µ)) ≥ 2δǫ, maxfl( βµ E 2 (µ)) ≥ 2δ √ ǫ, ( 5.15)
where δ = G 00 V and ǫ is the machine precision.

Proof. Let µ ∈ P select . We present the proof for E 1 (µ); the proof for E 2 (µ) is similar. We need to evaluate the right-hand side of (5.9). Let (ϕ ρ ) 1≤ρ≤N denote the basis of V, so that, for instance, G 00 = N ρ=1 (G 00 ) ρ ϕ ρ . In exact arithmetics, there holds E 1 (µ) = 0, so that

N i=1 d k=1 γ i (µ)α k (µ) (G k u i ) ρ = -(G 00 ) ρ for all 1 ≤ ρ ≤ N .
As a result, using (5.13), we obtain

maxfl   (G 00 ) ρ + N i=1 d k=1 γ i (µ)α k (µ)(G k u i ) ρ   ≈ 2|(G 00 ) ρ |ǫ.
Since computing the V-norm on the right-hand side of (5.9) can only increase the round-off errors, we infer the desired lower bound. where βmin = inf µ∈P select ( βµ ).

♦

Remark 5.9 (Inf-sup constant)

The computable positive lower bound βµ of the inf-sup constant suffers from round-off errors as well. However, since it is a multiplicative factor, the quality of its computation does not severely affect the quality of the error bound. Moreover, the value of the inf-sup constant does not depend on the size of the reduced basis, contrary to G µ ûµ V . Therefore, there is no phenomenon susceptible to degrade the accuracy of its computation with the increase of the size of the reduced basis. If the Successive Constraint Method is used, the procedure to compute βµ is carried out before the greedy algorithm of the RB method.

Remark 5.10 (Improved floating-point arithmetic)

Increasing the machine precision from ǫ to ǫ 2 (quadruple-precision) for computing the coefficients in (5.11), as well as for evaluating the multivariate polynomial in xµ , is a first solution to recover a good precision with the formula E 2 . Moreover, since current architectures are optimized for the double-precision format, changing the floating-point arithmetic can potentially degrade software performance. There are also methods allowing one to double the precision of the evaluation of a polynomial while keeping the doubleprecision format, namely compensated schemes. For instance, the compensated Horner scheme in double-precision [START_REF] Langlois | Compensated Horner Scheme[END_REF] doubles the precision and is faster than the full quadruple precision implementation. However, this requires to representing the result of the intermediate operations by two doubles, one for the value in double-precision and another one for the subsequent digits. These strategies are equivalent to quadruple precision (except for the computational savings in evaluating the error bound).

Remark 5.11 (Goal-oriented case, round-off errors)

The same analysis can be carriedout in the goal-oriented case. Let µ ∈ P select . There holds

maxfl( βd µ E go 1 (µ)) ≥ 2δςǫ 2 , maxfl( βd µ E go 2 (µ)) ≥ 2δςǫ, (5.17)
where ς := Q V ′ . We indeed observe in our simulations that the round-off errors on E go 1 scale like ǫ 2 , while the round-off errors on E go 2 scale like ǫ (see Section 5.5). If we suppose that the lower bounds are reached in (5.17), then the formulae E go 1 and E go 2 are valid for computing the error bound with tolerance tol if, respectively,

for E go 1 , 2 βd min -1 δςǫ 2 ≤ tol, for E go 2 , 2 βd min -1 δςǫ ≤ tol, (5.18)
where βd min = inf µ∈P select ( βd µ ).

Approximate reduced basis functions

In large-scale simulations, the accuracy of the RB procedure is also limited by the numerical method used for computing the reduced basis functions. We want here to illustrate this fact on a simple example where we suppose that the approximation of the reduced basis functions comes from an iterative solver with prescribed stopping criterion. We recall that for a given value µ ∈ P select , E µ consists in solving a linear system of size N of the form A µ U µ = B. Thus, for µ ∈ P trial , the formulae E 1 and E 2 for the error bound are based on the computation of the residual of E µ for the reduced solution ûµ . Indeed, it is easy to see that

G µ ûµ V = A µ Ûµ -B * V ′ , where for all Φ ∈ C N , Φ * V ′ = sup V ∈C N |(V,Φ) C N | N i=1 V i ϕ i V , recalling that (ϕ ρ ) 1≤ρ≤N are the basis func- tions in V, see [42, §9.1.5].
In this section, we suppose that the formulae E 1 and E 2 are free of round-off errors (therefore, for all µ ∈ P trial , E 1 (µ) = E 2 (µ)), but the problem E µ is not solved exactly, leading to approximate reduced basis functions such that the residuals do not vanish. Hence, for all µ ∈ P select , E 1 (µ) = E 2 (µ) and these error bounds are nonzero owing to inexact linear algebra solves. The reduced problems ʵ are supposed to be solved freely of round-off errors. Proposition 5.12 (Approximate reduced basis functions) If the reduced basis functions are computed using an iterative solver with the following stopping criterion on the normalized residual:

∀µ ∈ P trial , A µ U µ -B * V ′ B * V ′ ≤ ξ, ( 5
• * V norm, B * V ′ = sup V ∈C N b( N i=1 V i ϕ i ) N i=1 V i ϕ i V = b V ′ = G 00 V = δ. Then, G µ ûµ V = sup v∈V (Gµ ûµ,v)V v V = sup v∈V aµ(ûµ,v)-b(v) v V = sup V ∈C N (V,Aµ Ûµ-B) C N N i=1 V i ϕ i V = A µ Ûµ -B * V ′ . Therefore, E k (µ) = β-1 µ G µ ûµ V = β-1 µ A µ Ûµ -B * V ′ = β-1 µ A µ U µ -B * V ′ ≤ β-1 µ B * V ′ ξ = β-1 µ δξ ≤ β-

Synthesis

Taking into account the round-off errors in the computation of the error bound and the stopping criterion of an iterative solver, and supposing that the bounds (5.15) and (5.17) δγ max ξ 2 , ǫ ≤ tol.

(5.22)

Focusing on round-off errors, the formula E 1 for computing the error bound is valid for tolerances scaling as ǫ, but is not online-efficient, whereas the formula E 2 is online-efficient but is valid only for (significantly) higher tolerances, namely tolerances scaling as √ ǫ.

New procedures for accurate and online-efficient evaluation of the error bound

In this section, online-efficient methods, that are valid for tolerances scaling as ǫ, are devised to evaluate the error bound.

Procedure 1: rewriting E 2

We first present the procedure proposed in [Ar1]. We consider that a reduced basis of size N has been constructed. Let σ := 1 + 2d N + (d N ) 2 . For a given µ ∈ P trial and the resulting ûµ ∈ Span{u 1 , ..., u N } solving the reduced problem, we define X(µ) ∈ C σ as the vector with components (1, xµ I , x * µ I , x * µ I xµ J ), where xµ I = α k (µ)γ i (µ) (we recall that γ i (µ) are the coefficients of the reduced solution in the reduced basis, see (5.3), and α k (µ) the coefficients of the affine decomposition of a µ in (5.7)), with 1 ≤ I, J ≤ d N (with

I = i + N (k -1) such that 1 ≤ i ≤ N , 1 ≤ k ≤ d, and with J = j + N (l -1) such that 1 ≤ j ≤ N , 1 ≤ l ≤ d).
We can write the right-hand side of (5.11) as a linear form in X(µ) as follows:

δ 2 + 2Re(s t xµ ) + x * t µ S xµ = σ p=1 t p Xp (µ), (5.23)
where t p is independent of µ (as δ, s, and S are independent of µ) and Xp (µ) is the p-th component of X(µ). Now, in the offline stage, we take σ values (e.g. random values) µ r ∈ P trial , r ∈ {1, ..., σ}, of the parameter µ. Then, we compute the vectors X(µ r ) and the quantities

V r := σ p=1 t p Xp (µ r ).
(

Finally, we define T ∈ C σ×σ as the matrix whose columns are formed by the vectors X(µ r ), that is, T pr = Xp (µ r ) for all 1 ≤ p, r ≤ σ. We assume that T is invertible, which always happens to be the case in our simulations. Now, suppose that in the online stage we want to evaluate the error bound for the RB solution ûµ computed at a certain parameter µ ∈ P trial . Then, we evaluate the vector X(µ) and solve the linear system T λ(µ) = X(µ), (5.25) yielding λ(µ) ∈ C σ . We then obtain X(µ) = σ r=1 λ r (µ) X(µ r ) and

σ p=1 t p Xp (µ) = σ p,r=1 t p λ r (µ) Xp (µ r ) = σ r=1 λ r (µ)V r . (5.26)
This yields the following new formula for computing the error bound:

E 3 (µ) := β-1 µ σ r=1 λ r (µ)V r 1 2 , ( 5.27)
where the quantities V r = G µr ûµr 2 V can be precomputed. Thus, computing E 3 requires solving (5.25) and summing the σ precomputed quantities V r . Since the complexity of this procedure is independent of N , the formula E 3 is online-efficient for computing the error bound. Notice that in the linear combination in (5.27), the V r are expected to have the same magnitude as the result of the linear combination, preventing this formula from the loss of significance observed for E 1 .

Remark 5.14 (Goal-oriented case) For the goal-oriented case, the procedure is carried out independently on the two multivariate polynomials

G µ ûµ 2 V and G d µ vµ 2 V .
Notice that E 1 (µ), E 2 (µ), and E 3 (µ) are equal in exact arithmetic. As pointed out in [Ar1], the matrix T exhibits in practice large condition numbers, and there is no guarantee that T is actually invertible. We will see in Section 5.5 for a three-dimensional acoustic scattering problem that E 3 can be in practice as ill-behaved as E 2 . Moreover, there is no a priori method for selecting the parameters µ r for which the quantities V r are precomputed. In the next section, we propose a new procedure that solves these problems.

Procedure 2: improvement on Procedure 1 using the EIM

In the formula E 3 , a potentially ill-conditioned problem T λ(µ) = X(µ) is solved in order to exactly represent X(µ) by the linear combination σ r=1 λ r (µ) X(µ r ). Following a suggestion by Patera [84], we propose to approximate X(µ) by means of an interpolation procedure. We want to modify the formula E 3 by an interpolation formula relying on a better conditioned linear system. The price to pay is that the new formula E 4 will not be equal to E 1 in exact arithmetic; the interpolation errors are however marginal, as further discussed in Remark 5.20. We also look for a way to choose the parameters µ r for which the quantities V r have to be precomputed. We refer to these values for µ r as "interpolation points", and to the set of these points as P inter .

Consider the function of two variables (p, µ) → Xp (µ), for all p ∈ {1, ..., σ} and all µ ∈ P trial . We look for an approximation of this function in the form ∀µ ∈ P trial , ∀p ∈ {1, ..., σ}, Xp (µ) ≈ σ r=1 λ σ r (µ) Xp (µ r ), (5.28) for a certain parameter σ ≤ σ. The empirical interpolation method (EIM) (more precisely the discrete EIM since p is a discrete variable) provides a numerical procedure to construct this approximation and to choose the interpolation points (see [START_REF] Barrault | An 'empirical interpolation' method: application to efficient reduced-basis discretization of partial differential equations[END_REF][START_REF] Maday | A general multipurpose interpolation procedure: the magic points[END_REF]).

The EIM is an offline-online procedure. During the offline stage, σ basis functions are computed, denoted q j : P trial ∋ µ → q j (µ) ∈ C, for all j ∈ {1, ..., σ}. These basis functions will be used in the online stage to carry out the interpolation. We define q σ as the vector-valued map P trial ∋ µ → q σ(µ) := (q j (µ)) 1≤j≤σ ∈ C σ. During the offline stage, σ interpolation points µ r ∈ P trial are also selected; these points are collected in the set P inter . Notice that P select , the set of parameter values selected by the greedy algorithm of the RB method, is different from P inter . During the online stage, the matrix B σ ∈ C σ,σ , where B σ ij = q i (µ j ), for 1 ≤ i, j ≤ σ, is constructed. Letting µ ∈ P trial , we solve for λ σ(µ) ∈ C σ such that B σλ σ(µ) = q σ(µ), (5.29) and compute the rank-σ interpolation operators defined as follows.

Definition 5.15 Let 1 ≤ k ≤ σ. The rank-k interpolation operator I k is defined such that

I k X(µ) := k r=1 λ k r (µ) X(µ r ), (5.30
)

where λ k (µ) ∈ C k solves B k λ k (µ) = q k (µ).
(5.31) Equation (5.30) defines an interpolation in the sense that I k Xpr (µ) = Xpr (µ) for all 1 ≤ r ≤ k and all µ ∈ P trial . The formula Xp (µ) ≈ (I σ X) p (µ), for all µ ∈ P trial and all p ∈ {1, ..., σ}, provides the approximate interpolation formula searched for in (5.28). Definition 5. [START_REF] Björck | Loss and recapture of orthogonality in the modified Gram-Schmidt algorithm[END_REF] The residual operator δ σ is defined by

δ σ := Id -I σ.
(5.32)

Algorithm 2 presents the construction of the function q σ by a greedy algorithm during the offline stage. This EIM algorithm is a variant from the classical one, described in [START_REF] Maday | A general multipurpose interpolation procedure: the magic points[END_REF]. The differences stand in the definition of the interpolation operator (5.29), the linear system (5.31) to solve during the online calls, and the definition of the B k matrix. In particular, the present variant leads to the approximation (5.30), which is nonintrusive in the sens that I k X(µ) is obtained as a linear combination of evaluations of X at some parameter values µ r . The classical EIM can recover such a property, but to the price of an additional change of basis between q k (•) and Xp k (•). However, contrary to the classical EIM, the variant needs the additional change of basis to be able to compute an approximation between learning points, namely for µ ∈ P trial \P. We refer to Section 7.2 for more details about the differences between the EIM variant considered here and the classical algorithm. Compute p k+1 := argmax p∈{1,...,σ}

(δ k X)p(•) ℓ ∞ (P trial ) 9.
Compute µ k+1 := argmax

µ∈P trial |(δ k X)p k+1 (µ)| [(k + 1)-th interpolation point] 10.
Set Pinter := Pinter ∪ {µ k+1 } [Update of Pinter]

11.

Set

q k+1 (•) := (δ k X)p k+1 (•) (δ k X)p k+1 (µ k+1 ) [(k + 1)-th basis function] 12. B k+1 ij := qi(µj), 1 ≤ i, j ≤ k + 1 [(k + 1)-th B matrix] 13.
k ← k + 1 [Increment the size of the interpolation]

14. end while Definition 5.17 The new formula for computing the error bound is

E 4 (µ) := β-1 µ σ r=1 λ σ r (µ)V r 1 2 , ( 5.33)
where λ σ(µ) is the solution to (5.29). We recall that V r = G µr ûµr 2 V .

Proposition 5. [START_REF] Boyaval | Reduced basis techniques for stochastic problems[END_REF] The computation of the formula E 4 is well defined, and this formula is onlineefficient.

Proof. Owing to [73, Theorem 1], the matrix B is upper triangular with diagonal unity. Hence, det B = 1 and B is guaranteed to be invertible. The online procedure of EIM, consisting in solving a linear system defined by the matrix B, is thus well defined. Then, since the EIM procedure in carried out on Xp (µ), for all p ∈ {1, ..., σ} and all µ ∈ P trial , all the computations involved are of complexity independent of N , even the offline part of the EIM. Finally, the complexity of the online part of EIM only depends on σ. ♦

Remark 5.19 (Stopping criterion in Algorithm 2)

For ease of presentation, we chose a simple stopping criterion based on an a priori fixed maximum number of interpolation points. In practice, one possibility is to stop the algorithm when the maximal approximation error in the EIM is below a prescribed value, by monitoring the quantity (δ k X) p k+1 (µ k+1 ).

Remark 5.20 (Interpolation errors)

As already observed, E 4 does not equal E 1 in exact arithmetics owing to interpolation errors (when σ < σ). Thus, although Algorithm 2 yields an accurate approximation of Xp (µ), a given interpolation error on Xp (µ) does not directly translate into a bound on the difference between E 1 (µ) and E 4 (µ) (the latter depending also on δ, s, and S, as well as on βµ ). We observe in our numerical experiments that these latter errors are lower than the errors incurred in the evaluation of E 2 (due to round-off errors) and in the evaluation of E 3 (due to the poor conditioning of T ).

Remark 5.21 (Non affine dependence)

When the affine dependence assumption is not available (see Definition 5.4), one can look for an approximation of a µ in the following form:

a µ (u, v) ≈ d k=1 α k (µ)a k (u, v), ∀u, v ∈ V.
(5.34)

In the reduced basis context, this approximation is usually computed using the EIM. We saw that the formula (5.10) for E 2 makes use of this affine decomposition to ensure online efficiency, and therefore does not account for the approximation in the operator. On the contrary, the formulae (5.4) for E 1 and (5.27) for E 3 use the exact operator.

Illustration

Consider as in [Ar1] a one-dimensional linear diffusion problem, namely the boundary value problem -u ′′ + µu = 1 on ]0, 1[ with u(0) = u(1) = 0, with parameter µ ∈ P := [1, 100]. The analytic solution is

u(x) = - 1 µ (cosh ( √ µx) -1) + cosh √ µ -1 µ sinh √ µ sinh ( √ µx) .
(5.35)

The Lax-Milgram theory is valid, and the coercivity constant is bounded from below by 1 in the H 1 -norm. The error bound is given by E 1 (µ) = G µ ûµ H 1 (]0,1[) . Lagrange P 1 finite elements are used with uniform mesh cells of length 0.005. The set P trial consists of 1000 points uniformly distributed in P. The RB method is carried out until the formula E 2 suffers from round-off errors, which already happens for a reduced basis of size N = 7 (since d = 2, we obtain σ = 225). A direct solver is used, so that the only adverse phenomenon to compute the error bound are round-off errors.

In Figure 5.2, we see that the classical formula E 2 is not valid for computing the error bound with any tolerance below 10 -7 , whereas the formulae E 1 , E 3 and E 4 are valid with tolerances down to 10 -14 . The difference is of 7 orders of magnitude ; given that √ ǫ ≈ 10 -7 , this is consistent with Remark 5.8 and Section 5.4.1. In Figure 5.3, we observe that instabilities occur in the formula E 3 , especially for parameter values close to the elements of P select . This is due to the poor conditioning of the matrix T when solving (5.25). The new formula E 4 based on the EIM is seen to introduce much less numerical errors than E 3 .

Procedure 3: improvement of Procedure 2 using a stabilized EIM

In practice, round-off errors are accumulated during the loop in Algorithm 2, and if we keep increasing the number of interpolation points, the coefficients of the matrix B suffer from roundoff errors, so that the relation det(B) = 1 no longer holds. Even worse, the matrix B becomes non invertible at some stage. To solve this problem, we now propose a numerical stabilization of EIM based on the following property: Property 5.22 There holds ∀i < j, I j • I i = I i , (5.36) where the interpolation operators I j are defined by (5.30). Proof. Using [73, Lemma 1], I i X ∈ Span (q 1 , ..., q i ) and

I i v = v for all v ∈ Span (q 1 , ..., q i ).
Therefore, I j • I i X = I i X for all i < j. ♦

In our numerical experiments, we observe that, as the number of iterations of the greedy procedure for the EIM grows, the relation (5.36) is no longer verified numerically, due to accumulation of round-off errors. These numerical instabilities can be compensated in the same fashion as the Gram-Schmidt orthonormalization procedure is stabilized (see [50, chapter 5.2.8]). The Gram-Schmidt algorithm transforms a linearly independent family of vectors {v i } into an orthonormal basis {u i }. To simplify the presentation, we suppose in what follows that the normalization step is not carried out. Consider the orthogonalization step for the k-th vector. We denote by Π k the projection operator on Span(u 1 , ..., u k ), and δ k := Id -Π k . For the EIM, we suppose that (k -1) interpolation operators I i , 1 ≤ i ≤ k -1, have been constructed, and we wish to construct the k-th interpolation operator I k . A comparison between the stabilized Gram-Schmidt orthonormalization procedure and the proposed stabilization for the EIM is presented in Table 5.1.

Proposition 5.23

Let k ∈ N * . In exact arithmetic, the following relations hold for the residuals defined in Table 5.1:

δ k stab v = δ k v.
Proof. We prove by recursion that, for all i ≤ k, δ k,i stab = δ i . The case i = 1 is clear from the definition of the first intermediate residual in Table 5.1. Let i ≤ k and suppose that δ k,i-1 stab = Id -I i-1 for the EIM. There holds

δ k,i stab = δ k,i-1 stab -I i • δ k,i-1 stab = Id -I i-1 -I i + I i • I i-1 = Id -I i = δ i , (5.37) since I i • I i-1 = I i-1
owing to Property 5.22. The results follow from the case i = k. The same relation is proved likewise for the Gram-Schmidt procedure, for which

Π i • Π i-1 = Π i-1 holds as well. ♦ stabilized Gram-Schmidt stabilized EIM global input (v1, ..., vσ) basis of C σ v : Ptrial → C σ classical residual at step k δ k v k = v k -Π k v k (δ k v)(µ) = v(µ) -(I k v)(µ) intermediate residuals at step k δ k,1 stab v k = v k -Π 1 v k (δ k,1 stab v)(µ) = v(µ) -(I 1 v)(µ) δ k,2 stab v k = δ k,1 stab v k -Π 2 δ k,1 stab v k , (δ k,2 stab v)(µ) = (δ k,1 stab v)(µ) -I 2 (δ k,1 stab v)(µ), . . . . . . δ k,k stab v k = δ k,k-1 stab v k -Π k δ k,k-1 stab v k (δ k,k stab v)(µ) = (δ k,k-1 stab v)(µ) -I k (δ k,k-1 stab v)(µ) stabilized residual at step k δ k stab v k = δ k,k stab v k (δ k stab v)(µ) = (δ k,k stab v)(µ) global output (δ 1 stab v1, δ

Definition 5.24 (Stabilized EIM)

The stabilized EIM consists in the same offline procedure as the one described in Section 5.4.2, except that the residuals δ k are replaced by the stabilized residuals δ k stab defined in Table 5.1. The online stage is the same as that of the classical EIM.

The stabilized Gram-Schmidt procedure generates a set of vectors much less polluted by round-off errors (see [START_REF] Björck | Loss and recapture of orthogonality in the modified Gram-Schmidt algorithm[END_REF][START_REF] Giraud | When modified Gram-Schmidt generates a well-conditioned set of vectors[END_REF]). By analogy, we expect that the stabilized EIM produces a more accurate interpolation procedure than the classical EIM, that is, much less polluted by round-off errors. This is numerically verified in Figure 5.4, where det(B σ) and cond(B σ) are represented as a function of σ. We consider the test case described in Section 5.4.3, where we recall that N = 7, d = 2, and σ = 225. If the method is stable, then det(B σ) = 1 should hold throughout the process. Figure 5.4 shows that the stabilized EIM behaves as intended. The classical EIM curve stops since the matrix B σ becomes noninvertible at some point: a parameter already in P inter has been selected by the greedy algorithm. Invertibility can be recovered artificially by ensuring that the new interpolation point is not an element of the current set P inter . We call this procedure EIM with unique choice. However, this fix is not completely satisfactory, since det(B σ) = 1 is not satisfied. Moreover, cond(B σ) is much more ill-behaved with this procedure than with the stabilized EIM.

Remark 5.25 (Computational cost and variant of stabilized EIM) The computational cost of the stabilized EIM is more than that of the classical EIM, since the stabilized residual requires as many calls to a classical residual as the number of selected interpolation points (i.e. the scaling with σ is σ2 for the stabilized EIM as opposed to σ for the classical EIM). One can think of a cheaper procedure by monitoring det(B σ) and adding some intermediate residuals δ k,j

stab until det(B σ) is close enough to 1.

Summary

The advantages and drawbacks of the four considered formulae for computing the error bound are summarized in Table 5.2. To estimate the computational complexity of the methods, we keep only the leading order in operation count. We denote the complexity of the resolution of (5.12) by N sol . The linear systems of size σ, σ, and N are supposed to be solved by a direct solver, hence with complexity proportional to σ 3 , σ3 , and N 3 , respectively. For the offline stage of E 2 and E 3 , we have to evaluate respectively d N + 1 and σ times the functional G µ , which requires to solve (5.12). For the offline stage of E 4 , let M denote the cardinality of P trial . The k-loop in Algorithm 2 requires at each step to compute a maximum over σ different ℓ ∞ (P trial ) norms, and then to solve a linear system of size k, leading to a complexity of σ4 σM + σN sol . If the stabilized EIM is used instead for E 4 , each residual evaluation in the k-loop requires solving k linear systems of size 1 to k, leading to a complexity of σ5 σM + σN sol . For the online stage, all the formulae require to solve the problem ʵ of size N . Moreover, E 2 additionally requires a linear combination of size σ, whereas E 3 and E 4 require to solve a linear system of size σ and σ respectively. We notice that if N sol ≫ σ4 σM and σ < d N + 1, then the offline stage of E 4 with stabilized EIM requires less precomputations than the offline stage of E 2 .

Property

Application to a three-dimensional acoustic scattering problem

Formulation of the problem

We refer to Section 2.2 for more details. We consider a ball Ω -⊂ R 3 with boundary Γ and Ω + := R 3 \Ω -, see Figure 5.5. We consider a monopole source located in Ω + . The surface of the ball is impedant, meaning that any incident wave will be partially absorbed and partially scattered. The proportion of absorbed and scattered parts is quantified by the impedance coefficient µ, which is used in a Robin boundary condition at Γ . We are interested in the computation of the scattered field p sc in Ω + . We denote p inc the known pressure field created by the source in the absence of the sphere; the total acoustic field in Ω + is the sum of p inc and p sc .

   N -ik 2µ I D D -S -iµ 2k I       χ λ    =    γ - 1 p inc -γ - 0 p inc    , (5.39)
where k is the wave number of the monopole source, N , D, D and S are classical boundary integral operators (see [START_REF] Sauter | Boundary Element Methods[END_REF]), and γ - 0 p inc and γ - 1 p inc are respectively the interior Dirichlet and Neumann traces of the known function p inc . The software we are using, ACTIPOLE (see [START_REF] Delnevo | Code acti3s, justifications mathématiques, Partie II : presence d'un écoulement uniforme[END_REF][START_REF] Delnevo | Code acti3s harmonique, justification mathématique, Partie I[END_REF]), deals with the block system defined in (5.39), which presents the advantage of being invertible for all frequencies of the source when the surface Γ is Lipschitz, see Thoerem 2.2. We denote A µ the block operator defined by the left-hand side of (5.39). From [START_REF] Hsiao | Boundary Element Methods: Foundation and Error Analysis[END_REF][START_REF] Mclean | Strongly elliptic systems and boundary integral equations[END_REF][START_REF] Sauter | Boundary Element Methods[END_REF], we infer that A µ is a bounded bijective operator from

H 1 2 (Γ ) × L 2 (Γ ) into H -1 2 (Γ ) × L 2 (Γ ). The variational form is as follows: find (χ, λ) ∈ H 1 2 (Γ ) × L 2 (Γ ) such that for all ( χ, λ) ∈ H 1 2 (Γ ) × L 2 (Γ ),        N χ - ik 2µ χ, χ + Dλ, χ = (γ 1 p inc , χ) , λ, Dχ -λ, Sλ + iµ 2k λ = -λ, γ 0 p inc , (5.40)
where (•, •) denotes the H

1 2 (Γ ) × H -1 2 (Γ ) duality product and •, • denotes the L 2 (Γ ) inner product.
Let M be a shape-regular triangular mesh of Γ with meshsize h, and let V 1 h and V 0 h be respectively the spaces spanned by continuous piecewise affine polynomials on M and piecewise constant polynomials on M. Let (φ i ) 1≤i≤P and (ψ j ) 1≤j≤P ′ be the usual bases of V 1 h and V 0 h of size P and P ′ , respectively. The product space

V 1 h × V 0 h is a conforming approximation of H 1 2 (Γ )×L 2 (Γ ).
The discrete problem is derived from a Galerkin procedure on V 1 h ×V 0 h using the boundary element method (BEM). The obtained discrete approximation of the problem (7.42) is inf-sup stable for h small enough, see Proposition 2.4. A direct solver is used, in double-precision format.

Application of the RB method

The RB method has recently been applied to problems solved by means of integral equations in electromagnetism, see [START_REF] Fares | The reduced basis method for the electric field integral equation[END_REF][START_REF] Chen | Certified reduced basis method for electromagnetic scattering and radar cross section estimation[END_REF]. In these works, the classical a posteriori error bounds were used. We are here interested in the application of our improved a posteriori error bounds to such problems. We take as parameter for the RB method the value of the impedance µ, which is supposed here to be a positive real number. To recover an affine dependence on the parameter µ, we write the BEM matrix in the form A µ = a 1 (µ)A 1 + a 2 (µ)A 2 + a 3 (µ)A 3 , so that d = 3 in the affine decomposition (5.7) with a 1 (µ) = 1, a 2 (µ) = 1 µ and a 3 (µ) = µ. Specifically,

A 1 =          (N φ i , φ j ) 1 ≤ i ≤ P 1 ≤ j ≤ P Dψ j , φ i 1 ≤ i ≤ P 1 ≤ j ≤ P ′ Dφ j , ψ i 1 ≤ i ≤ P ′ 1 ≤ j ≤ P -Sψ i , ψ j 1 ≤ i ≤ P ′ 1 ≤ j ≤ P ′          , ( 5
.41)

A 2 =          -ik 2 (φ i , φ j ) 1 ≤ i ≤ P 1 ≤ j ≤ P (0) 1 ≤ i ≤ P 1 ≤ j ≤ P ′ (0) 1 ≤ i ≤ P ′ 1 ≤ j ≤ P (0) 1 ≤ i ≤ P ′ 1 ≤ j ≤ P ′          , A 3 =          (0) 1 ≤ i ≤ P 1 ≤ j ≤ P (0) 1 ≤ i ≤ P 1 ≤ j ≤ P ′ (0) 1 ≤ i ≤ P ′ 1 ≤ j ≤ P ′ -i 2k ψ i , ψ j 1 ≤ i ≤ P ′ 1 ≤ j ≤ P ′          .
(5.42)

In the general-purpose RB, the quantity of interest is the pair of potentials (χ, λ) on Γ . For the goal-oriented case, we consider the value of the pressure at a given point in Ω + . If this point is far enough from Γ , approximations can be made in the representation formula for the pressure. This is the far-field approximation, which consists in a linear form Q acting on the solution pair (χ, λ) as

Q(χ, λ) =       -ik e -ik x 2 4π x 2 e -iky• x x 2 x x 2 • n(y), χ(y) ik e -ik x 2 4π x 2 Γ e -iky• x x 2 , λ(y)       ∈ C 2 .
(5.43)

For simplicity, we take the Euclidian norm of vectors in C P +P ′ instead of the H 1 2 (Γ ) × L 2 (Γ ) norms of the reconstructed functions in (5.9) and (5.10). This way, the Riesz isomorphism J is simply the identity. Therefore, the computation of the terms G µ u µ , as well as that of the terms G k u i , does not require to invert the stiffness matrix as in (5.12). The Successive Constraint Method (see Section 5

Error bound curves

We present the error bound curves for test case (i) with a general-purpose RB, #P trial = 100, ( N , σ, σ) = (2, 7, 49), [START_REF] Amiet | The aerodynamic noise of small-perturbation subsonic flows[END_REF][START_REF] Beldi | Some new results for the study of acoustic radiation within a uniform subsonic flow using boundary integral method[END_REF][START_REF] Tartar | An Introduction to Sobolev Spaces and Interpolation Spaces[END_REF], [START_REF] Ammar | A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids[END_REF][START_REF] Brakhage | Über das Dirichletsche Außenraum Problem für die Helmholtzsche Schwingungsgleichung[END_REF]169), and (5, 30, 256) in Figure 5.8 and for test case (ii) with a goal-oriented RB, #P trial = 225, N = 8, σ = 60, and σ = 1681 in Figure 5.9.

In test case (i), the classical formula E 2 exhibits quite poor performances, since it cannot compute values below 10 -4 . This is explained by the values of the inf-sup constant which are around 10 -6 . Furthermore, in agreement with Remark 5.8, the lowest computable values of E 1 and E 2 differ by 8 orders of magnitude. In test case (ii), the behavior of formula E 3 is quite poor, and we do not observe the level of accuracy we observed so far for E 3 . Here, the matrix T defined in (5.25) is so ill-conditioned that the numerical errors introduced by its resolution are larger than the ones introduced by the formula E 2 . Furthermore, the formula E 4 exhibits, as before, a very good performance. We see in Figure 5.9 that argmax µ∈P select (E 4 (µ)) = (1, 1) and E 4 (1, 1) ≈ 10 -16 ; therefore, the formula E 4 with σ = 60 is valid for computing the error bound in Algorithm 1 with tol = 10 -16 .

The behavior of E 4 when σ increases is investigated in Figure 5.10 for test case (i). We consider the values σ = 14, 30, 40 and 50. These four values lead to the same local maxima, and increasing σ allows the formula E 4 to be valid for smaller tolerances (respectively 5 × 10 -8 , 10 -8 , 8 × 10 -9 and 2 × 10 -9 ). Another interesting observation comes from considering the fourth plot in Figure 5.8 and the first plot in Figure 5.10: the classical formula E 2 requires 16 offline resolutions of (5.12) and stagnates at 10 -4 while the formula E 4 with σ = 14 only requires 14 offline resolutions of (5.12) and is valid for tolerances down to 5 × 10 -8 . This shows that at least in some regimes, the new formula E 4 is valid for lower tolerances than the classical formula E 2 , and requires less precomputations. However, contrary to E 2 , using E 4 requires that all the quantities V r defined in (5.24) be recomputed when adding a new vector to the reduced basis.

Conclusion

In this work, we have extended the ideas of [Ar1] by proposing a more stable numerical procedure, using the empirical interpolation method, to represent the a posteriori error bound in the reduced basis method as a linear combination of its values at given parameter values, called interpolation points. Moreover, the proposed method provides a way of choosing the interpolation points, and yields better accuracy levels than the classical a posteriori error bound and than the procedure proposed in [Ar1]. Besides, our new procedure may require less precomputations than the classical a posteriori error bound. The new error bound derived herein can be of particular interest in three situations: (i) when the stability constant of the original problem is very small (this is the case in many practical problems), (ii) when very accurate solutions are needed, (iii) when considering a nonlinear problem (for which, in some cases, no error bound is possible until a very tight tolerance is reached, see [START_REF] Yano | A space-time Petrov-Galerkin certified reduced basis method: Application to the boussinesq equations[END_REF]).

The Successive Constraint Method

The Successive Constraint Method (SCM) enables the computation of a lower bound of the inf-sup constant using an optimization problem, by means of an offline-online procedure. It has been introduced in [START_REF] Huynh | A successive constraint linear optimization method for lower bounds of parametric coercivity and inf-sup stability constants[END_REF]. We can also find detailed presentations in [START_REF] Chen | Improved successive constraint method based a posteriori error estimate for reduced basis approximation of 2d Maxwell's problem[END_REF]. A complex-valued operator version has been proposed in [START_REF] Chen | Certified reduced basis method for electromagnetic scattering and radar cross section estimation[END_REF]. Consider T µ : V → V such that, for all µ ∈ P and all u ∈ V, (T µ u, v) V = a µ (u, v) ∀v ∈ V. It is shown in [START_REF] Sen | Natural norm" a posteriori error estimators for reduced basis approximations[END_REF] that

β 2 µ = inf v∈V (T µ v, T µ v) V v 2 V .
(5.45) Thus, computing β µ requires to solve an eigenvalue problem. Since T µ naturally inherits the affine dependence on µ from a µ , we inject its affine decomposition in (5.45), where we can identify the functions z q and the sesquilinear forms âq , 1 ≤ q ≤ Q, such that

β 2 µ = inf v∈V Q q=1 z q (µ) âq (v, v) v 2 V . (5.46)
It is shown in [START_REF] Fares | The reduced basis method for the electric field integral equation[END_REF] that such a decomposition can be obtained in the case of complex-valued operators with z q being real-valued functions of µ and âq being Hermitian forms on V × V.

Then, β µ can be evaluated by solving a constrained minimization problem in R Q . Defining

I : P × R Q → R (µ, y) → Q q=1
z q (µ)y q , (5.47) there holds β 2 µ = min y∈Y I(µ, y), (5.48) where

Y = y = (y 1 , ...y Q ) ∈ R Q ∃v ∈ V such that y q = âq (v, v) v 2 V , 1 q Q . (5.49)
Finally, we observe that solving the problem (5.48) on a subset and a superset of Y, we can obtain respectively an upper bound and a lower bound of β 2 µ as follows:

β 2 µ LB := min y∈Y LB I(µ, y) β 2 µ β 2 µ UB := min y∈Y UB I(µ, y), (5.50)
where Y UB ⊂ Y ⊂ Y LB .

Algorithm

The SCM consists in a greedy procedure to construct nested spaces verifying

Y 1 UB ⊂ Y 2 UB ⊂ ... ⊂ Y n UB ⊂ ... ⊂ Y ⊂ ... ⊂ Y n LB (µ) ⊂ ... ⊂ Y 2 LB (µ) ⊂ Y 1 LB (µ), (5.51)
where the supersets Y n LB (µ) defining lower bounds are µ-dependent, whereas the subsets Y n UB defining upper bounds are µ-independent. Before presenting the algorithm to construct these subsets and supersets, we need some preliminary definitions.

Definition 5.26 (Eigenvector associated to the smallest eigenvalue) We denote by y(µ) the vector of coordinates

y(µ) q = âq (w, w) w 2 V , 1 ≤ q ≤ Q, (5.52)
where w is the generalized eigenvector corresponding to the smallest generalized eigenvalue λ min of the following generalized eigenproblem of unknowns (λ, v):

(T µ v, T µ v) V = λ v 2 V .
(5.53)

The inf-sup constant for the parameter µ is then known:

β µ := √ λ min .
Notice that Definition 5.26 requires solving a large eigenvalue problem. In some cases, there exist efficient algorithms to compute the eigenpair associated to the smallest eigenvalue. For instance, the Locally Optimal Block Preconditioned Conjugate Gradient Method [START_REF] Knyazev | Toward the optimal preconditioned eigensolver: Locally optimal block preconditioned conjugate gradient method[END_REF] is an iterative algorithm suitable for symmetric positive definite matrices.

Suppose that we are at step n of the offline stage of the SCM: n parameter values µ are selected from P trial and stored in a set denoted C n . For all µ ′ ∈ C n , the smallest eigenvalue of Problem (5.53) is computed and stored (therefore β 2 µ ′ is known), and the corresponding eigenvector is computed and stored in Y n UB . Hence, Y n UB is a set of vectors from R Q , and is of cardinal n. At the end of each step of the offline stage of the SCM, a lower bound of β 2 µ is computed for all µ ∈ P trial . At each step, this lower bound becomes sharper. In particular at step n, the lower bound computed at step n -1 is available for all µ ∈ P trial , and is denoted

β 2 µ n-1

LB

. Let µ ∈ P trial , the nested superset at step n is defined as:

Y n LB (µ) := y ∈ B I(µ ′ , y) β 2 µ ′ ∀µ ′ ∈ P M 1 (µ, C n ) and I(µ ′ , y) β 2 µ ′ n-1 LB ∀µ ′ ∈ P M 2 (µ, P trial ) , where B = Q q=1 σ - q , σ + q with σ - q = inf v∈V âq(v,v) v 2 V , σ + q = sup v∈V âq(v,v) v 2 V , 1 ≤ q ≤ Q ; M 1 and M 2 are
positive integers, and, for any subset E of P trial , (5.56)

P M (µ, E) := M closest points to µ in E if card(E) > M, E if card(E) M. (5.54) Since y → I(µ, y) is linear, Y n LB (µ) is the intersection of 2Q + M 1 + M 2 half-spaces of R Q : 2Q half-spaces for y ∈ B, M 1 half-spaces for I(µ ′ , y) β 2 µ ′ ∀µ ′ ∈ P M 1 (µ, C n ), and M 2 half-spaces for I(µ ′ , y) β 2 µ ′ n-1 LB ∀µ ′ ∈ P M 2 (
The offline construction of the nested spaces is described in Algorithm 3.

Once the offline stage of the SCM is done, the online problem consists in solving (5.55), which provides a lower bound of the inf-sup constant, where the sharpness is controlled by η n (µ) over P trial .

Remark 5.28 (Choice of M 1 and M 2 ) The positive integers M 1 and M 2 hide a tradeoff in computational complexity: if M 1 and M 2 are large, the bounds β 2 µ ′ n-1 LB are sharp and the offline of the SCM is supposed to converge fast, but the optimization problem (5.55) contains more constraints, and therefore is more complex to solve. If M 1 and M 2 are smaller, the number of steps of the SCM is larger, but the optimization problem (5.55) is faster to solve. In particular, the complexity of the online problem of the SCM depends on M 1 and M 2 . Compute η n+1 (µ) for all µ ∈ Ptrial using (5.56), with

Algorithm 3 Offline stage of the SCM algorithm

1. Compute σ - q = inf v∈V âq (v,v) v 2 V and σ + q = sup v∈V âq (v,v) v 2 V , 1 ≤ q ≤ Q, and set B = Q q=1 σ - q , σ +
Y n+1 LB (µ) := y ∈ B I(µ ′ , y) β 2 µ ′ ∀µ ′ ∈ PM 1 (µ, Cn+1) and I(µ ′ , y) β 2 µ ′ n LB ∀µ ′ ∈ PM 2 (µ, Ptrial) 10.
n ← n + 1 11. end while

A nonintrusive EIM to approximate linear systems with nonlinear parameter dependence

This chapter is based on the preprint [Pr1].

Summary. We consider a family of linear systems Aµα = C with system matrix Aµ depending on a parameter µ and for simplicity parameter-independent right-hand side C. These linear systems typically result from the finitedimensional approximation of a parameter-dependent boundary-value problem. We derive a procedure based on the Empirical Interpolation Method to obtain a separated representation of the system matrix in the form Aµ ≈ m βm(µ)Aµ m for some selected values of the parameter. Such a separated representation is in particular useful in the Reduced Basis Method. The procedure is called nonintrusive since it only requires to access the matrices Aµ m . As such, it offers a crucial advantage over existing approaches that instead derive separated representations requiring to enter the code at the level of assembly. Numerical examples illustrate the performance of our new procedure on a simple one-dimensional boundary-value problem and on three-dimensional acoustic scattering problems solved by a boundary element method.

Introduction

In industrial projects, decisions are often taken after a series of complex computations using computer codes of various origins. To simplify the overall computation, surrogate models can be used to replace some parts of the computation. Some of these surrogates are constructed using only a series of input/output couples. With some hypotheses on the input, confidence intervals can be derived, see e.g. [START_REF] Stein | Interpolation of spatial data: some theory for kriging[END_REF] for the kriging method. When additional knowledge on the underlying mathematical formulation is available, model reduction methods can be used. For instance, the Reduced Basis Method (RBM) enables fast resolutions on a basis of precomputed solutions, rather than on a finite element basis (see [START_REF] Machiels | Reliable real-time solution of parametrized partial differential equations: Reduced-basis output bound methods[END_REF] for a detailed presentation and [START_REF] Binev | Convergence rates for greedy algorithms in reduced basis methods[END_REF] for some convergence results). We consider a family of linear systems A µ α = C of order n, where n is large. For simplicity, we assume that the right-hand side is independent of the parameter µ.

The RBM consists first in an offline stage, where a reduced basis of n functions u j , 1 ≤ j ≤ n are computed using a greedy algorithm. These functions are solution of the original problem for some values (µ j ) 1≤j≤n of the parameter µ, which are selected using a greedy algorithm. The functions u j thus write u j (x) = n i=1 α i (µ j )θ i (x), where (θ i ) 1≤i≤n is the finite element basis and the vector α(µ j ) = (α i (µ j )) 1≤i≤n is such that A µ j α(µ j ) = C. In practice, the dimension of the reduced basis is much smaller than the dimension of the finite element basis: n ≪ n. Denote by U the rectangular matrix of size n × n such that (U ) i,j = α i (µ j ). Second, in the online stage, for a given value of the parameter µ, a reduced problem is constructed as µ α(µ) = Ĉ, where µ = U t A µ U and Ĉ = U t C. Solving this reduced problem for a certain value of µ leads to the approximate solution ûµ (x) = n j=1 αj (µ)u j (x). To efficiently construct the online problems, a separated representation (also known as an affine decomposition in the RBM literature) of the matrix assembled by the code is needed in the form

A µ ≈ d m=1 γ m (µ)A m , (6.1) so that µ ≈ d m=1 γ m (µ)U t A m U, (6.2)
where the matrices U t A m U are of small size n × n and can be precomputed during the offline stage. The separated representation (6.1) thus enables online problems to be constructed in complexity independent of n, as long as the functions µ → γ m (µ) are also computed in complexity independent of n. Standard techniques (see [START_REF] Maday | A general multipurpose interpolation procedure: the magic points[END_REF]) to obtain the separated representation (6.1) require in general nontrivial modifications of the assembling routines of the computational code in order to access separately various terms of the variational formulation at hand (See Remark 6.4 below for more details).

The present work provides a step forward in this context, since a procedure that yields a separated representation of A µ in the form

A µ ≈ d m=1 β m (µ)A µm (6.3)
is derived, where (µ m ) 1≤m≤d are some selected values of the parameter. Since the separated representation (6.3) only uses the complete system matrix at the selected parameter values, this representation requires no implementation effort in the assembly routines of the computational code under the (mild) assumptions that we can indeed access the system matrix A µ and that we can identify the functional dependencies on µ in the variational formulation under consideration (see below for more details). For this reason, the procedure is called nonintrusive.

In Section 6.2, we present the approximation problems investigated in this work, first a simple introductory example and then problems with a more complex parameter dependence. In Section 6.3, we briefly recall the Empirical Interpolation Method. In Section 6.4, we present our nonintrusive procedure for the introductory example and test it on a one-dimensional boundaryvalue problem. The procedure is extended to more complex parameter dependence in Section 6.5 where it is also applied to two three-dimensional scattering problems. Some conclusions are drawn in Section 6.6 where, in particular, we observe that our procedure can be extended to the approximation of other quantities.

The approximation problem

We first present an introductory example. Let V be a Hilbert space and consider the following weak formulation: Find u ∈ V such that for all u t ∈ V,

Ω g(µ, x)∇u(x) • ∇u t (x)dx + Ω µu(x)u t (x)dx = b(u t ), (6.4)
where Ω is the domain of computation, µ a parameter belonging to a given parameter set P, g(µ, x) a given function defined on P × Ω and b a bounded linear form on V. Consider now a conforming n-dimensional approximation of the space V denoted by V h (the subscript h refers to an underlying mesh), and a basis of V h denoted by (θ i ) 1≤i≤n . The finite element approximation of (7.42) requires the computation of the matrix A µ of size n × n with entries

(A µ ) i,j := Ω g(µ, x)∇θ j (x) • ∇θ i (x)dx + µ Ω θ j (x)θ i (x)dx i,j . ( 6.5)
The notation A µ is adopted to stress the fact that the matrix A µ depends on the value of the parameter µ. The problem solved by the computational code is

A µ α = C, ( 6.6)
where (C) i = b(θ i ) for all 1 ≤ i ≤ n, and where an approximation of the solution u to (7.42) is obtained in the form u(x)

≈ n i=1 α i θ i (x). Let A 1 µ i,j := Ω g(µ, x)∇θ j (x) • ∇θ i (x)dx i,j and A 0 i,j := Ω θ j (x)θ i (x)dx i,j (6.7) so that A µ = A 1 µ + µA 0 . ( 6.8)
Definition 6.1 (Intrusivity) A procedure leading to a separated representation of A µ in the general form (6.1) is called -intrusive if it requires to implement new integral terms, -weakly-intrusive if it only requires to precompute independently A 1 µ for some values of µ and A 0 , -nonintrusive if it only requires to precompute A µ for some values of µ.

The term "weakly-intrusive" comes from the fact that the user has to enter the routines of the code and to insert switches at the right places to save the terms in A 1 µ independently from the terms in A 0 . In the context of industrial codes, this is not always possible. The notion of nonintrusivity in Definition 6.1 is different from the notion of black-box, which requires only the computation of input / output couples. Our purpose is to develop a nonintrusive procedure leading to the separated representation (6.3) of A µ .

The above example can be generalized to a class of engineering problems requiring to compute a large, parameter-dependent matrix A µ for many values of the parameter µ where A µ is of the form

A µ = R ̺=1 A ̺ µ + S s=1 ψ s (µ)A s , ( 6.9)
where A ̺ µ are matrices that require to integrate some functions g ̺ (µ, x) over Ω, ψ s are given functions of µ and A s are µ-independent matrices resulting from some integration over Ω. The introductory example corresponds to R = 1, S = 1, and ψ 1 (µ) = µ. To simplify the presentation of the main ideas, we consider the setting of (6.8) in Sections 6.3 and 6.4 and return to the more general setting of (6.9) in Section 6.5.

Empirical Interpolation Method

The Empirical Interpolation Method (EIM) is a procedure to approximate two-variable functions. In particular, it can be used to approximate the two-variable function g(µ, x), for all µ ∈ P and all x ∈ Ω. Denote by EIM g this particular procedure. EIM g leads to an interpolation operator I g d g such that

I g d g g (µ, x) ≈ g(µ, x), ∀µ ∈ P, ∀x ∈ Ω, ( 6.10)
where d g is the number of interpolation points (called magic points in the context of RBM, see [START_REF] Maday | A general multipurpose interpolation procedure: the magic points[END_REF]). EIM g is composed of two stages: (i) an offline stage, where a matrix B g of size d g × d g , a set of d g x-dependent basis functions {q g k } 1≤k≤d g , a set of d g points {x k } 1≤k≤d g in Ω, and a set a d g parameter values {µ k } 1≤k≤d g in P are constructed, (ii) an online stage, where the quantities computed in the offline stage are used to carry out the approximation (6.10) (see Section 6.4.2 for more details on the offline / online stages for the whole procedure).

The offline stage of EIM g is detailed in Algorithm 6. This variant corresponds to the classical EIM, described in [START_REF] Maday | A general multipurpose interpolation procedure: the magic points[END_REF]. In the loop on k in Algorithm 6, the residual operator δ g k is defined by δ g k = Id -I g k , where the interpolation operator I g k is such that

I g k g (µ, x) := k m=1 λ g m (µ)q g m (x), (6.11)
and for a given µ ∈ P, the λ g m (µ)'s are defined by

k m=1 B g l,m λ g m (µ) = g(µ, x g l ), ∀1 ≤ l ≤ k. ( 6.12)
After d g iterations, the interpolation formula (7.7) leads to the following approximation for A µ :

A µ ≈ d g m=1 λ g m (µ)M m + µA 0 , ( 6.13)
where (M m ) i,j = Ω q g m (x)∇θ j (x) • ∇θ i (x)dx. This representation of A µ is of the form (6.1).

Property 6.2 (Interpolation

) ∀x ∈ Ω, ∀ 1 ≤ m ≤ d g , I g d g g (µ g m , x) = g(µ g m , x).
Proof. See [73, Lemma 1]. ♦ Property 6.2 means that, at the parameter values µ g k 1≤k≤d g selected by EIM g , the approximation (6.13) is exact since

d g m=1 λ g m (µ g k )M m = A 1 µ g k for all 1 ≤ k ≤ d g . Since Vect 1≤k≤d g q g k (x) = Vect 1≤k≤d g g(µ g k , x
) holds in Algorithm 6, the functions q g k (x) can be expressed in terms of the functions g(µ g k , x) in the following form: there exist

γ l,k , 1 ≤ l ≤ k ≤ d g such that q g k (x) = d g l=1 γ l,k g(µ g l , x
), for all 1 ≤ k ≤ d g . Letting (λ g m (µ)) 1≤m≤d g solve (7.8) for k = d g , we obtain after exchanging the summations

(I g d g g)(µ, x) = d g m=1 d g l=1 γ m,l λ g l (µ) g(µ g m , x). (6.14) Algorithm 4 Offline stage of EIM g 1. Choose d g > 1 [Number of interpolation points] 2. Set k := 1 3. Compute µ g 1 := argmax µ∈P g(µ, •) L ∞ (Ω) 4. Compute x g 1 := argmax x∈Ω |g(µ g 1 , x)| [First interpolation point] 5. Set q g 1 (•) := g(µ g 1 , •) g(µ g 1 , x g 1 )
[First basis function]

6. Set B g 1,1 := 1 [Initialize B g matrix] 7. while k ≤ d g do 8.
Compute

µ g k+1 := argmax µ∈P (δ g k g)(µ, •) L ∞ (Ω) 9.
Compute

x g k+1 := argmax x∈Ω |(δ g k g)(µ g k+1 , x)| [(k + 1)-th interpolation point] 10. Set q g k+1 (•) := (δ g k g)(µ g k+1 , •) (δ g k g)(µ g k+1 , x g k+1 ) [(k + 1)-th basis function] 11.
Set B g i,k+1 := q g k+1 (x g i ), for all 1

≤ i ≤ k + 1 [Increment matrix B g ] 12.
k ← k + 1 [Increment the size of the interpolation]

13. end while Define η g m (µ) :=

d g l=1 γ m,l λ g l (µ).
The following property then holds: Property 6.3 (Weak-intrusivity) EIM g leads to a weakly-intrusive procedure, since the resulting approximation of A µ can be written If the considered variational formulation contains only one term, the above procedure was already proposed in the RBM literature as a nonintrusive method to obtain a separated representation of the linear system under consideration, see [START_REF] Maday | A general multipurpose interpolation procedure: the magic points[END_REF]. For instance in (6.15), if

A µ ≈ d g m=1 η g m (µ)A 1 µ g m + µA 0 . ( 6
A 0 = 0, then A 1 µ g m = A µ g m .
In the general setting of (6.9), this corresponds to R = 1 and S = 0. In any other case, the classical EIM needs to access independently matrices associated to each term of the variational formulation and thus cannot deliver a separated representation solely based on the A µ matrices.

The nonintrusive procedure

Description of the procedure

Denote by G g (µ) the vector-valued function with d g components such that G g m (µ) = g(µ, x g m ), for all 1 ≤ m ≤ d g . Then, from (7.8), λ g (µ) = (λ g m (µ)) 1≤m≤d g can be concisely written as λ g (µ) = (B g ) -1 G g (µ). Notice that the computation of λ g (µ) only requires the matrix B g and the set of points {x g m } 1≤m≤d g . Let (z p (µ)) 1≤p≤dmax with d max := d g + 1, be such that

z p (µ) := λ g p (µ) 1 ≤ p ≤ d g , µ p = d g + 1. (6.16)
Recalling the notation (M m ) i,j := Ω q g m (x)∇θ j (x) • ∇θ i (x)dx for all 1 ≤ m ≤ d g , we infer from (6.13) that

A µ ≈ d g m=1 λ g m (µ)M m + µA 0 = dmax p=1 z p (µ)T p , ( 6.17)
where the matrices

T p := M p 1 ≤ p ≤ d g , A 0 p = d g + 1 = d max , (6.18)
are independent of µ. Note that d max is the number of matrices to precompute and store when using the approximation (6.13).

The key idea is now to apply a second EIM to approximate z p (µ), where z is seen as a function depending on the two variables p and µ. The EIM procedure to approximate z p (µ) is denoted by EIM z and its offline stage is detailed in Algorithm 5. This implementation correspond to the variant from the classical EIM (see Algorithm 2). As explained in Section 5.4.2, this variant leads to a nonintrusive without resorting to an additional change of basis. We refer to Section 7.2 for more details about the differences between the EIM variant considered here and the classical algorithm. The number of interpolation points is denoted by d z ≤ d max . In the loop on k in Algorithm 5, the residual operator δ z k is defined by δ z k = Id -I z k , where

(I z k z) p (µ) := k m=1 β z m (µ)z p (µ z m ), (6.19)
and

k m=1 B z m,l β z m (µ) = q z l (µ), 1 ≤ l ≤ k. ( 6.20)
Owing to the interpolation property, there holds (I z d z z) p z k (µ) = z p z k (µ) for all 1 ≤ k ≤ d z and all µ ∈ P. If d z = d max , all the indices p are selected in Algorithm 5 and (I z dmax z) p (µ) = z p (µ) for all 1 ≤ p ≤ d max and all µ ∈ P. Observe that we can stop EIM z before d z = d max interpolation matrices have been computed, see Sections 6.5.2 and 6.5.3 for some illustrations.

Injecting the approximation (6.19) with k = d z into the right-hand side of (6.17) with z p (µ) replaced by (I z d z z) p (µ) yields

A µ ≈ dmax p=1 T p d z m=1 β z m (µ)z p (µ z m ) = d z m=1 β z m (µ) dmax p=1 T p z p (µ z m ) ≈ d z m=1 β z m (µ)A µ z m , (6.21)
where β z m (µ) is obtained from (6.20). The right-hand side of (6.21) is the desired separated representation of A µ that can be built in a nonintrusive way.

Practical implementation

To compute the L ∞ -norms and determine the argmax in Algorithms 6 and 5, it is convenient to consider finite subsets of P and Ω, denoted respectively by P trial and Ω trial . This

Algorithm 5 Offline stage of EIM z 1. Choose d z > 1 [Number of interpolation points] 2. Set k := 1 3. Compute p z 1 := argmax 1≤p≤d g +1 (z)p(•) L ∞ (P) 4. Compute µ z 1 := argmax µ∈P |(z) p z 1 (µ)| [First interpolation point] 5. Set q z 1 (•) := (z) p z 1 (•) (z) p z 1 (µ z 1 )
[First basis function]

6. Set B z 1,1 := 1 [Initialize B z matrix] 7. while k ≤ d z do 8.
Compute p z k+1 := argmax

1≤p≤d g +1 (δ z k z)p(•) L ∞ (P) , 9.
Compute

µ z k+1 := argmax µ∈P |(δ z k z) p z k+1 (µ)| [(k + 1)-th interpolation point] 10. Set q z k+1 (•) := (δ z k z) p z k+1 (•) (δ z k z) p z k+1 (µ z k+1 ) [(k + 1)-th basis function] 11. B z i,k+1 := q z k+1 (µ z i ), for all 1 ≤ i ≤ k + 1 [Increment matrix B z ] 12.
k ← k + 1 [Increment the size of the interpolation]

13. end while becomes necessary when, for instance, the function g(µ, x) is not known analytically, but only for some elements of P and Ω. It seems natural to take for Ω trial the set of Gauss points on which the quadrature formulae to compute the integrals in (7.42) are defined. However, this supposes to know and manipulate the set of the Gauss points associated with the mesh. Since the functions q g defined in Algorithm 6 are only used to construct the matrix B g and are not directly integrated with respect to x to carry out the interpolation (6.21), it is possible to write the procedure with any set Ω trial sampling the geometry. Such an approach is considered in the numerical example of Section 6.5.3. More generally, the sets P trial and Ω trial should be fine enough to capture all the phenomena, but not too fine to limit the overall computational cost.

The numerical examples of Section 6.5 indicate that high accuracy can be obtained with simple choices for P trial and Ω trial on nontrivial cases.

In addition to the two sets P trial and Ω trial , the number of interpolation points d g and d z for each EIM have to be chosen. The choice we made is to stop the two EIM's when respectively (δ g k g)(µ g k+1 , x g k+1 ) and (δ z k z) p z k+1 (µ z k+1 ) have reached a prescribed threshold, typically set at the level of the machine precision.

Finally, we specify the offline and online stages of our procedure when used within the RBM. EIM g and the offline stage of EIM z are part of the offline stage of the RBM. During the online stage of the RBM, the reduced matrix is constructed as

µ ≈ d z m=1 β z m (µ)U t A µ z m U, ( 6.22)
so that only the online stage of EIM z (i.e., the resolution of (6.20)) is needed.

Illustration

As a first illustration, we consider the following boundary-value problem:

- d dx exp(µx) du dx (x) + µu(x) = 1 in Ω := (-3, 3), (6.23)
with the following Dirichlet boundary condition u(-3) = u(3) = 0. The weak form reads: Find u ∈ H 1 0 (Ω) such that for all u t ∈ H 1 0 (Ω),

a µ (u, u t ) = Ω u t (x)dx, (6.24)
with

a µ (u, u t ) := Ω exp(µx) du dx (x) du t dx (x)dx + Ω µu(x)u t (x)dx. (6.25)
First-order continuous Lagrange finite elements are used, with a three-point quadrature formula in each mesh cell. The mesh is uniform with h x = 0.015. Ω trial is taken to be the set of Gauss points on the obtained mesh, and P trial = {1, 1 + h µ , 1 + 2h µ , ..., 3} with h µ = 0.005. To derive the separated approximation (6.21), EIM g is first applied to g(µ, x) := exp(µx). Then, the vector-valued function z(µ) is constructed using (6.16). The quality of the whole procedure is measured, for various values of d g and d z = d g + 1 using two error criteria: (i) the relative Frobenius norm error on the matrix A µ and (ii) the relative L 2 (Ω)-norm error on the solution, see Figures 6.1 and 6.2. We conclude from this first test case that the present method allows for a very good approximation of the matrix and the solution.

z p (µ) :=                                                  (λ g m ) 1 (µ), 1 ≤ p ≤ (d g ) 1 , m = p, . . . (λ g m ) R (µ), 1 + R-1 ̺=1 (d g ) ̺ ≤ p ≤ R ̺=1 (d g ) ̺ , m = p - R-1 ̺=1 (d g ) ̺ , ψ 1 (µ), p = R ̺=1 (d g ) ̺ + 1, . . . ψ S (µ), p = R ̺=1 (d g ) ̺ + S, (6.27)
and let EIM z be applied to z p (µ), with d z interpolation points, such that

d z ≤ d max = R ̺=1 (d g ) ̺ + S,
to obtain an approximation of A µ in the same form as (6.21). Note that d max is the number of matrices to precompute and store when using the approximation (6.13), while the number of matrices to precompute and store when using (6.21) is d z ; in our numerical examples (see below), accurate representations of A µ are already achieved for d z smaller than d max . Notice also that in total, there are (R + 1) EIM procedures to be applied.

Sound-hard scattering in the air at rest

The problem of interest is the sound-hard scattering of an acoustic monopole source of wave number µ by an aircraft (whose boundary is denoted by Γ ) in the air at rest, in the timeharmonic case. To simulate the noise created by one of the engines, the monopole is located under the left wing of the plane. This is a classical Helmholtz exterior problem, for which one possible weak formulation is:

Find u ∈ H 1 2 (Γ ) such that for all u t ∈ H 1 2 (Γ ), a µ (u, u t ) = Γ f incµ (x)u t (x)dx, (6.28)
where

a µ (u, u t ) := 1 4π Γ Γ exp (iµ |x -y|) |x -y| --→ curl Γ u(x) • --→ curl Γ u t (y) dxdy - µ 2 4π Γ Γ exp (iµ |x -y|) |x -y| u(x)u t (y) ( -→ n x • -→ n y ) dxdy, (6.29)
where --→ curl Γ denotes the surfacic curl on Γ , -→ n x the unit normal vector on Γ pointing towards the medium of propagation, and f incµ is the incident acoustic field created by the source. We refer to [START_REF] Nédélec | Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems[END_REF]Section 3.4] for details on the derivation of (6.28), and justifications on the wellposedness of the integral in (6.29). The parameter of interest is the wave number µ of the acoustic monopole source. The Boundary Element Method (BEM) is used to approximate problem (6.28). This leads to a dense µ-dependent matrix (A µ ) i,j = a µ (θ j , θ i ), where (θ i ) 1≤i≤n denote the basis functions of the considered finite element space on Γ . Two different meshes, on which the matrices are assembled, are considered, see Table 6.1 and Figure 6.3. The in-house code ACTIPOLE developed by EADS-IW and Airbus [START_REF] Delnevo | Code acti3s harmonique, justification mathématique, Partie I[END_REF][START_REF] Delnevo | Code acti3s, justifications mathématiques, Partie II : presence d'un écoulement uniforme[END_REF] is used. This test case is a challenging benchmark for

A µ ≈ 1 + µ 2 d g m=1 λ g m (µ)M m ,
where the matrices M m have been defined in Section 6.4.1, so that the approximation (6.21) can be written using

z p (µ) := λ g m (µ), 1 ≤ m ≤ d g , p = m, µ 2 λ g m (µ), 1 ≤ m ≤ d g , p = m + d g . ( 6.31)
Note that we exploited the links in the functional dependence on µ for the two terms on the right-hand side of (6.29) to carry out only one EIM g procedure.

EIM g and EIM z are carried out with respectively d g = 30 and d z = 32 interpolation points (notice that d max = 60). To check the accuracy of the approximation, we compute the relative Frobenius norm error on the matrix A µ and the relative Euclidian norm error on the acoustic pressure computed using the approximate matrix, on a network of 400 points located behind the aircraft. Figure 6.5 presents the results on Mesh 1. In this figure, the relative differences are computed on 100 values of µ, namely one tenth of the considered parameter values, explaining why only 7 minima are achieved on the left plot. On the right plot concerning the acoustic pressure behind the aircraft, a large number of values are at the level of machine precision. Note that the right-hand side of (6.28) also depends on the parameter µ. To compute the right plot of Figure 6.5, we computed the exact values of this right-hand. Figure 6.6 shows the solution to the problem on Mesh 1 and the relative difference of the solution using the exact matrix and its approximation for µ = 2.47.

The simulation is repeated on Mesh 2, with d g = 50 and d z = 50. A twice as large frequency interval is considered since Mesh 2 has a better spatial resolution than Mesh 1. Figure 6.7 shows the relative Frobenius norm error on the matrix A µ , confirming the accuracy of the approximation.

Sound-hard scattering in a non-uniform flow

Consider an ellipsoid with major axis directed along the z-axis. This object is included inside a larger ball, see Figure 7.3. The external border of the ball after discretization is denoted by Γ ∞ . The complement of the ellipsoid in the ball is denoted by Ω -. A potential flow is precomputed around the ellipsoid and inside the ball, such that the flow is uniform outside the ball, of Mach number 0.3 and directed along the z-axis. The flow is fixed, and does not depend on the parameter µ. An acoustic monopole source lies upstream of the ball, on the z-axis as well. The parameter is again the wave number of the monopole source. The considered formulation is a coupled Finite Element Method (FEM) -BEM formulation (3.45) derived in Section 3.3.4. It consists in (i) applying a change of variable to transform the convected Helmholtz equation into the classical Helmholtz equation outside the ball, in order to apply a standard BEM, and (ii) stabilizing the formulation to avoid resonant frequencies associated with the eigenvalues of the Laplacian inside the ball of border Γ ∞ . The formulation depends on the wave number of the source in a complex way, but we will see in our numerical tests that our nonintrusive procedure provides an accurate approximation of the resulting matrix as a linear combination of a few snapshots of the complete matrix at some wave numbers of the source.

Consider the product space H

:= H 1 (Ω -) × H -1 2 (Γ ∞ ) × H 1 (Γ ∞ ) with inner product (Φ, λ, p) , Φ t , λ t , p t H := Φ, Φ t H 1 (Ω -) + λ, λ t H -1 2 (Γ∞) + p, p t H 1 (Γ∞) . The weak formulation is: Find (Φ, λ, p) ∈ H such that ∀ Φ t , λ t , p t ∈ H, V µ (Φ, Φ t ) + N µ (γ - 0 Φ), γ - 0 Φ t Γ∞ + Dµ - 1 2 I (λ), γ - 0 Φ t Γ∞ = γ 1 f incµ , γ - 0 Φ t Γ∞ , (6.32a) λ t , D µ - 1 2 I (γ - 0 Φ) Γ∞ -λ t , S µ (λ) Γ∞ -i λ t , p Γ∞ = -λ t , γ 0 f incµ Γ∞ , (6.32b) N µ (γ - 0 Φ), p t Γ∞ + Dµ + 1 2 I (λ), p t Γ∞ -δ Γ∞ (p, p t ) = γ 1 f incµ , p t Γ∞ , (6.32c)
where (•, •) Γ∞ denotes the extension of the L 2 (Γ ∞ )-inner product to the duality pairing on

H -1 2 (Γ ∞ ) × H 1 2 (Γ ∞ )
, and where δ Γ∞ (p, q) := (∇ Γ∞ p, ∇ Γ∞ q) Γ∞ + (p, q) Γ∞ , (

with ∇ Γ∞ the surfacic gradient on Γ ∞ , and

V µ (Φ, Φ t ) := Ω - Ξ∇Φ • ∇Φ t -µ 2 Ω - βΦΦ t + iµ Ω - V • Φ∇Φ t -Φ t ∇Φ , (6.34)
where

β := r ς + γ 2 ∞ P 2 -γ 4 ∞ M 2 ∞ , V := r ς + γ 2 ∞ P N M -γ 3 ∞ M ∞ , Ξ := rN ON with r := ρ ρ∞ , ς := c∞ c , γ ∞ := 1 √ 1-M 2 ∞ , P := M • M ∞ , N := I + C ∞ M ∞ M T ∞ , O := I -M M T , and C ∞ := γ∞-1 M 2 ∞ .
In the above notation, the subscript ∞ is used for quantities outside the ball, ρ is the density of the flow, c is the speed of sound when the flow is at rest and M = v c , where v is the velocity of the flow. The operators γ 0 and γ 1 are Dirichlet and Neumann traces on the coupling surface Γ ∞ . The operators N µ , D µ , Dµ , and S µ are boundary integral operators, expressed in terms of the Green kernel G µ (x, y) = exp(iµ|x-y|) 4π|x-y| associated with the Helmholtz equation at wave number µ. The next step is to identify the dependencies in µ in the formulation (7.47). It turns out that the functions of µ involved in the integrals of the formulation (7.47) are µ, µ 2 , exp(iµr), µ exp(iµr), µ 2 exp(iµr), and µ 2iπµ c µ -1 exp(iµr). As in the previous test case, EIM g is carried out to approximate the function g(µ, r) = exp(iµr), r = |x -y|, x, y ∈ Γ ∞ . We choose µ ∈ P trial := {10, 10.03, ..., 40}, a set of 1000 values for the wave number, so that the highest wave number of the source corresponds to a wavelength 5 times larger than the mean edge of the mesh. This time, instead of considering a subset of |x -y| where x and y are the Gauss points associated with the mesh, we take r ∈ {0, h, ..., N h}, where N = 10000 and h = D N , D being the diameter of the sphere Γ ∞ . With this choice, we no longer need to know the position of the Gauss points, but simply the diameter of the geometry of the test case.

Then, the functions (λ g m (µ)) 1≤m≤d g , µ ∈ P trial , are computed using (7.8), and the functions z p (µ), 1 ≤ p ≤ d max := 3d g + 3, are defined by

z p (µ) :=                        λ g m (µ), 1 ≤ p ≤ d g , m = p, µλ g m (µ), d g + 1 ≤ p ≤ 2d g , m = p -d g , µ 2 λ g m (µ), 2d g + 1 ≤ p ≤ 3d g , m = p -2d g , 1, p = 3d g + 1, µ, p = 3d g + 2, µ 2 , p = 3d g + 3. (6.35)
EIM g and EIM z are carried out with respectively 17 and 20 interpolation points (notice that d max = 71). Figure 6.9 shows the relative Frobenius norm error on the matrix A µ and the relative Euclidian norm error on the acoustic pressure computed using the approximate matrix on a network of 400 points located behind the scattering ellipsoid. In this test case, an excellent accuracy is obtained with only 20 precomputed matrices. frequency (Hz) relative error on the solution Fig. 6.9. Log10 of the relative error on the Frobenius norm of the matrix Aµ (left) and on the acoustic pressure computed using the approximate matrix computed using (6.21) on a network of 400 points located behind the object in Euclidian norm (right), with d g = 17 and d z = 20.

Outlook

The method described herein provides an efficient nonintrusive approximation of parameterdependent linear systems, provided that the considered code can return the assembled matrix and that the corresponding weak formulation is known. The method offers a crucial practical advantage over existing methods since it avoids significant implementation efforts. In the present work, the choice has been made to approximate the whole matrix A µ assembled by the code, but the procedure applies in the same way to the approximation of any linear functional l of the matrix A µ , whereby

l(A µ ) ≈ d z m=1 β z m (µ)l(A µ z m ), (6.36)
where the storage of A µ z m for all 1 ≤ m ≤ d z is replaced by the storage of l(A µ z m ) for all 1 ≤ m ≤ d z , which may be much lighter in terms of memory usage. The efficient construction of the reduced matrix µ in the RBM corresponds to l(A µ ) = U t A µ U , as explained in the introduction.

Finally, we observe that in the case where the right-hand side C of the problem (6.6) is also dependent on the parameter µ (then written C µ ), the same procedure can be applied to derive a separated representation of C µ .

A nonintrusive Reduced Basis Method applied to aeroacoustic simulations

This chapter is based on the preprint [Pr3]

Summary. The Reduced Basis Method can be exploited in an efficient way only if the so-called affine dependence assumption on the operator and right-hand side of the considered problem with respect to the parameters is satisfied. When it is not, the Empirical Interpolation Method is usually used to recover this assumption approximately. In both cases, one must be able to access and modify the assembly routines of the corresponding computational code, leading to an intrusive procedure. In this work, we derive variants of the EIM algorithm and explain how they can be used to provide a nonintrusive procedure to compute affine decompositions of the operator and of the right-hand side of the problem while being computable for all values of the parameter (and not only at training points). We explain how this algorithm can be effectively applied in various contexts. We present various examples of aeroacoustic problems solved by integral equations using the reduced basis method and show how this algorithm can benefit from the linear algebra tools available in the considered code.

Introduction

In many problems such as optimization, uncertainty propagation and real-time simulations, one needs to solve a (complex) parametrized problem for many values of the parameters. Among the various available methods to reduce the computational cost, the Reduced Basis Method (RBM) has received increased interest over the last decade (see [START_REF] Machiels | Reliable real-time solution of parametrized partial differential equations: Reduced-basis output bound methods[END_REF][START_REF] Maday | A general multipurpose interpolation procedure: the magic points[END_REF][START_REF] Patera | A posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations[END_REF] for a detailed presentation and [START_REF] Devore | Greedy algorithms for reduced bases in Banach spaces[END_REF] for some convergence results). Consider the following problem: Find

u µ ∈ V such that a µ (u µ , v) = c µ (v), ∀v ∈ V, (7.1)
where µ ∈ P is the parameter, a µ is a sesquilinear form, c µ is a linear form, and V is a finitedimensional functional space of size n, where n is typically very large. Since the linear problem (7.1) is written on a finite-dimensional space, we can consider the following matrix form:

A µ U µ = C µ , ( 7.2)
where A µ ∈ C n×n and C µ ∈ C n . We refer to the solutions to (7.1) as truth solutions.

The RBM allows one to compute very fast an approximation of the truth solution u µ by means of an offline/online procedure. The online stage is a Galerkin procedure written on a basis of truth solutions u µ j , 1 ≤ j ≤ n ≪ n, rather than on a basis of V. The parameter values µ j are selected by a greedy algorithm in the offline stage, where the functions u µ j of the reduced basis are also precomputed. Denote by U the rectangular matrix of size n × n such that (U ) i,j = γ i (µ j ), where γ i (µ j ), 1 ≤ i ≤ n, are the coefficients of u µ j on the basis of V. Then, the RBM approximation is computed by solving the reduced problem µ γ(µ) = Ĉµ , where µ = U t A µ U and Ĉµ = U t C µ , so that ûµ (x) := n j=1 γj (µ)u µ j (x) ≈ u µ (x). The efficiency of the RBM hinges on the assumption of an affine dependence of the operator and the right-hand side with respect to the parameter. This assumption states that

A µ = d i=1 α i (µ)A i . (7.3)
We only discuss the case of the operator A µ , the right-hand side C µ being treated in the same way. Owing to the separated representation (7.3), the assembly of the reduced problems and the computation of the a posteriori error bound are performed in complexity independent of n. When affine dependence does not hold, the Empirical Interpolation Method (EIM) can be used to recover it approximately. In any case, replacing A µ by the right-hand side of (7.3) so as to assemble the reduced problems in complexity independent of n requires in general nontrivial modifications of the assembling routines of the computational code since various terms of the variational formulation at hand corresponding to the matrices A i in (7.3) have to be accessed separately.

When considering the approximation of A µ and C µ by the EIM in a Reduced Basis context, some requirements are crucial in practical applications: (i) the assembly of the online problems must be of complexity independent of n to ensure online-efficiency, (ii) the approximation must be computable for all µ ∈ P, and not only on some training points in P, (iii) the procedure should be nonintrusive in the sense that it should not require to assemble new terms such as the matrices A i , but just the matrices A µ for selected values of the parameter µ. In this work, we derive various variants for the classical EIM algorithm and discuss their advantages, drawbacks, and properties. We show that some of the derived approximation procedures meet the requirements (i) and (ii), but not (iii). Meeting requirement (iii) hinges on deriving a separated representation of A µ in the form

A µ ≈ r m=1 β m (µ)A µm , ( 7.4)
where µ m , 1 ≤ m ≤ r, are some selected values of the parameter (which are different from the parameter values µ j , 1 ≤ j ≤ n, selected by the greedy algorithm in the offline stage of the RBM). In this work, we derive such nonintrusive approximation procedures, to the price of an additional EIM approximation of which we can control the accuracy. One of these was first considered in Chapter 6; we explain that among all the possible choices, this approximation property is optimal in terms of computational savings and, at a certain limit, interpolant with respect to the parameter.

In Section 7.2, we briefly recall the classical EIM algorithm, and present some new variants that are useful in the present context. Then, procedures to approximate A µ meeting the requirements (i), (ii), and (iii) are derived in Section 7.3. Finally, numerical simulations are presented on aeroacoustic problems solved by integral equations in Section 7.4, where the use of the nonintrusive formulae is crucial.

Classical EIM and variants

Consider a function g(µ, x) defined over P × Ω for two sets P and Ω. We look for an approximation of this function in a separated form with respect to µ and x. There are different possible ways to achieve such an approximation using EIM-like algorithms. An EIM algorithm consists in an offline stage, where some quantities are precomputed within a greedy procedure, and an online stage (where the approximation is computed making use of these precomputed quantities).

First, we recall the classical EIM as defined in [START_REF] Maday | A general multipurpose interpolation procedure: the magic points[END_REF]. We denote the offline stage of this algorithm by EIM S1 , S1 refering to Slice 1, since the obtained approximation is interpolant with respect to the first variable (see Proposition 7.2 below). Fix an integer d > 1 (the total number of interpolation points). For all 1 ≤ k < d, the rank-k approximation operator is defined as

I S1 k g (µ, x) := k m=1 λ S1 m (µ)q S1 m (x), (7.5)
where the functions λ S1 m (µ), 1 ≤ m ≤ k, solve the linear system

k m=1 B S1 l,m λ S1 m (µ) = g(µ, x S1 l ), ∀1 ≤ l ≤ k. ( 7.6)
The notation λ always refers to the solution of the online problem, which can be a function of µ or of x in what follows. The functions q S1 m (•) and the matrices B S1 ∈ R k×k are constructed as described in Algorithm 6, where δ S1 k = Id -I S1 k . Note that this algorithm constructs a set of points {x S1 l } 1≤l≤d in Ω used in (7.6), and also a set of points {µ S1 l } 1≤l≤d in P.

Algorithm 6 Offline stage EIM S1 1. Choose d > 1 [Number of interpolation points] 2. Set k := 1 3. Compute µ S1 1 := argmax µ∈P g(µ, •) L ∞ (Ω) 4. Compute x S1 1 := argmax x∈Ω |g(µ S1 1 , x)| [First interpolation point] 5. Set q S1 1 (•) := g(µ S1 1 , •) g(µ S1 1 , x S1 1 )
[First basis function] Consider now the following online problem, denoted by O2 for Online problem 2. Like O1, O2 is based on quantities precomputed in Algorithm 6:

6. Set B S1 1,1 := 1 [Initialize matrix B S1 ] 7. while k < d do 8. Compute µ S1 k+1 := argmax µ∈P (δ S1 k g)(µ, •) L ∞ (Ω) 9. Compute x S1 k+1 := argmax x∈Ω |(δ S1 k g)(µ S1 k+1 , x)| [(k + 1)-th interpolation point] 10. Set q S1 k+1 (•) := (δ S1 k g)(µ S1 k+1 , •) (δ S1 k g)(µ S1 k+1 , x S1 k+1 ) [(k + 1)-th basis function] 11. Set B S1 i,k+1 := q S1 k+1 (x S1 i ), for all 1 ≤ i ≤ k + 1 [Increment matrix B S1 ]
I S1O2 d g (µ, x) := d m=1 λ S1O2 m (x)g(µ, x S1 m ), (7.9)
where the functions λ S1O2 m (x), 1 ≤ m ≤ d, solve the linear system A variant of Algorithm 6 is obtained by switching the roles of x and µ in the offline stage. We denote this variant by EIM S2 , S2 refering to Slice 2, since the obtained approximation is interpolant with respect to the second variable (see Proposition 7.3 below). Fix an integer d > 1 (the total number of interpolation points). Then, for all 1 ≤ k < d, the rank-k approximation operator is defined as

d m=1 (B S1 ) t l,m λ S1O2 m (x) = q S1 l (x), ∀1 ≤ l ≤ d. ( 7
I S2 k g (µ, x) := k m=1 λ S2 m (x)q S2 m (µ), (7.12)
where the functions λ S2 m (x), 1 ≤ m ≤ k, solve the linear system

k m=1 B S2 l,m λ S2 m (x) = g(µ S2 l , x), ∀1 ≤ l ≤ k. (7.13)
The functions q S2 m (•) and the matrices B S2 ∈ R k×k are constructed during the offline stage, described in Algorithm 7, where δ S2 k = Id -I S2 k . Note that this algorithm constructs a set of points {µ S2 l } 1≤l≤d in P used in (7.13), and also a set of points {x S2 l } 1≤l≤d in Ω.

Algorithm 7 Offline stage EIM S2 1. Choose d > 1 [Number of interpolation points] 2. Set k := 1 3. Compute x S2 1 := argmax x∈Ω g(•, x) L ∞ (P) 4. Compute µ S2 1 := argmax µ∈P |g(µ, x S2 1 )| [First interpolation point] 5. Set q S2 1 (•) := g(•, x S2 1 ) g(µ S2 1 , x S2 1 )
[First basis function]

6. Set B S2 1,1 := 1 [Initialize matrix B S2 ] 7. while k < d do 8. Compute x S2 k+1 := argmax x∈Ω (δ S2 k g)(•, x) L ∞ (P) 9.
Compute

µ S2 k+1 := argmax µ∈P |(δ S2 k g)(µ, x S2 k+1 )| [(k + 1)-th interpolation point] 10. Set q S2 k+1 (•) := (δ S2 k g)(•, x S2 k+1 ) (δ S2 k g)(µ S2 k+1 , x S2 k+1 ) [(k + 1)-th basis function] 11. Set B S2 i,k+1 := q S2 k+1 (µ S2 i ), for all 1 ≤ i ≤ k + 1 [Increment matrix B S2 ] 12.
k ← k + 1 [Increment the size of the decomposition]

13. end while

The online stage O1 of EIM S2 is computed by solving (7.12)-(7.13) for k = d. This yields

I S2O1 d g (µ, x) := d m=1 λ S2O1 m (x)q S2 m (µ), (7.14)
where the functions λ S2O1 m (x), 1 ≤ m ≤ d, solve the linear system

d m=1 B S2 l,m λ S2O1 m (x) = g(µ S2 l , x), ∀1 ≤ l ≤ d. (7.15)
The online stage O2 is defined as

I S2O2 d g (µ, x) := d m=1 λ S2O2 m (µ)g(µ S2 m , x), ( 7.16)
where the functions λ S2O2 m (µ), 1 ≤ m ≤ d, solve the linear system . We refer to Table 7.1 for notation of the four possible approximation procedures.

d m=1 (B S2 ) t l,m λ S2O2 m (µ) = q S2 l (µ), ∀1 ≤ l ≤ d. ( 7
The classical interpolation property from [73, Lemma 1] (corresponding to the choice S1O1) and Proposition 7.1 yield the following propositions.

Proposition 7.2 (Interpolation with S1)

The approximation procedures S1O1 and S1O2 are interpolant with respect to the first variable: for all x ∈ Ω,

(I S1O1 d g)(µ S1 m , x) = (I S1O2 d g)(µ S1 m , x) = g(µ S1 m , x), ∀1 ≤ m ≤ d. ( 7
(I S2O1 d g)(µ, x S2 m ) = (I S2O2 d g)(µ, x S2 m ) = g(µ, x S2 m ), ∀1 ≤ m ≤ d. (7.19)

Nonintrusive procedure

The goal of this section is to obtain an approximation of the following objects:

Q t (µ) = ς s=1 Ω g s (µ, x)Ψ s,t (x)dx, ∀1 ≤ t ≤ N, ( 7.20)
where ς ≥ 2 while N is supposed to be large, using an offline-online procedure. We want the procedure to be robust with respect to N . This means that EIM algorithms can only be carried out to approximate the functions (µ, x) → g s (µ, x) and not the functions (µ, x) → g s (µ, x)Ψ s,t (x). The index t refers to basis functions or couples of basis functions when evaluating the entries of the vector C µ and the matrix A µ in (7.2), see Section 7.4 for various examples.

Applying the four approximation formulae from Table 7.1 to the ς functions g s (µ, x) leads to the construction of ς sets of points x, points µ, matrices B and vector-valued functions q(•). We denote these quantities with an additional index s; for instance, EIM S1 carried out on g s (µ, x) leads to the construction of the vector-valued functions q S1 s (•), of components q S1 s,m : x → q S1 s,m (x), for all 1 ≤ m ≤ d. For simplicity, we assume that each EIM algorithm stops at the same rank d. This leads to the following approximations for Q t (µ):

-S1O1:

(I S1O1 d Q t )(µ) := ς s=1 d m=1 λ S1O1 s,m (µ) Ω q S1 s,m (x)Ψ s,t (x)dx, (7.21)
where λ S1O1 s,m (µ) solves (7.8), -S1O2:

(I S1O2 d Q t )(µ) := ς s=1 d m=1 g s (µ, x S1 s,m ) Ω λ S1O2 s,m (x)Ψ s,t (x)dx, (7.22)
where λ S1O2 s,m (x) solves (7.10), -S2O1:

(I S2O1 d Q t )(µ) := ς s=1 d m=1 q S2 s,m (µ) Ω λ S2O1 s,m (x)Ψ s,t (x)dx, (7.23)
where λ S2O1 s,m (x) solves (7.15), -S2O2:

(I S2O2 d Q t )(µ) := ς s=1 d m=1 λ S2O2 s,m (µ) Ω g s (µ S2 s,m , x)Ψ s,t (x)dx, (7.24)
where λ S2O2 s,m (µ) solves (7.17). We require that the approximation formula for Q t (µ) is (i) online-efficient (in the sense that integrations over Ω are not allowed during the online calls), (ii) computable for all µ ∈ P, and not only on some training points in P, and (iii) nonintrusive (in the sense that the only allowed integration over Ω in the offline stage of the procedure is the quantity Q t (µ) itself). We first moderate the requirement of nonintrusivity by that of weak-intrusivity saying that the only integrations over Ω in the offline stage of the procedure are Ω g s (µ, x)Ψ s,t (x)dx, 1 ≤ s ≤ ς, for all µ ∈ P.

We show in Section 7.3.1 that S1O1 and S2O2 lead to online-efficient procedures, while S1O2 and S2O1 do not. This suggests to discard the approximation procedures S1O2 and S2O1. In Section 7.3.2, we show that S2O2 is weakly intrusive while S1O1 is not, and that S1O1 allows to compute the approximation for all µ ∈ P while S2O2 only allows the computation at the training points. Then, we show in Section 7.3.3 that to the price of an additional linear system to be solved online, S1O1 can be turned to be weakly intrusive as well and that to the same price, the approximation using S2O2 can be computed for all µ ∈ P as well. However, none of the two procedures is nonintrusive. Finally, we show in Section 7.3.4 that following an idea from Chapter 6 and to the price of an additional EIM approximation of which we can control the accuracy, we can devise a nonintrusive procedure for both S1O1 ans S2O2.

Online-efficient procedures

We notice from (7.22) and (7.23) that S1O2 and S2O1 need to integrate solutions to the online problems over Ω. Thus, these procedures are not online-efficient. On the contrary, we notice from (7.21) and (7.24) that for S1O1 and S2O2, the integrals over Ω can be precomputed once and for all during the offline stage and reused in the online call. Therefore, the procedures S1O1 and S2O2 are online-efficient.

Computation between training points and weak intrusivity

In practice, the EIM algorithms are carried out on finite-dimensional subsets Ω trial ⊂ Ω and P trial ⊂ P. This means that the functions q S1 s,m (•) and q S2 s,m (•) are constructed respectively over Ω trial and P trial . We suppose that the functions g s (µ, x) are known for all x ∈ Ω and all µ ∈ P. For instance, for S1O1, λ S1O1 s,m (µ) such that d m=1 (B S1 s ) l,m λ S1O1 s,m (µ) = g s (µ, x S1 s,l ) can be computed for all µ ∈ P, but the approximation (I S1O1 d g s )(µ, x) = d m=1 λ S1O1 s,m (µ)q S1 s,m (x) can be computed only for x ∈ Ω trial , due to the evaluation of q S1 s,m (x). As a result, the approximation of g s (µ, x) is available on -P × Ω trial for S1O1, -P trial × Ω for S2O2. In practice, the integrations over Ω are computed numerically on a mesh, using for instance quadrature formulae. The values of g s (µ, x) are then needed only on the Gauss points corresponding to the quadrature: we can take the set of these Gauss points as Ω trial , so that S1O1 provides an approximation computable at all the needed points. Instead, S2O2 provides an approximation computable only at the training points in P trial and not for all µ ∈ P.

Concerning intrusivity, we see from (7.21) that S1O1 is not weakly intrusive since the integration of the functions q S1 s,m (•) over Ω is required. On the contrary, we see from (7.24) that S2O2 is weakly intrusive.

To sum up, the two online-efficient procedures enjoy the following exclusive properties: -S1O1: intrusive approximation, computable for all µ ∈ P, -S2O2: weakly intrusive approximation, computable only for µ ∈ P trial .

Modification of the online problems

Consider the offline stage EIM S1 described in Algorithm 6. By construction, it is clear that for all 1 ≤ s ≤ ς, Vect 1≤m≤d q S1 s,m (•) = Vect 1≤m≤d g s (µ S1 m , •) . Therefore, there exists a matrix Γ S1 s ∈ R d×d such that, for all 1 ≤ l ≤ d,

d m=1 (Γ S1 s ) l,m q S1 s,m (x) = g s (µ S1 s,l , x), ∀x ∈ Ω trial . (7.25)
Lemma 7. [START_REF] Ammar | A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids[END_REF] The matrix Γ S1 s can be constructed recursively in the loop in k of Algorithm 6 in the following way:

-

k = 1: (Γ S1 s ) 1,1 = g s (µ S1 s,1 , x S1 s,1 ), (7.26
)

-k → k + 1: (Γ S1 s ) k+1,k+1 = (δ S1 k g s )(µ S1 s,k+1 , x S1 s,k+1 ), (Γ S1 s ) l,k+1 = 0, ∀1 ≤ l ≤ k, (Γ S1 s ) k+1,l = κ S1 s,l , ∀1 ≤ l ≤ k, (7.27)
where the vector

κ S1 s is such that k m=1 (B S1 s ) l,m κ S1 s,m = g s (µ S1 s,k+1 , x S1 s,l ), for all 1 ≤ l ≤ k.
Proof. The case k = 1 results from line 5 of Algorithm 6. Suppose that the assertion holds at rank k. Using the definition (7.25) of Γ S1 s at rank (k + 1), for all 1 ≤ l ≤ k and all x ∈ Ω trial , there holds (Γ S1 s ) l,k+1 q S1 s,k+1 (x)+ k m=1 (Γ S1 s ) l,m q S1 s,m (x) = g s (µ S1 s,l , x). Using the same definition at rank k leads to (Γ S1 s ) l,k+1 = 0 for all 1 ≤ l ≤ k. Then, using the same definition for l = k + 1, it is inferred that (Γ S1 s ) k+1,k+1 q S1 s,k+1 (x) + k m=1 (Γ S1 s ) k+1,m q S1 s,m (x) = g s (µ S1 s,k+1 , x). Using line 10 of Algorithm 6, we identify (Γ S1 s ) k+1,k+1 = (δ S1 s,k g)(µ S1 s,k+1 , x S1 s,k+1 ) and k m=1 (Γ S1 s ) k+1,m q S1 s,m (x) = (I S1 k g s )(µ S1 s,k+1 , x). From (7.7)-(7.8), there holds

k m=1 (Γ S1 s ) k+1,m q S1 s,m (x) = k l=1 k m=1 (B S1 s ) -1 m,l q S1 s,m (x)g(µ S1 s,k+1 , x S1 s,l ). (7.28) Therefore, (Γ S1 s ) k+1,m = k l=1 (B S1 s ) -1 m,l g s (µ S1 s,k+1 , x S1 s,l
), finishing the proof. ⊓ ⊔ ♦ Proposition 7. [START_REF] Anderson | An Introduction to Multivariate Statistical Analysis[END_REF] To the price of an additional online problem, the procedure S1O1 is weakly intrusive.

Proof. From Lemma 7.4, q S1 s,m (x) = d l=1 (Γ S1 s ) -1 m,l g s (µ S1 s,l , x) holds for all 1 ≤ m ≤ d, so that (7.21) can be written

(I S1O1 d Q t )(µ) = ς s=1 d m=1 λ S1O1 s,m (µ)σ m,s,t , (7.29)
where the coefficients σ m,s,t , for all 1 ≤ m ≤ d, solve the linear system

d l=1 (Γ S1 s ) m,l σ s,l,t = Ω g s (µ S1 s,m , x)Ψ s,t (x)dx, (7.30)
and λ S1O1 s,m (µ) solves (7.8). ⊓ ⊔ ♦

Consider now the offline stage EIM S2 described in Algorithm 7. There exists a matrix

Γ S2 s ∈ R d×d such that d m=1 (Γ S2 s ) l,m q S2 s,m (µ) = g s (µ, x S2 s,l ), ∀µ ∈ P trial . (7.31)
Notice that even if the function q S2 s,m (•) is constructed on P trial , the above expression allows one to extend the definition of q S2 s,m (•) to P. In the same fashion as Lemma 7.4, we obtain the following result.

Lemma 7.6 The matrix Γ S2

s can be constructed recursively in the loop in k of Algorithm 7 in the following way:

-

k = 1: (Γ S2 s ) 1,1 = g s (µ S2 s,1 , x S2 s,1
), (7.32)

-k → k + 1: (Γ S2 s ) k+1,k+1 = (δ S2 k g s )(µ S2 s,k+1 , x S2 s,k+1 ), (Γ S2 s ) l,k+1 = 0, ∀1 ≤ l ≤ k, (Γ S2 s ) k+1,l = κ S2 s,l , ∀1 ≤ l ≤ k, (7.33)
where the vector

κ S2 s is such that k m=1 (B S2 s ) m,l κ S2 s,m = g s (µ S2 s,l , x S2 s,k+1
), for all 1 ≤ l ≤ k.

Proposition 7.7

To the price of an additional online problem, the approximation of Q t (µ) using the procedure S2O2 can be computed between training points.

Proof. Let µ ∈ P. Compute first q S2 s,m (µ) solving (7.31), and compute then λ S2O2 s,m (µ) solving (7.17). Then, (I S2O2

d Q t )(µ) = ς s=1 d m=1 λ S2O2 s,m (µ) Ω g s (µ S2 s,m , x)Ψ s,k (x)dx is computable for all µ ∈ P. ⊓ ⊔ ♦
To sum up, with the above-discussed modifications of the online problems, the two onlineefficient procedures S1O1 and S2O2 are now weakly intrusive and computable for all µ ∈ P. Unfortunately, none of the two procedures is nonintrusive in the sense defined above.

The nonintrusive procedures

The observation made in Chapter 6 is that the expressions (7.21) and (7.24) are linear forms in a vector z ∈ R ςd , whose components, denoted by z p (µ), 1 ≤ p ≤ ςd, contain all the µ-dependencies. Notice that this is still the case when considering the additional online problems of Section 7.3.3. For instance, consider the procedure S1O1 and define

z p (µ) :=                λ S1O1 1,m (µ), 1 ≤ p ≤ d, m = p, λ S1O1 2,m (µ), 1 + d ≤ p ≤ 2d, m = p -d, . . . λ S1O1 ς,m (µ), 1 + (ς -1)d ≤ p ≤ ςd, m = p -(ς -1)d. (7.34) Q t,p :=                        Ω q S1 1,m (x)Ψ 1,t (x)dx, 1 ≤ p ≤ d, m = p, Ω q S1 2,m (x)Ψ 2,t (x)dx, 1 + d ≤ p ≤ 2d, m = p -d, . . . Ω q S1 ς,m (x)Ψ ς,t (x)dx, 1 + (ς -1)d ≤ p ≤ ςd, m = p -(ς -1)d. (7.35)
Then,

(I S1O1 d Q t )(µ) = ςd p=1 z p (µ)Q t,p . (7.36)
The idea consists in applying another EIM to z p (µ) seen as the two-variable function (µ, p) → z p (µ). Here again, the four approximation procedures S1O1(z), S1O2(z), S2O1(z), and S2S2(z) are possible for the approximation of z p (µ), where now p plays the role that x played in Section 7.2, and where we indicate specifically in the notation that these procedures are related to the approximation of z p (µ). We consider an approximation of z p (µ) using either the procedure S1O1(z) or the procedure S2O2(z) (see Remark 7.8 below). We then inject the approximation of z p (µ) in the right-hand side of (7.36). This leads to -S1O1(z):

(I S1O1 d Q t )(µ) ≈ d z m=1 β S1O1(z) m (µ) ςd p=1 q S1(z) m (p)Q t,p , (7.37)
where β

S1O1(z) m (µ) solves d z m=1 B S1(z) l,m β S1O1(z) m (µ) = z p S1(z) l (µ), ∀1 ≤ l ≤ d z , ( 7.38)
-S2O2(z):

(I S2O2 d Q t )(µ) ≈ d z m=1 β S2O2(z) m (µ) ςd p=1 z p (µ S2(z) m )Q t,p , (7.39)
where β

S2O2(z) m (µ) solves d z m=1 (B S2(z) ) t l,m β S2O2(z) m (µ) = q S2(z) l (µ), ∀1 ≤ l ≤ d z . (7.40)
In the above equations, we now denote the solution to the online problem of the EIM procedures to approximate z p (µ) by β. We keep the same notation for the constructed matrices B and vectorvalued functions q m (•), and the selected points µ m , while we introduced in (7.38) the indices p m selected by the EIM procedures to approximate z p (µ).

With S2O2(z), (7.39) can be rewritten as

Q t (µ) ≈ d z m=1 β S2O2(z) m (µ)Q t (µ S2(z) m ), (7.41)
which provides a nonintrusive procedure for the approximation of Q t (µ). As explained in Section 7.3.3, we need to solve an additional online problem to retrieve this approximation for all µ ∈ P. In the same fashion, we can prove that with S1O1(z), we can derive from (7.37), to the price of an additional online problem, another nonintrusive formula for the approximation of Q t (µ), which is directly available for all µ ∈ P.

Remark 7.8 Like before, we want to be able to precompute as many terms as possible in the offline stage. To be able to precompute summations of size ςd in (7.37) and (7.39), we can discard S1O2(z) and S2O1(z). Note however that in general, ςd is not as large as N .

We can repeat this work for the case where the EIMs on g s (µ, x) are performed with S2O2, and achieve the same kind of conclusions. Actually, we have four possible choices for the complete procedure at our disposal, namely the product of the choices S1O1 and S2O2 for the approximation of g s (µ, x), times the choices S1O1(z) and S2O2(z) for the approximation of z p (µ). The overall choice is driven by (approximate) interpolation properties we wish our formula to exhibit and by avoiding additional online problems if a certain choice permits it. Other performance considerations can be considered, for instance if #P trial ≫ #Ω trial , one may prefer to construct functions q defined on Ω trial rather than on P trial . In our simulations, we chose S1O1 for the EIM on g s (µ, x) and S2O2(z) for the EIM on z p (µ). For the EIM on g s (µ, x), S1O1 enables the λ S1O1 s,m (µ) involved in the online problems to be computed for all µ ∈ P, but the weak intrusivity allowed by S2O2 is not useful for the approximation of g s (µ, x). For the EIM on z p (µ), S2O2(z) provides (7.41), a nonintrusive procedure for the approximation of Q t (µ), while S1O1(z) allows to compute the approximation of Q t (µ) with no additional online problem. When nonintrusivity is crucial for practical purposes, the additional online problem needed for S1O1(z) to be nonintrusive is always required, while the additional online problem needed for S2O2(z) to be computed is only needed for µ ∈ P\P trial . Hence, the choice S1O1/S2O2(z) requires no additional problem in the offline stage (when the approximation operators are evaluated), and a single additional online problem for computing the approximation for µ ∈ P\P trial . Table 7.2 presents the number of additional online problems to be solved when considering an offline call for the approximation operators (i.e. for µ ∈ P trial ) and an online call (i.e. for µ ∈ P\P trial ). Among the two choices that minimize the computation cost, S1O1/S2O2(z) was chosen since it provides an approximation for Q t (µ) that is interpolant with respect to µ at the limit d z = ςd.

S1O1/S1O1(z) S1O1/S2O2(z) S2O2/S1O1(z) S2O2/S2O2(z) Offline stage 1 0 1 0 Online stage 1 1 1 1 Table 7.2.
Number of additional online problems for each evaluation of the approximation operator in the offline and online stages, for the four EIM combinations, when a nonintrusive procedure is required. Offline stage refers to an approximation for µ ∈ P trial and online stage refers to an approximation for µ ∈ P\P trial

Nonintrusive RBM for aeroacoustic problems

In this section, we use the nonintrusive formula (7.41) for the approximation of the matrix and the right-hand side of discrete variational formulations arising in aeroacoustic problems modelled by the Helmholtz equation or the convected Helmholtz equation. The finite element method (FEM) and the boundary element method (BEM) are used to obtain the matrix and right-hand side of the problem. Both quantities are of the form Q t (µ) as defined in (7.20). For the matrix, the index t in Ψ s,t (x) refers to the product of two finite element basis functions, while for the right-hand side, the index t refers to the finite element functions themselves. Using the nonintrusive approximation (7.41), we only need to compute matrix-vector products involving A µ and scalar products to precompute in the offline stage all the quantities needed to construct efficiently the reduced problem and compute the error bound in the online stage.

Implementation of the RBM

For simplicity, we take the Euclidian norm of the discretized vectors in the computation of the a posteriori error bound in the RBM, to avoid dealing with the computation of dual norms. When considering reduced basis strategies for the (convected) Helmholtz equation approximated by BEM with the frequency as a parameter, the approximate affine decomposition (7.41) has a quite large number of terms. When applying the Successive Constraints Methods (SCM, see [START_REF] Huynh | A successive constraint linear optimization method for lower bounds of parametric coercivity and inf-sup stability constants[END_REF]) as an online-efficient procedure for computing the inf-sup constant, we have to solve constrained linear optimization problems, with a number of constraints proportional to the square of the number of terms in the decomposition (7.41). For simplicity, we do not look for an online-efficient way to compute the inf-sup constant. In Sections 7.4.2 and 7.4.3, we compute a single value of this constant (for centered values of the parameters) and use it for any error bound evaluation. discretization P trial ) is of the order of 10 -6 , the online stage takes 2.8 × 10 -3 s to compute a reduced solution and the error bound, while the full direct problem is solved in about 30 s in parallel on 4 processors, which corresponds to an acceleration factor of 10 4 .

To formulate the optimization problem, we consider as an illustration a set of values f r i , 1 ≤ i ≤ p, for the frequency of the source and we denote by J i (µ 1 , µ 2 , µ 3 ), the quantity of interest computed for the frequency f r i of the source and depending on the three impedance coefficients. Consider the following cost function:

(µ 1 , µ 2 , µ 3 ) → J (µ 1 , µ 2 , µ 3 ) := p i=1 α i J i (µ 1 , µ 2 , µ 3 ) + h(µ 1 , µ 2 , µ 3
). The goal of the study is to find the values of the impedance coefficients that minimize the cost function. With such a cost function, we can minimize the far-field acoustic pressure scattered by the object, taking into account that some frequencies are more harmful than others for the human ear (through the weights α i ), and that some treatments of the object surface to modify the impedance coefficients are more expensive than others (through the function h). To illustrate, we choose p = 20, α i = 2 for 1 ≤ i ≤ 7, α i = 1 for 8 ≤ i ≤ 13 and α i = 3 for 14 ≤ i ≤ 20, and h(µ 1 , µ 2 , µ 3 ) = 1 6 (0.2µ -0.5 1 + 0.3µ -0.8 1 + 0.5µ -1 1 ) -8. The cost function is computed for 1000 values of the impedance coefficients (each coefficient being sampled by 10 values). Notice that for each evaluation of the cost function, we need to compute the solution of the aeroacoustic problem for 20 values of the frequency. Using the online stage, the minimum of the cost function over this sample of impedance coefficients is 0.366, reached for (µ 1 , µ 2 , µ 3 ) = (2.8, 1, 1.9), and is found in less than 24 s. Figure 7.2 shows a screenshot of a java applet computing the quantity of interest at 50 values of the frequency, and at values of the impedance coefficients selected by the user.

An uncertainty quantification problem for an object surrounded by a potential flow

Consider an ellipsoid with major axis directed along the z-axis. This object is included inside a larger ball, see Figure 7.3. The external border of the ball after discretization is denoted by Γ ∞ . The complement of the ellipsoid in the ball is denoted by Ω -. A potential flow is precomputed around the ellipsoid and inside the ball, such that the flow is uniform outside the ball, of Mach number 0.3 and directed along the z-axis. An acoustic monopole source lies upstream of the object, on the z-axis as well.

The considered formulation is a coupled BEM-FEM formulation. It consists in (i) applying a change of variables to transform the convected Helmholtz equation into the classical Helmholtz equation outside the ball, in order to apply a standard BEM on Γ ∞ , and (ii) stabilizing the formulation to avoid resonant frequencies associated with the eigenvalues of the Laplacian inside the ball of boundary Γ ∞ . Consider the product space

H := H 1 (Ω -) × H -1 2 (Γ ∞ ) × H 1 (Γ ∞ ) with inner product (Φ, λ, p) , Φ t , λ t , p t H := Φ, Φ t H 1 (Ω -) + λ, λ t H -1 2 (Γ∞) + p, p t H 1 (Γ∞) . The weak formulation is: Find (Φ, λ, p) ∈ H such that ∀ Φ t , λ t , p t ∈ H,
:= r ς + γ 2 ∞ P 2 -γ 4 ∞ M 2 ∞ , V := r ς + γ 2 ∞ P N M -γ 3 ∞ M ∞ , Ξ := rN ON with r := ρ ρ∞ , ς := c∞ c , γ ∞ := 1 √ 1-M 2 ∞ , P := M • M ∞ , N := I + C ∞ M ∞ M T ∞ , O := I -M M T , and C ∞ := γ∞-1 M 2 ∞ .
In the above notation, the subscript ∞ is used for quantities outside the ball, ρ is the density of the flow, c is the speed of sound when the flow is at rest and M = v c , where v is the velocity of the flow. The operators γ 0 and γ 1 are Dirichlet and Neumann traces on the coupling surface Γ ∞ . We refer to Chapter 3 for more details on this formulation and its well-posedness. The considered finite-dimensional approximation of (7.47) has 1711 unknowns.

The potential flow, represented in the right panel of Figure 7.3, is part of the data for the problem. We perturb this flow uniformly in space. Although the boundary condition on the solid surface Γ and transmission condition on Γ ∞ are violated as soon as the perturbation of the flow is nonzero, the present study can be viewed as a first step towards quantifying uncertainties on the potential flow and their impact on a quantity of interest. The flow perturbation takes the form δM = µ 1 e x + µ 2 e y + µ 3 e z . The quantity of interest is the acoustic pressure at a point located on the axis of symmetry, downstream of the object. The parameters of the problem are the frequency of the source, and the magnitude of the uniform perturbations of the potential flow in each Cartesian direction. The frequency varies from 487 to 1082 Hz, and the magnitude of the uniform perturbations of the flow varies from 0 to 0.1. Denote by µ the wavenumber of the source (so that the frequency of the source is 340µ 2π in the air at rest), and by µ 1 , µ 2 , µ 3 the three magnitudes of the perturbation. A goal-oriented reduced basis method is carried out to select a basis of n = 20 truth solutions using the nonintrusive formula (7.41) to approximate the matrix of the problem, the right-hand side of the direct problem, and the right-hand side of the adjoint problem corresponding to our quantity of interest. For the matrix, the approximation procedure S1O1 is applied to g(µ, r) := exp (iµr) , r = |x -y| , x, y ∈ Γ ∞ , (7.50) and the procedure S2O2(z) is applied to

z p (µ, µ 1 , µ 2 , µ 3 ) :=                                                      λ S1O1 m (µ), 1 ≤ m ≤ d, p = m, µλ S1O1 m (µ), 1 ≤ m ≤ d, p = m + d, µ 2 λ S1O1 m (µ), 1 ≤ m ≤ d, p = m + 2d, 1, p = 3d + 1, µ, p = 3d + 2, µ 2 , p = 3d + 3, µ 2 µ 3 , p = 3d + 4, µ 2 µ 2 3 , p = 3d + 5, µµ i , 1 ≤ i ≤ 3, p = 3d + 5 + i, µµ i µ 3 , 1 ≤ i ≤ 3, p = 3d + 8 + i, µ i µ j , 1 ≤ i, j ≤ 3, p = 3d + 11 + i + 3(j -1), (7.51)
where these parameter dependencies have been identified upon injecting M → M + δM in (7.47), while using that M ∞ is collinear to e z . For the right-hand side of the direct and dual problems, the approximation procedure S1O1 is applied to

g(µ, x) := exp (iµ|x -x 0 |) , x ∈ Γ ∞ , (7.52)
where x 0 is respectively the position of the source and the point where the quantity of interest is computed, and the approximation procedure S2O2(z) is applied to

z p (µ) := λ S1O1 m (µ), 1 ≤ m ≤ d, p = m µλ S1O1 m (µ), 1 ≤ m ≤ d, p = m + d. (7.53)
The EIM algorithms are carried out with d = 13 and d z = 25 for the matrix, and d = 13 and d z = 18 for the right-hand side of the direct and dual problems. Over the considered parameter values, the relative error for the three nonintrusive formulae is of the order of 10 -12 (in Frobenius norm for the matrix and Euclidian norm for the vectors). The maximum error bound (over a discretization P trial ) is of the order of 10 -7 , the online stage takes 2.8 × 10 -3 s to compute a reduced solution and the error bound, while the full direct problem is solved in about 14 s, which corresponds to an acceleration factor of 5 × 10 3 .

To illustrate, we suppose that the perturbation of the potential flow is modelled by random variables: the law of µ 1 is a truncated Gaussian, that of µ 2 is a uniform law, and that of µ 3 is a truncated log-normal low. The goal is to compute the probability density function of the quantity of interest. Figure 7.4 shows a screenshot of a java applet computing an histogram of the values taken by the quantity of interest, at a frequency selected by the user.

A scalable RBM implementation applied to an industrial test case of an impedant aircraft in the air at rest

In BEM implementations for the Helmholtz equation, the Fast Multipole Method (FMM) allows one to approximately compute matrix-vector products, and then approximately solve linear systems using iterative methods, in complexity scaling with n log n, where n denotes the With n = 30 basis vectors selected by the greedy algorithm, the relative error between the direct solution and the reduced solution, in Euclidian norm, at the value of the parameters that maximizes the error bound, is less than 3%. Notice that the procedure is scalable with respect to the number of available CPUs: the matrix-vector products in FMM and the exploration of P trial by the greedy algorithm, which are the two steps with high computational complexity (dependent on the number of unknowns and on the cardinality of P trial ), are parallel. Therefore, the procedure is expected to highly benefit from large clusters.

We now consider the finer mesh. Each time a vector U µ j is added to the reduced basis, we have to compute the d z = 50 matrix-vector products

A µ S2(z) m U µ j , 1 ≤ m ≤ d z , where the µ S2(z) m
's are the values of the parameter in the nonintrusive approximation formula (7.41). Therefore, in addition to the resolution of the direct problem, 50 matrices have to be assembled at each step of the greedy algorithm, which is time-consuming. However, once a matrix is constructed, it is relatively cheap to compute many matrix-vector products with the same matrix. Hence, a greedy algorithm is not considered on the finer mesh, but the values of the parameters selected by the greedy algorithm on the coarser mesh are directly used to build the reduced basis. This way, the 50 matrices are constructed once, and the 30 matrix-vector products (corresponding to n = 30 values of the parameter selected by the greedy algorithm on the coarser mesh) are carried out at once. The simulations have been performed on a laptop with a quadricore CPU, and 4 Go of RAM. The formula (7.41) allows us to directly use the FMM. Without the FMM, this simulation on this computer would have been impossible, since one matrix needs 60 Go to be stored. An approach consisting in computing and storing the 50 matrices of the decomposition would need 3 To of memory.

The online stage takes 1.5 × 10 -2 s to compute a reduced solution and the error bound, while the full direct problem is solved in about 40 minutes, which corresponds to an acceleration and the reduced solution, in Euclidian norm, is 1%. On the array of 1681 points located behind the aircraft, the relative error for the scattered acoustic field is 1.4%. Figure 7.7 shows the corresponding acoustic pressure fields, and the difference between the reduced basis and direct solutions.

Conclusion

In this work, we derived a nonintrusive procedure for the reduced basis method. Its implementation is relatively simple: it has been successfully and easily applied to the approximation of various matrices and right-hand sides. In particular, this procedure allows for the direct use of advanced linear algebra tools, since we are only dealing with quantities already assembled by the computational code at hand.

Introduction

In the civil aircraft industry, one of the main stakes is fuel efficiency. The use of composite materials, to replace aluminum alloys, enables the manufacturers to lighten the plane while keeping the required mechanical properties. However, these materials present lower thermal conductivity, leading to new air conditioning problems. The goal of the present work is to develop fast tools to compute the temperature in an aircraft cabin in flying conditions, with presence of heat sources: mainly the electronic components.

A closer look will be taken at two problems: -the passenger comfort in which case the output of interest is the temperature in the cabin; -the equipment failure in which case the output of interest is the maximum of the temperature in the electronic components. After presenting three levels of modeling for the physical phenomena, we present numerical simulations of the model providing the best trade-off between physical realism and computational accuracy. Then, a reduced basis approach will be developed for the electronic component problem to further speed up the computations.

Physical modeling

In both cabin and equipment problems, the flow as well as the temperature have to be computed. We first introduce three models of increasing complexity to solve this physical prob-lem. The standard continuum model for natural convection phenomenon is the compressible Navier-Stokes system (CNS). This model present both theoretical and numerical difficulties in the sense that the equations of conservation are strongly coupled and nonlinear. We choose to consider a hierarchy of simplifications of (CNS).

A hierarchy of models

Consider a bounded domain Ω ⊂ R 2 representing the cabin or the electronic components. The domain Ω is split in two parts:

Ω = Ω solid ∪ Ω air ,
where Ω solid stands for solid structures in the cabin or in the electric component. Thus, the velocity field is considered as non-zero on Ω \ Ω solid and extended by 0 on Ω solid . As a first simplification, we will consider the Boussinesq equations, neglecting density variations except in the body force so that the fluid is divergence free. The coupling between the velocity and temperature fields appears in the body force terms in the equations of the fluid and the advective term in the heat equation. This model expresses conservation of momentum and mass of the fluid coupled to the heat equation. The unsteady equations of conservation reads, for a time t S > 0, for all t ∈ [0, t S ]:

             ρ 0 ∂u ∂t + u • ∇u = ρ 0 1 - T -T 0 T 0 g -∇p + η∆u in Ω air , div (u) = 0 in Ω air , ρ 0 c p (x) ∂T ∂t + u • ∇T = div (κ(x)∇T ) + Q(x) in Ω, ( 8.1)
Here, u denote the velocity field, p the pressure and T the temperature. T 0 is a reference temperature (300 Kelvin), ρ 0 is the air density at temperature T 0 , g the gravity constant, η the air dynamic viscosity. Then, c p and κ are space dependent discontinuous functions and represent the heat capacity and the heat conductivity of the considered medium (e.g air, aircraft structure). Eventually, Q is a space dependent function representing a source term in the heat problem. In the following, the function κ, c p , Q have the form

κ(x) = i κ i 1 Ω i , c p (x) = i c i p 1 Ω i , Q(x) = i Q i 1 Ω i , ( 8.2)
where indices i refer to the air and different solid parts depending on the considered problem (aircraft structure, electronic component part,...).

If we assume that the variations of temperature do not modify the velocity field, we can decouple the fluid and heat problems. Moreover, we consider the fluid at steady state thereby neglecting any feedback of the temperature on the convection of the air. The conservation equations writes for a time t S > 0, for all t ∈ [0, t S ]:

         ρ 0 u NS • ∇u NS = ρ 0 g -∇p + η∆u NS in Ω air , div (u NS ) = 0 in Ω air , ρ 0 c p (x) ∂T ∂t + u NS • ∇T = div (κ(x)∇T ) + Q(x) in Ω. (8.
3)

The last level of modeling consists, as in the previous case, in taking a stationary regime and in decoupling the fluid and heat problems for the same reasons. The next simplification is to consider a basic model for the fluid equation, namely a potential flow. All the phenomena induced by the viscosity are not captured by this type of model (recirculation zones, boundary layers).

The conservation equations writes:

-∆ψ = 0 in Ω air , ρ 0 c p (x) (∂ t T + u pot • ∇T ) = div (κ(x)∇T ) + Q(x) in Ω. (8.4)
Here, u pot = ∇ψ is the velocity field associated with the potential ψ. All the three models are supplemented with initial and boundary conditions. A nice consequence for the numerical calculations is that for the two last systems (8.3) and (8.4), a single evaluation of the velocity field is required.

Geometry and boundary conditions

Now, we describe the different boundary conditions that we consider in the numerical experiments. Assume that the boundary of Ω is partitioned as follow:

∂Ω = Γ in ∪ Γ out ∪ Γ wall .
The portions Γ in/out represent parts of the domain where there are exchange of air (fans and evacuations) and Γ wall are solid adiabatic walls. We enforce non-penetration and no-slip boundary conditions for the fluid flow on the walls: u = 0 on Γ wall , (8.5) and an inflow of air is imposed on Γ in through a Dirichlet boundary condition:

u • n = u in on Γ in , η (∇u • n) • τ = 0 on Γ in , ( 8.6)
where u in is a scalar function that we will specify for the numerical calculations and τ a tangent vector to Γ in . In the case of systems (8.1) and (8.3), we enforce natural boundary condition, requiring:

η (∇u • n) = pn on Γ out . (8.7)
In the case of system (8.4), in order to have a well posed problem, we impose exact conservation of mass enforcing:

u pot • n = 0 on Γ out , ( 8.8)
so that the Poisson problem for the potential flow has to be solved with non-homogeneous Neumann boundary conditions. With this setting, the potential is determined up to an additive constant. For the numerical experiments we fix Ω air ψ = 0. This condition ensure the wellposedness of the problem.

For the temperature field, we assume Dirichlet boundary conditions in the inflow part. Since we assume that the wall are adiabatic, the boundary conditions for the temperature reads:

T = T in on Γ in , ∇T • n = 0 on Γ out ∪ Γ wall .
(8.9)

Time and space discretization

We now describe the numerical implementation used to solve the previous systems of conservation equations. We describe the numerical algorithm used to solve unsteady Navier-Stokes equations and the heat equation. We use those algorithms in an iterative process to solve the Boussinesq equations. We start by describing the Navier-Stokes solver. The discretization is based on finite elements in space and implicit Euler scheme in time. Let δt be the time step, taken to be constant for simplicity. We denote by t n = nδt the n-th discrete time. We introduce V h (Ω air ) 2 and M h (Ω air ) the finite elements spaces for velocity and pressure. We define the fluid problem at time t n+1 for a temperature T by: given

u n ∈ V h (Ω air ) 2 , we seek (u n+1 , p n+1 ) ∈ V h (Ω air ) 2 × M h (Ω air ) such that for all (v, q) ∈ V h (Ω air ) 2 × M h (Ω air ), Fluid(n+1, T ) :=                  1 δt Ω air ρ 0 u n+1 • v dx + Ω air ρ 0 u n+1 ∇u n+1 • v dx + η Ω air ∇u n+1 : ∇v dx - Ω air p n+1 div (v) dx = 1 δt Ω air ρ 0 u n • v dx + Ω air ρ 0 1 - T -T 0 T 0 g • v dx, Ω air qdiv (u n+1 ) dx = 0.
(8.10) Let us denote ũ the extension by zero of u on Ω. We search the temperature in the same finite element space than the velocity components. We define the heat problem at time t n+1 for a fluid velocity ũ by: given

T n ∈ M h (Ω), we seek T n+1 ∈ M h (Ω) such that for all Θ ∈ M h (Ω): Heat(n + 1, ũ) :=        1 δt Ω ρ 0 c p (x)T n+1 Θ dx + Ω ρ 0 c p (x) ũ • ∇T n+1 Θ dx+ Ω k(x)∇T n+1 • ∇Θ dx = Ω Q(x)Θ dx + 1 δt Ω ρ 0 c p (x)T n Θ dx. (8.11)
The variational formulation for steady Navier-Stokes equations and equation can be straightforwardly obtained from (8.10) and (8.11). Note that the time scheme for the system (8.10) is nonlinear and we resort to a Newton-Raphson algorithm to solve this nonlinear problem at each time step. In the case of the potential flow, one can resort to a variational formulation of the Poisson equation with Neumann boundary conditions in conjunction with a Lagrange multiplier to ensure that Ω air ψ = 0. For instance, we take the inf-sup stable pair of discrete finite elements spaces V h (Ω air ) 2 × M h (Ω air ) = (P 2 ) 2 × P 1 for the fluid equations and M h (Ω) = P 1 for the heat equation. Let us depict the iterative process allowing to solve the Boussinesq system: Initialize with u 0 ∈ V h (Ω air ) 2 and T 0 ∈ M h (Ω); For all n ≥ 0, (i) solve Fluid(n + 1, T n ) to get u n+1 ; (ii) solve Heat(n + 1, u n+1 ) to get T n+1 ; All calculations are performed using the finite element solver FreeFEM++ (see [START_REF] Hecht | freeFEM++[END_REF]). As a matter of illustration, the computationnal code was tested on a simple configuration: consider a square box, with initial state uniform temperature and fluid at rest. A source term Q for the heat problem is localized in the lower part of the box. As we can see in Figure 8.1, the temperature dependent density in the gravitational term lightens hot air and weighs down cold air. This enable a nonzero velocity field to arise, and a convective dissipation of the heat. for all, and is assumed not to depend on the temperature. The air viscosity produces recirculation zones. Without Boussinesq effect, the cooling down is only ensured by diffusion and convection from the precomputed velocity field. These two effects are too low, and the temperature reaches very high values.

-Unsteady decoupled potential/heat: See Figure 8.6. The precomputed flow is very simple, viscosity is neglected. Recirculation zones are not obtained and therefore the cooling of the components (away from the air conditioning main stream) is even worse. Diffusion and convection are way too low, and the temperature diverges. For this situation, only the unsteady Boussinesq can capture all the physical phenomena, and is therefore the model to be considered. In the two other models, the convection brought by the Boussinesq term is not captured, and the components are not cooled down enough. Note that there do not seem to be any steady state to reach for this problem. We simulated up to 40 min, and the velocity and temperature fields appear to reach a periodic in time behavior.

In the present section, we simply modeled the presence of electronic components by a constant surfacic heat source term. We will now develop a model to simulate the velocity and the temperature inside the electronic component.

The electronic component

Consider a 2D section of an electronic component (see Figure 8.7). The green area represent the support board for the red-colored integrated circuits. The blue zone is filled with air, pushed from the bottom part by a fan, and leaving the box through the top part. The red components will heat up by Joule effect while functioning. Periodic boundary conditions are enforced at Γ per .

In this case, the steady decoupled incompressible Navier-Stokes / heat model gives the same result as the long time Boussinesq (see Figure 8.8). Actually, the Péclet number is large (≈ 10 4 ), indicating that convective effects are dominant. Moreover, the flux is strongly guided inside a channel, which is different from the previous case where air was blown inside a large volume at rest. Forced convection dominates convection induced by local thermal fluxes. The unsteady decoupled potential / heat is not satisfactory: the absence of boundary layer does not

Computation aspects and construction of the basis with a greedy algorithm

Offline stage: precomputation and greedy algorithm

The certified RB method consists in iteratively construct a basis with solutions of the considered problem, computed at particular values of the parameters µ.

Ideally, we would like to choose a tolerance ǫ for the error |s RB, * RB µ s F E µ | and make use of the error estimate ∆ µ to construct a low dimensional basis where the error is guaranteed to be smaller than ǫ for all µ in the parameter space D. Unfortunately, such a result does not exist. One can refer to recent work by Buffa, Maday, Patera, Prud'homme and Turinici [START_REF] Buffa | A priori convergence of the greedy algorithm for the parametrized reduced basis method[END_REF] for theoretical results on greedy algorithms convergence performances.

In the present setting, we have an affine dependence of the operator on the parameters (thanks to equations (8.2)): for all w, v ∈ M h ,

a µ (w, v) = a 0 (w, v) + n i=1 µ i a i (w, v) = n i=0 µ i a i (w, v).
In the same fashion, the residuals are written: for all w, v ∈ M h ,

           g µ (w, v) = -l(v) + n i=0 µ i a i (w, v), g * µ (w, v) = l(w) + n i=0 µ i a i (w, v),
where the a i do not depend on µ.

Define now G

i , G * i : ∀w, v ∈ M h , ∀i = 1, ...n, (G i w, v) H 1 0 = a i (w, v) and (w, G * i v) H 1 0 = a * i (w, v
). Take g 0 and g * 0 , solutions of (g 0 , v) H 1 0 = -l(v) and (w, g * 0 ) H 1 0 = l(w) (note that in this case, g 0 = -g * 0 ). In practice, we only need to compute G i w and G * i v for given vectors w, v ∈ M h . This is done by solving variational problems. Consider for instance

G i w ∈ M h : Let w ∈ M h . Find u ∈ M h such that ∀v ∈ M h : Ω ∇u • ∇v = a i (w, v), w being fixed, v -→ a i (w, v) is a continuous linear form, and G i w := u.
The other quantities defined above are computed in the same way. This will speed up the online stage since lots of quantities will be precalculated during the offline stage, letting to the online stage simple low dimension algrebra operations (see section 8.3.3). Take a finite set D trial ⊂ D. The greedy algorithm is given in Algorithm 8.

Note that we did not explain how to chose the finite set D trial . In our case, we simply discretized each direction of D with a constant step: see Figure 8.10. The efficient evaluation of ∆ µ is explained in the next section 8.3.3.

Conclusion

We developped different models for solving the velocity and the temperature of the air under different conditions. The full Boussinesq model yields satisfactory results in the cases we considered, whereas decoupled incompressible Navier-Stokes / heat was satisfactory for the electronic component case (forced convection in a pipe).

We then tested a certified RB method for the heat problem with non homogeneous convection. Our code makes use of an a posteriori error estimate to build a basis iteratively using a greedy algorithm. The affine parametric assumption allows to precompute many terms, and online calls are reduced to add and inverse low dimensional matrices. This enables us to drastically speed up the resolution in the electric component, with a physically satisfying model, and controlled approximation errors. where λ J (µ) can be computed for all µ ∈ P trial using the online stage of the EIM, and µ J ∈ P trial , 1 ≤ J ≤ D is a set of interpolation points. We suppose in this section that d = #P trial , so that, for all µ ∈ P trial , where t J := d i=1 b i λ J (µ i ) is independent of µ. Then, using the symmetry of K,

f1 (µ) = D J=1 K(µ, µ J )t J ≈ D J=1 D J ′ =1 λ J ′ (µ)K(µ J ′ , µ J )t J = D J ′ =1 λ J ′ (µ) D J=1 K(µ J , µ J ′ )t J ≈ D J ′ =1
λ J ′ (µ) f1 (µ J ′ ).

(A.7)

Proposition A.2 For all 1 ≤ J ′ ≤ D, f1 (µ J ′ ) = f (µ J ′ ).

Proof. Using the interpolation property of the EIM, there holds, for all µ ∈ P trial , for all 1 ≤ J ′ ≤ D, In this case, the interpolation is computed by a linear combination of values of f . Providing that the values λ J ′ (µ) are of order 1, the formula (A.10) is then mush less sensitive to round-off errors than the formula (A.1).

Remark A. [START_REF] Amiet | The aerodynamic noise of small-perturbation subsonic flows[END_REF] We are not constrained by the use of the interpolation points selected by EIM. The same method can be derived on any set of interpolation points µ J .

A.3 Simple numerical illustration

We apply the two previously defined interpolation method to the function µ → sin(10µ), 0 ≤ µ ≤ 1. P trial is obtained by a uniform discretization of (0, 1) using 501 points, and the same set of interpolation points is used for both methods, namely the points selected by EIM. The gaussian kernel K(µ, µ ′ ) := exp (µ-µ ′ ) 2 the numerical pollution due to the ill-conditioning of the matrix K increases with the number of interpolation points for the first method. For the second method, no such pollution is observed, and the error at the interpolation points is always zero in all the simulations.

B Étude d'un modèle d'incertitude non paramétrique

Dans cette annexe, nous nous intéressons à la relation entre le modèle mécanique des plaques constitutives du fuselage d'un avion et la puissance acoustique transmise de l'extérieur de l'avion vers les passagers. Nous considérons la modélisation des plaques mal connue ou aléatoire. Dans cette situation, une première possibilité est de supposer que les coefficients classiques de modélisation (rigidité, vitesse des ondes) sont des variables aléatoires dont nous imposons une loi, basée sur l'expérience ou l'expertise. Une deuxième possibilité, que nous développons dans cette annexe, consiste à modéliser le comportement aléatoire des plaques par une approche probabiliste non paramétrique où les aléas sont quantifiés par un scalaire δ, qui représente une variance de l'opérateur mécanique complet. Enfin, nous considérons une fonction coût dépendant de δ et nous étudions l'optimisation de cette fonction coût sous une contrainte en probabilité de dépassement de seuil de la norme en énergie de la solution du système mécanique (qui correspond à l'énergie acoustique transmise à travers une plaque du fuselage). Dans le cas de plusieurs plaques reliées à leurs bords, on peut s'intéresser au problème d'allocation des incertitudes entre les plaques, sous la même contrainte en probabilité exprimée sur l'énergie acoustique totale transmise à travers le réseau de plaques.

Pour simplifier la présentation, nous considérons une corde vibrante tendue soumise à une force linéique. Ce problème peut être remplacé sans difficulté par le problème d'une plaque en vibration, car les mêmes types de matrices interviennent.

Dans la section B.1, nous introduisons le problème modèle et les notations. Dans la section B.2, nous détaillons la modélisation probabiliste de la matrice du problème obtenu, basée sur les travaux de Soize [START_REF] Soize | A nonparametric model of random uncertainties for reduced matrix models in structural dynamics[END_REF]. Enfin, le problème d'optimisation sous contrainte en probabilité est présenté dans la section B.3 et une illustration numérique est fournie en section B.4. Dans la section B.5, nous donnons les premiers éléments pour étendre le raisonnement au cas de plusieurs cordes reliées à leurs extrémités. L'introduction de la nouvelle inconnue Q peut s'apparenter à un changement de base. Nous considérons que les composantes de Q sont les coordonnées de la solution de (B.4) dans cette base, appelée base modale. Dans cette base, la matrice de masse vaut l'identité, et la matrice de rigidité est diagonale, et contient les valeurs propres de M -1 2 KM -1 2 . On peut aussi réduire le nombre de degrés de liberté en ne prenant que les modes de basse énergie, c'est-à-dire en réduisant D à une matrice diagonale ne contenant que les n plus petites valeurs propres. Dans ce cas, la matrice de passage B est rectangulaire, et on ne résout qu'une approximation de (B.2).

B.1 Problème modèle

B.2 Modélisation probabiliste

Nous nous appuyons sur les travaux de Soize [START_REF] Soize | A nonparametric model of random uncertainties for reduced matrix models in structural dynamics[END_REF] pour modéliser la méconnaissance ou les aléas sur la corde en considérant que la matrice A du problème (B.5) est une matrice aléatoire suivant une loi de Wishart. En invoquant un principe de maximisation d'entropie en se fixant des contraintes a priori (matrices simulées symétriques définies positives, moyenne et dispersion imposées), Soize montre que le problème se réduit (à une constante près) à une loi de Wishart dont les paramètres sont une matrice symétrique définie positive (le paramètre d'échelle), que l'on peut relier à la matrice moyenne, et un entier (le nombre de degrés de liberté), que l'on peut relier à la dispersion de la loi [START_REF] Soize | A nonparametric model of random uncertainties for reduced matrix models in structural dynamics[END_REF]Section 3]. Dans cette section, nous détaillons d'abord comment les paramètres de la loi de Wishart sont reliés aux propriétés de moyenne et dispersion des matrices aléatoires suivant cette loi, puis nous étudions la loi de la norme d'énergie de la solution Q de (B.5) lorsque A suit une loi de Wishart. L'entier m nous permet de contrôler la dispersion de ces matrices aléatoires autour de A moy par la formule (B.13). La solution Q est notre inconnue sur la base modale, et nous prenons pour variable d'intérêt la norme d'énergie de ce vecteur induite par A : Â-1 11 . Il nous reste à déterminer la loi du premier coefficient de l'inverse d'une matrice de Wishart. Soit B = Â-1 . Il est connu, sous le nom de décomposition de Bartlett, que  ∼ Z t Z, où Z est une matrice triangulaire supérieure telle que tous les coefficients non nuls suivent des lois indépendantes: pour tout 1 ≤ i ≤ n, Z 2 ii suit une loi du χ 2 à ni + 1 degrés de liberté et tous les coefficients au dessus de la diagonale suivent des loi normales centrées réduites. Nous n'avons pour l'instant pas d'information sur la monotonie de δ → P (V δ > S). Dans le cas représenté à la figure B.2, l'équation P (V δ > S) = ǫ a plusieurs solutions, notamment δ opt et δ + , et {δ, t.q. P (V δ > S) ≤ ǫ} est une union de deux intervalles disjoints. Tout algorithme d'optimisation serait alors inutilisable, car selon l'initialisation, il pourrait renvoyer δ + , par exemple, qui n'est pas dans l'ensemble des états admissibles de (B. [START_REF] Boyaval | A reduced basis approach for variational problems with stochastic parameters: Application to heat conduction with variable Robin coefficient[END_REF]). Nous voulons ∆f α l ≥ 0. Il nous reste à prouver que (A l ) l est une suite majorée par 1 pour montrer que l → f α l est croissante ∀α ∈ 0, 1 2 .

B.2.1 La loi de Wishart

V δ = Q 2 A = Q t AQ = (A -1 B t M -1 2 F ) t A(A -1 B t M -1 2 F ) = (B t M -1 2 F ) t A -1 (B t M -1 2 F ).

Remarque

(2l + n -1) i 2 i i!k!   = (2l + n + 1) l l! -2 l   e l-1 i=0 (2l + n -1) i 2 i i! + l-1 k=0 1 k!   l-1 i=l-k (2l + n -1) i 2 i i! - l-1 i=0 (2l + n -1) i 2 i i!     = (2l + n + 1) l l! -2 l e l-1 i=0 (2l + n -1) i 2 i i! - l-1 k=0 1 k! l-k-1 i=0 (2l + n -1) i 2 i i! = (2l + n + 1) l l! -2 l e l-1 i=0 (2l + n -1) i 2 i i! - l-1 i=0 (2l + n -1) i 2 i i! l-i-1 k=0 1 k! = (2l + n + 1) l l! -2 l l-1 i=0 (2l + n -1) i 2 i i! +∞ k=l-i
A l = l l (l + 1) l l-1 i=0 l!l i-l i! +∞ k=l-i 1 k! = 1 + 1 l -l l-1 i=0 l!l i-l i! 1 0 t l-i-1 (l -i -1)! e 1-t dt = 1 + 1 l -l

B.5 Le problème d'optimisation sous contraintes stochastiques dans le cas de deux objets : décomposition de domaines

On utilise la décomposition de Craig-Bampton, qui permet de coupler deux cordes par une extrémité (ou deux objets mécaniques par leur interface sur les noeeuds du maillage commun), tout en gardant une résolution modale de chacune des deux sous-parties, voir [START_REF] Soize | Random uncertainties model in dynamic substructuring using a nonparametric probabilistic model[END_REF] pour la description détaillée. Notons avec un indice i les éléments relatifs à la sous-partie i.

Soit E i = M -1 i K i -ω 2 I et G i = M -1 i F i . Nous obtenons le système E i E C i E C t i E Σ i Y i Y Σ = G i G Σ , (B.40)
où Y i est l'inconnue déplacement dans les noeuds du maillage intérieurs au sous-domaine i, et Y Σ est l'inconnue déplacement à l'interface. Considérons la transformation modale de la façon suivante :

Y i Y Σ = B i S i 0 I Q i Y Σ , (B.41)
où B i est la matrice de changement de base intervenant dans la diagonalisation de M -1 i K i (éventuellement rectangulaire si l'on ne garde pas toutes les valeurs propres de M -1 i K i , comme discuté dans la Section B.1), et où S = -K i t (1,1) K i(1,2) , tel que

K i = K i(
    A 1 0 Ω 1 0 A 2 Ω 2 Ω t 1 Ω t 2 ∆ 1 + ∆ 2         Q 1 Q 2 Y Σ     =     B t 1 G 1 B t 2 G 2 S t 1 G 1 + S t 2 G 2 + G Σ     , (B.44) où ∆ i = S t i E i S i + E i S i + E t C i S i + S t i E C i + E Σ i , Ω i = B 1 t (E 1 S 1 + E C 1 ), et A i = B t i E i B i = D i + ω 2 I
, avec D i la matrice diagonale contenant les (n premières) valeurs propres de E i . Réécrivons (B.44) sous la forme de système :

       A 1 Q 1 + Ω 1 Y Σ = B t 1 G 1 , A 2 Q 2 + Ω 2 Y Σ = B t 2 G 2 , Ω t 1 Q 1 + Ω t 2 Q 2 + (∆ 1 + ∆ 2 ) Y Σ = S t 1 G 1 + S t 2 G 2 + G Σ . (B.45)
Nous avons donc Nous choisissons comme quantité d'intérêt

           Y Σ = Ω t 1 A -1 1 Ω 1 + Ω 2 A -1 2 Ω 2 -(∆ 1 + ∆ 2 ) -1 Ω t 1 A -1 1 B t 1 G 1 + Ω t 2 A -1 2 B t 2 G 2 -S t 1 G 1 -S t 2 G 2 -G Σ , Q 1 = A -1 1 B t 1 G 1 -Ω 1 Y Σ , Q 2 = A -1 2 B t 2 G 2 -Ω 2 Y Σ . (B.
V δ 1 ,δ 2 = Y t Y, (B.47) où Y =     Y 1 Y 2 Y Σ     =     B 1 0 S 1 0 B 2 S 2 0 0 I         Q 1 Q 2 Y Σ     , (B.48) avec Q 1 , Q 2 et Y Σ donnés par (B.46).
Nous supposons maintenant que la dispersion sur la loi de A 1 est connue (δ 1 est fixé), et nous définissons le problème d'optimisation sous contraintes en probabilité suivant :

2 )

 2 Introduisons la dérivée convective DDt := ∂ ∂t + v • ∇. Les équations de Navier-Stokes peuvent être réécrites sous la forme masse :∇ • q -p∇ • v + τ : ∇v, • q + τ : ∇v, (1.3d) entropie : ρT D Dt s = -∇ • q + τ : ∇v,(1.3e)où (1.1a) et (1.1b) ont été utilisées pour obtenir (1.3b), et les définitions de e, h et s ont été utilisées pour obtenir (1.3c)-(1.3d)-(1.3e). Les trois dernières équations ne sont pas indépendantes. En pratique, c'est celle sur l'entropie (1.3e) qui est utilisée pour des applications en acoustique. Dans les conditions normales de température et de pression (T = 273 K et p = 10 5 P a), l'air vérifie la loi des gaz parfaits avec une grande précision. Cette approximation consiste à considérer que les molécules du gaz sont suffisamment éloignées les unes des autres pour négliger leurs interactions électrostatiques. Un gaz parfait vérifie la loi d'état p = ρRT, (1.4) où R est la constante spécifique du gaz parfait. Nous définissions C P et C V , respectivement, les capacités thermiques à volume et pression constants, par les relations suivantes : de = C V dT, (1.5a) dh = C P dT.

2 0

 2 + Ec Re τ : ∇ṽ, (1.9c) où Re = ρ 0 v 0 L µ est le nombre de Reynolds, Pe = ρ 0 C P v 0 L κ est le nombre de Péclet et Ec = v C P ∆T est le nombre d'Eckert. Comme Pe = PrRe, où Pr = µCp κ est le nombre de Prandtl, qui est de l'ordre de 1 dans la plupart des gaz et fluides, Pe et Re sont du même ordre de grandeur. Ainsi, si Re est élevé et Ec n'est pas trop élevé, nous pouvons faire les approximations suivantes pour des simulations de propagation acoustique (nous revenons ici à des équations dimensionnées) : où la dernière ligne signifie que l'entropie reste constante le long des lignes de courant. Nous faisons toujours l'hypothèse de gaz parfait de Laplace, ce qui conduit à compléter le système (1.10) par les relations (1.6) et (1.7). D'après (1.6), il vient ρ p dp dt = C P C V dρ dt et en utilisant (1.7), dρ dt = c 2 dp dt . Si l'écoulement est homentropique (s est uniformément constante), il vient ∂ρ ∂t = c 2 ∂p ∂t , et il existe une constante K telle que p = Kρ γ . (1.11)

22 )ρ 0 .

 220 En utilisant (1.11) et (1.15), il vient Kγρ γ-2 0 ρ ′ = γ p 0 ρ ′ La linéarisation de (1.21) conduit donc à ∂ ∂t ϕ ′ + v 0 • ∇ϕ ′ + c 2 0 ρ ′ ρ 0 = 0. (1.23)La formule (1.23) correspond à la relation de Bernoulli linéarisée pour les petites perturbations.

  1.1], une solution rayonnante par morceaux de l'équation d'Helmholtz u est entièrement déterminée par le saut de ses traces de Dirichlet et de Neumann à travers Γ . Plus précisément, u = -S([γ 1 u] Γ ) + D([γ 0 u] Γ ), dans Ω + ∪ Ω -, (1.30) où (1.30) est connue sous le nom de formule de représentation. La trace de Dirichlet du premier terme du membre de droite est continue à travers Γ , et il en est de même de la trace de Neumann du deuxième terme. Nous définissons les opérateurs intégraux de simple couche S, double couche D, le dual de l'opérateur de double couche D et l'opérateur hypersingulier N par Sλ := γ 0 (Sλ) , Dλ := {γ 0 (Dλ)} Γ , Dλ := {γ 1 (Sλ)} Γ , N λ := -γ 1 (Dλ) , (1.31) où λ est une fonction définie sur Γ . D'après [80, Theorem 3.1.2], une solution rayonnante par morceaux de l'équation d'Helmholtz u vérifie

  1] d et supposons qu'il existe une matrice rectangulaire A de taille m × d, m < d, et une fonction û telles que pour tout x ∈ [0, 1] et tout µ ∈ [0, 1] d , |u(x, µ) -û(x, Aµ)| ≤ ǫ.

  et où on suppose que Span (D) = H. Le problème (1.42) peut être écrit comme le problème de minimisation u = argmin v∈H E(v),

  sont calculés en complexité indépendante de N dans la phase online en résolvant (1.47)-(5.3). Il est ainsi possible de calculer û N µ et une borne supérieure de l'erreur û N µu µ H en complexité indépendante de N .

28. while max µ∈P trial r N µ αcoer ≥ tolérance do 29 .

 29 Trouver µ N +1 ∈ argmax µ∈P trial r N µ [Selection du paramètre qui maximise l'estimateur d'erreur] 30.

  ) où σ N (F ) représente l'erreur d'approximation faite lorsque V N est construit par l'algorithme glouton considéré ici ; d N (F ) désigne alors la plus petite de ces erreurs d'approximation parmi les sous-espaces de F de dimension N . Le résultat le plus récent et le plus fort connu à ce jour est le suivant ([36, Corollary 3.3 (i)]) :

  et l'estimation (1.63) peut être appliquée aux problèmes (1.70) et (1.71) avec C = cc ′ .

[ 6 .

 6 Fig. 1.6. Géométrie du cas test pour le problème impédant

Le chapitre 3

 3 reprend l'article [Ar4] et traite le problème de diffraction d'onde acoustique par un objet solide dans un écoulement potentiel dans un domaine borné Ω -de l'espace et uniforme à l'extérieur de ce domaine, voir la figure 1.7. L'équation étudiée est l'équation d'Helmholtz convectée (1.27). La transformation de Prandtl-Glauert consiste en un changement de variable et un changement de fonction inconnue. Lorsque l'équation d'Helmholtz est convectée par un écoulement uniforme, il est connu qu'une transformation de Prandtl-Glauert particulière Enfin, le chapitre 4 présente plusieurs expériences numériques à vocation de validation des formulations obtenues dans le chapitre 3 et d'illustration sur des cas tests industriels. En particulier, une application industrielle de la prépublication [Pr2] est présentée. La partie II regroupe les travaux effectués sur la méthode des bases réduites. Le chapitre 5 reprend la publication [Ar3], qui propose une amélioration de la méthode introduite dans la publication [Ar1] (cette dernière n'est pas reproduite dans le présent manuscrit).

Fig. 2 . 1 .

 21 Fig. 2.1. Geometry for the three-dimensional acoustic scattering problem.

Theorem 2 . 2

 22 Problem (2.13) has a unique solution.

Fig. 2 . 4 .

 24 Fig. 2.4. Condition number (log scale) of the matrices of three boundary integral problems for a sphere (left) and a cube (right).

  The convective flow is continuous through Γ ∞ and tangential on Γ . Hence ρ, k and M are continuous through Γ ∞ , and M • n = 0 on Γ .

Theorem 3 . 2

 32 which, by continuity of the convective flow across Γ ∞ , is just (3.21e). Finally, (3.18b) and (3.18c) are simply (3.21c) and (3.21f).♦ Problem (3.21) is well-posed.

. 26 )

 26 Let now f solve (3.21) and let v be the function defined by v| Ω + := f sc and v| R 3 \Ω + := -f - inc . The function v is a radiating piecewise Helmholtz solution (on Ω + this follows from (3.21b) and (3.21f), and on

  where the question of the inversibility of S has to be addressed. Two other examples are detailed in Sections 3.3.3 and 3.3.4 below.

  are dense. The block A unstab 2,3 is the transpose of the block A unstab 3,2 , and the block A unstab 3,3 is symmetric.

  Figure 3.2 displays Mesh 1 and the rescaled velocity M 0 of the potential flow.

Fig. 3 . 2 .

 32 Fig. 3.2. Left: representation of Mesh 1, Right: potential flow around the ellipsoid.

Figure 3 .

 3 Figure 3.6 presents the condition numbers of the matrices resulting from the formulations (3.36) and (3.45) with respect to the frequency. In the left plot, the curves are centered at the resonant frequencies. The finer the mesh, the higher the condition number explodes. The width

Fig. 3 . 3 .

 33 Fig. 3.3. Mesh 1, 1500 Hz. Top : real part of the total pressure; left: unstable formulation (3.36), right: stable formulation (3.45). Bottom : real part of the scattered pressure; left: unstable formulation (3.36), right: stable formulation (3.45). At this non-resonant frequency, both formulations yield similar results.

Fig. 3 . 4 .

 34 Fig. 3.4. Mesh 1, 1509.849 Hz. Top : real part of the total pressure; left: unstable formulation (3.36), right: stable formulation (3.45). Bottom : real part of the scattered pressure; left: unstable formulation (3.36), right: stable formulation (3.45). At this resonant frequency, the two formulations yield different results.

Fig. 3 . 7 .

 37 Fig. 3.7. Mesh 1 at resonance 1509.849 Hz; relative residual (left) and relative error (right) with respect to the number of iterations.

6 100 Fig. 3 . 8 .

 610038 Fig. 3.8. Condition number of the matrix for the stable formulation (3.45) centered around the resonant frequency at 1535.704 Hz for Mesh 4. In this case, the chosen value η = 1 leads to the minimal condition numbers.

  . Then, recalling that (3.21) is the original transmission problem, (3.36) the unstable formulation, and (3.45) the stable formulation, the mathematical results are obtained in the following order: Proposition 3.17 =⇒ Link between (3.21) and (3.45) (Proposition 3.9) Link between (3.21) and (3.36) (Proposition 3.5) Uniqueness for (3.45) =⇒ Well-posedness of (3.45) (Theorem 3.10) =⇒ Well-posedness of (3.21) (Theorem 3.2) Conditional uniqueness for (3.36) =⇒ Conditional well-posedness of (3.36) (Theorem 3.7)

. 74 )

 74 Consider now ũ := S(x) + D(iηM x). From[58, p. 113], the single-layer and double-layer potentials are radiating piecewise Helmholtz solutions. In particular, ũ| R 3 \Ω + solves the Helmholtz

. 75 )Proof. (of Proposition 3 . 9 )

 7539 From the trace relations(3.25), there holds γ - 0 ũ-= γ - 0 (S(x)+D(iηM x)) = Sx+iη D -1 2 M x. Then, using the first line of (3.74), γ -0 ũ-= -iηM x. Likewise, γ - 1 ũ-= x. From (3.75), -iη (x, M x) Γ∞ ∈ R. However, since (x, M x) Γ∞ = M x 2 H 1 (Γ∞) ∈ Rand Re(η) = 0, there holds (x, M x) Γ∞ = 0. Therefore x = 0, and p = M x = 0, leading to λ = γ - 1 Φ. Finally, (3.68c)-(3.68d) are directly obtained from (3.72a)-(3.72b) using λ = γ - 1 Φ and p = 0. ♦ The first part of the proposition is clear from the results of Section 3.3. Let (Φ, λ, p) solve (3.45). Since the single-layer and double-layer potentials are radiating piecewise Helmholtz solutions, R(Φ, λ) satisfies (3.21b) and (3.21f). From Proposition 3.17, p = 0, λ = γ - 1 Φ, and (3.68) holds. (3.21a) and (3.21c) are just (3.68a) and (3.68b). Then, using the definition of R given in Proposition 3.5 and the trace identities (3.25), the exterior Dirichlet trace

Proof. (of Theorem 3 . 10 )

 310 λ), from which we infer that the first transmission condition (3.21d) holds. The second transmission condition (3.21e) is obtained in the same fashion from the exterior Neumann trace of R(Φ, λ), (3.68c), and using the fact that λ = γ - 1 Φ. ♦ Proposition 3.18 Problem (3.45) has at most one solution.Proof. Let (Φ, λ, p) solve (3.45) with γ 0 f inc = 0 and γ 1 f inc = 0. From Proposition 3.17, p = 0 and λ = γ - 1 Φ. From Proposition 3.9, R(Φ, λ) solves (3.21). From the mapping properties of S and D (Section 3.3.2), R(Φ, λ) ∈ H 1 loc (Ω + ∪Ω -). Then, from the transmission conditions (3.21d) and (3.21e), R(Φ, λ) ∈ H 1 loc (Ω). Then, Proposition 3.16 implies that R(Φ, λ) = 0 in Ω. As a result, there holds Φ = R(Φ, λ)| Ω -= 0, and λ = γ - 1 Φ = 0. ♦ Consider the two sesquilinear forms a 1 and a 2 on H × H such that

  .76) where S 0 , D 0 , D0 and N 0 are the boundary integral operators S, D, D and N for k∞ = 0. Consider the linear form b on H such that b

2 ) 5 )

 25 and the corresponding results hold for the other boundary integral operators. Since the injection ofH 1 (Γ ∞ ) in H 1 2 (Γ ∞ ) is compact, then, up to subsequences, p in →p i in H 1 2 (Γ ∞ ), so that (λ 2n , p 1n ) Γ∞ → (λ 2 ,p 1 ) Γ∞ . The last two terms converge as well by continuity of N , γ - 0 and D, as well as the strong convergence of p 2n in H 1 2 (Γ ∞ ) up to subsequences. This concludes the proof. ♦ Proof. (of Theorem 3.This a direct consequence of Proposition 3.9 and Theorem 3.10. ♦ Proof. (of Proposition 3.The first part of the proposition is clear from the results of Section 3.3. Let (Φ, λ) solve (3.36). In the same fashion as in the proof of Proposition 3.17, (3.21a) and (3.21c) hold, as well as

Fig. 4 . 1 .

 41 Fig. 4.1. Cross section of the geometry of the test case

Fig. 4 . 2 .

 42 Fig. 4.2. Pressure fields comparison. Left: formulation (3.36), right: reference version of ACTIPOLE. Top: total pressure, bottom: scattered pressure. Notice that the poor quality of the scattered pressure field in Ω -is due to technical issues of the visualization software: the total pressure field has to be converted from cell data to point data, inducing an averaging of the field over each tetrahedron, before subtracting the incoming field.

Figure 4 . 3 .Fig. 4 . 3 .

 4343 Fig. 4.3. Geometry of the test case

Fig. 4 . 4 .

 44 Fig. 4.4. Real part of the pressure measured at the visualization point with respect to the frequency -red: IMACS, dashed black: formulation (3.36), blue: reference. The first two curves superimpose.

Fig. 4 . 5 .

 45 Fig. 4.5. Real part of the difference between the pressure computed at the visualization point and the reference with respect to the frequency of the source -blue: formulation (3.36), red: IMACS.

Fig. 4 . 8 .Figure 4 .

 484 Fig. 4.8. (P ressure(Jet) -P ressure(N o Jet)) db with respect to θ and φ defined in Figure 4.7, 30 m away from the source. Left: results obtained in [90, 88], right: results obtained using the formulation (3.36)

Fig. 4 . 9 .

 49 Fig. 4.9. (P ressure(Jet) -P ressure(N o Jet)) db , left: cross-section at θ = 60 • , right: cross-section at φ = 0 • , for several lengths of the cylinder and several velocity profiles in the shear layer

Fig. 4 . 12 .

 412 Fig. 4.12. Source located downstream of the object. Top: uniform flow in the interior domain, bottom: potential flow, left: scattered field, right: total field

and 4 . 17 .

 417 For each mode, in the top part of the figure, the pressure is obtained with the uniform flow model and in the bottom part of the figure the pressure is obtained with the potential model the Schur complement and 6.5 h on 120 processors and 204 iterations with the Schur complement, for an achieved residual of 10 -6 .

Fig. 5 . 1 .

 51 Fig. 5.1. Behavior of the formula E k with respect to the parameter value (schematic illustration of Definition 5.6, with Pselect = {µ1, ..., µ4}). Left: the formula E k is valid for computing the error bound with tolerance tol ; right: the formula is not valid as E k (µ2) > tol.

Algorithm 2 2 . 3 . 4 .| 5 . 6 .

 223456 Offline stage of the EIM 1. Choose σ > 1 [Number of interpolation points] Set k := 1 Compute p1 := argmax p∈{1,...,σ} Xp(•) ℓ ∞ (P trial ) Compute µ1 := argmax µ∈P trial Xp 1 (µ)| and set Pinter = {µ1} [First interpolation point] Set q1(•) := Xp 1 (•) Xp 1 (µ1) [First basis function] Set B 1 11 := 1 [Initialize B matrix] 7. while k < σ do 8.

Fig. 5 . 2 .

 52 Fig. 5.2. Error bound curves with respect to the parameter. The formula E4 is computed with σ = 23.

Fig. 5 . 3 .

 53 Fig. 5.3. Comparison of the formulae E3 and E4, with respect to the formula E1.

Fig. 5 . 4 .

 54 Fig. 5.4. Determinant (left) and condition number (right) of the matrix B σ as a function of σ, for the classical EIM, the classical EIM with unique choice, and the stabilized EIM. The classical EIM curves stop at 21 interpolation points since B σ becomes non invertible at 22 points.

Fig. 5 . 5 .

 55 Fig. 5.5. Geometry for the three-dimensional acoustic scattering problem

  .6) is used to compute a lower bound of the inf-sup constant, which is around 10 -6 in the present examples. We define two test cases: (i) one impedant sphere (d = 3), with N = 584 and µ ∈ P := [0.9, 1.1], (ii) two impedant spheres (d = 5), with N = 1561 and µ ∈ P := [0.99, 1.01] 2 . We present visualizations of the scattered pressure field, at a random value of the parameter µ, for test case (i) with #P trial = 100 and N = 10 in Figure 5.6 and for test case (ii) with #P trial = 225 and N = 10 in Figure 5.7.

Fig. 5 . 6 .

 56 Fig. 5.6. Real part of the pressure field for the BEM solution (left) and the RB solution (right), with a basis of size 10. The difference between the two fields is less than 10 -15 in infinity norm.

Fig. 5 . 7 .

 57 Fig. 5.7. Real part of the pressure field for the BEM solution (left) and the RB solution (right), with a basis of size 10. The difference between the two fields is less than 10 -15 in infinity norm.

Fig. 5 . 8 .

 58 Fig. 5.8. Error bound curves with respect to the impedance coefficient, with N equal to 2, 3, 4, and 5 (from left to right and top to bottom). The curve for E2 computed in quadruple precision superimposes to E1.

Fig. 5 . 10 .

 510 Fig. 5.10. Error bound curve for E4 with respect to the impedance coefficient, with N = 5 and σ equal to 14, 30, 40, and 50 (from left to right and top to bottom).

  µ, P trial ). Consider now the problem min y∈Y n LB (µ) I(µ, y). (5.55) The problem (5.55) belongs to the class of "linear program" optimization problems with Q variables and 2Q + M 1 + M 2 inequality constraints. Notice that since β 2 µ ′ n-1 LB ∀µ ′ ∈ P M 2 (µ, P trial ) and β 2 µ ′ ∀µ ′ ∈ P M 1 (µ, C n ) have been computed in previous steps of the offline stage of the SCM, the problem (5.55) is of complexity independent of N . Definition 5.27 (Error measure) The convergence of the offline stage of the SCM is controlled by monitoring η n (µ) defined by: η n (µ) := 1 -

q 2 . 3 . 4 . 7 . 8 .

 23478 Set n = 1 and fix M1, M2 ∈ N * Choose µ1 ∈ P trial randomly and set C1 := {µ1} Compute y1 = y(µ1) and βµ 1 using Definition 5.26, and set Y 1 UB := {y1} 5. Compute η 1 (µ) for all µ ∈ Ptrial using (5.56), with Y 1 LB (µ) := y ∈ B I(µ1, y) β 2 µ 1 and I(µ ′ , y) 0 ∀µ ′ ∈ PM 2 (µ, Ptrial) 6. while max µ∈P trial η n (µ) ≥ tolerance do Set µn+1 = argmax µ∈P trial η n (µ) and Cn+1 := {µn+1} ∪ Cn Compute yn+1 = y(µn+1) and βµ n+1 using Definition 5.26, and set Y n+1 UB := {y n+1 } ∪ Y n UB 9.

. 15 ) 6 . 4 (

 1564 Remark Comparison with the standard EIM procedure in the RBM literature)

Fig. 6 . 1 .

 61 Fig. 6.1. Log10 of the relative Frobenius norm error on the matrix Aµ for d g = 3, 6, 9, 12, 14, 16, and d z = d g + 1.

Fig. 6 . 4 .

 64 Fig. 6.4. Histograms: discrete values of r = |x -y| over the Gauss points from Mesh 1 (left), and chosen set of size 10 5 (right).

Fig. 6 . 5 .

 65 Fig. 6.5. Mesh 1, log10 of the relative error on the Frobenius norm of the matrix Aµ (left), and on the acoustic pressure computed using the approximate matrix on a network of 400 points located behind the plane in Euclidian norm (right), with d g = 30 and d z = 32.

Fig. 6 . 6 .

 66 Fig. 6.6. Mesh 1: total acoustic field on the plane and on the network of points (left), and difference between the exact and approximate solution (right), for µ = 2.47.

Fig. 6 . 7 .

 67 Fig. 6.7. Mesh 2, log10 of the relative error on the Frobenius norm of the matrix Aµ, with d g = 50 and d z = 50.

Fig. 6 . 8 .

 68 Fig. 6.8. Representation of the mesh.

12 .k

 12 ← k + 1 [Increment the size of the decomposition] 13. end while The online stage of the classical EIM S1 amounts to (7.5)-(7.6) for k = d. This yields , 1 ≤ m ≤ d, solve the linear system d m=1 B S1 l,m λ S1O1 m (µ) = g(µ, x S1 l ), ∀1 ≤ l ≤ d. (7.8) In (7.7)-(7.8), the exponent O1 refers to Online problem 1.

. 10 ) 7 . 1

 1071 Proposition For all x ∈ Ω and all µ ∈ P, The expression λ S1O1m (µ) = d l=1 (B S1 ) -1 m,l g(µ, x S1 l ), 1 ≤ m ≤ d, holds from (7.8), leading to (I S1O1 d g)(µ, x) = d m=1 d l=1 (B S1 ) -1 m,l g(µ, x S1 l )q S1 m (x).In the same fashion, the expressionλ S1O2 l (x) = d m=1 (B S1 ) -1 m,l q S1 m (x), 1 ≤ l ≤ d, holds from (7.10), leading to (I S1O2 d g)(µ, x) = d l=1 d m=1 (B S1 ) -1 m,l q S1 m (x)g(µ, x S1 l ). We recognize the expression of (I S1O1 d g)(µ, x) by switching the two dummy indices l and m. ⊓ ⊔ ♦ Although the approximations I S1O1 d g and I S1O2 d g are equal and both based on Algorithm 6 for the offline stage, they rely on different online problems (7.8) and (7.10), which induces different properties when considering the approximation of certain quantities based on g(µ, x), see Section 7.3.

. 17 )OnlineTable 7 . 1 .

 1771 Similarly to Proposition 7.1, we infer that for all x ∈ Ω and all µ ∈ P, (I S2O1 d g)(µ, x) = (I S2O2 d g)(µ, x). Since the roles of x and µ are not symmetric, the algorithms EIM S1 and EIM S2 Notation for the four possible approximation procedures lead to different approximations of the function g: in general, I S1O1 d

Fig. 7 . 3 .

 73 Fig. 7.3. Test case 2. Left: representation of the mesh. Right: potential flow around the ellipsoid

Fig. 7 . 5 .

 75 Fig. 7.5. Second impedant surface, with the finest mesh for test case 3

Fig. 7 . 7 .

 77 Fig. 7.7. Test case 3. Left: acoustic pressure fields on the aircraft and on an array of 1681 points located behind the aircraft computed solving the direct problem. Right: difference between the reduced basis and the direct solution

Fig. 8 . 3 .Fig. 8 . 4 .Fig. 8 . 5 .

 838485 Fig. 8.3. Temperature field in the Boussinesq model, with the cluster at the right of the bay, at times 0, 100, 200,... 1700

Fig. 8 . 6 .

 86 Fig. 8.6. Plane test case in the decoupled potential/heat model. Velocity and temperature fields.

Fig. 8 . 12 .

 812 Fig. 8.12. Evolution of the maximum error estimate ∆µ for µ ∈ D trial (a), the corresponding value of the error |s RB, * RB µ

Fig. 8 . 13 .

 813 Fig. 8.13. Temperature maps for the electronic component case. Left: RB online call u RB , center: FE calculation u FE , right: difference u FEu RB

λ

  J (µ i )K(µ J , µ) = D J=1 K(µ J , µ) d i=1 b i λ J (µ i ), (A.6)

λλ

  J (µ)K(µ J , µ J ′ ) = K(µ, µ J ′ ). J (µ i )K(µ J , µ J ′ ) = d i=1 b i K(µ i , µ J ′ ). (A.9)Then, since µ J ′ ∈ P trial and d = #P trial , the assertion follows from(A.5). ♦This leads to the new interpolation formula:f2 (µ) := D J ′ =1 λ J ′ (µ)f (µ J ′ ). (A.10)

2

  has been chosen. The results are in Figures A.1-A.2:

Fig. A. 1 .

 1 Fig. A.1. Comparison of the interpolation formulae for 5, 10, 11 and 12 interpolation points. KT refers to the kernel trick (A.1), and KT+EIM refers to the new formula (A.10).

Fig. A. 2 .

 2 Fig. A.2. Comparison of the interpolation formulae for 13, 14, 15, and 20 interpolation points.

3 ) 2 , 2 F

 322 Dans le domaine fréquentiel, l'équation de la corde vibrante s'écrit -y ′′ -ω 2 a 2 y = f, dans ]0, 1[, (B.1) où l'inconnue y est le déplacement vertical, a la vitesse locale de propagation des ondes et ω la fréquence. Nous prenons des conditions aux limites de Dirichlet homogènes en 0 et en 1, et considérons pour simplifier que le terme source vaut f = 1. En écrivant une formulation variationnelle sur H 1 0 (]0, 1[), puis en prenant les éléments finis P1 en dimension 1, constitués des fonctions continues affines par morceaux sur ]0, 1[, nous obtenons Kω 2 M Y = F, (B.2) où ]0, 1[ est discrétisé en segments de même taille h = 1 N +1 , avec N la taille des matrices obtenues. On a Les composantes du vecteur Y sont les composantes de la fonction approchant y dans la base des éléments finis considérés. La matrice M étant symétrique définie positive, on peut écrire que l'on diagonalise. La matrice H étant symétrique définie positive, il existe une matrice orthogonale B et une matrice diagonale D telles que B t HB = D. En posant Q = B t M 1 2 Y , (B.2) devient HBQω 2 BQ = M -1 , et en posant A = -ω 2 I + D, il vient AQ = B t M -1 2 F. (B.5)

L 6 )E 2 ijComme par ailleurs on a A moy 2 F 2 F||A moy || 2 F 13 )B. 2 . 2

 622221322 Nous supposons que A ∼ W n 1 m A moy , m , c'est-à-dire que A suit une loi de Wishart de paramètre d'échelle 1 m A moy et de degrés de liberté m > n -1, où n est la taille de la matrice A moy , voir[START_REF] Anderson | An Introduction to Multivariate Statistical Analysis[END_REF] pour des détails sur cette loi. Dans la suite, nous notons l'égalité en loi par ∼. Considérons la décomposition de Cholesky A moy = L A L t A , et prenons m vecteurs {U i } 1≤i≤m ∈ R n , dont les coefficients sont des variables aléatoires gaussiennes centrées réduites indépendantes. Les vecteurs L A U i suivent la loi normale n-dimensionnelle centrée de matrice de covarianceA moy , et 1 m m i=1 (L A U i )(L A U i ) t suit la loi W n 1 m A moy , m . Cela fournit un moyen pratique de générer des réalisations de la loi de Wishart. Considérons  = L -1 A AL -t A A U i (L A U i ) t L -tLa matrice m i=1 U i U t i a ses termes diagonaux qui suivent des lois du χ 2 à m degrés de liberté (d'espérance m) et des termes hors diagonale qui sont des sommes de produits de normales centrées réduites indépendantes (donc d'espérance nulle). Ainsi, E  = I etE (A) = L A E  L t A = A moy , (B.7)ce qui relie simplement le paramètre d'échelle de la loi de Wishart à l'espérance des matrices aléatoires. Il paraît alors naturel de définir une dispersion δ pour la loi de A par la formule|A ij -A moy ij | 2 = i,j Var (A ij ) . (B.9)En repassant par la matrice  et en utilisant la formule de la variance de la somme et du produit de variables aléatoires indépendantes, on montre queVar (A ij ) = 1 m (A moy ) ii (A moy ) jj + (A moy ) 2 ij . + A moy ii A moy jj ) = tr A moy A t moy = tr A 2 moy ,il vient δ 2 = E ||A -A moy || qui permet de relier le degré de liberté de la loi de Wishart à la dispersion des matrices aléatoires par la formule Dans la suite, par souci de concision, nous notons ς := La loi de la norme d'énergie de la solution Nous supposons que la matrice A du problème (B.5) est une matrice aléatoire de loi 1 m W n (A moy , m). La matrice A moy = -ω 2 I + D est connue : ω est la pulsation de la source et D est fixée à partir des connaissances a priori du modèle tel que décrit dans la section B.1.

(B. 14 ) 2 F 2 1e t 1 Â-1 e 1 =

 142211 Comme A moy est diagonale à coefficients positifs, nous pouvons prendreL A = diag A moy ii . Comme  := L A -1 AL A -t et en posant F = L A -1 B t M -1 , la quantité d'intérêt s'écrit V δ = F t Â-1 F . (B.15) Lorsque la taille de la base réduite est n = 1,  est scalaire et  ∼ c(m) m , où c(m) est une variable aléatoire suivant une loi du χ 2 à m degrés de liberté, et V δ ∼ m|| F || 2 2 1 c(m) , où || • || 2 est la norme Euclidienne. Proposition B.1 En dimension n > 1, V δ ∼ m|| F || 2 c(mn + 1) . (B.16) Preuve. Rappelons que  ∼ W n 1 m I, m . La loi de  est invariante par rotation: pour toute matrice de rotation R, R t ÂR ∼ W n 1 m I, m . Donc V δ ∼ || F || 2 2 || F || 2 2

B 11 = Â-1 11 = e t 1 Â 2 .Fig. B. 1 .B. 3 Proposition B. 2 2 || F || 2 2 S -1 2 1+ςδ 2 - 2 1|| F || 2 2 1+P 2 S 2 e 2 2 2 (Proposition B. 3 P V δ ≥ m|| F || 2 2 mProposition B. 4 || F || 2 2 S 2 m

 111112132222222222232422 Fig. B.1. Représentation des divers vecteurs considérés

Fig. B. 2 .

 2 Fig. B.2. Représentation de δ → P (V δ > S) (bleu) et de δ → e -f (m(δ)) (rouge) avec les informations de la Proposition B.4

B. 5 C.Proposition B. 7 2 .|| F || 2 2 2S 2 ) 2 ) 2 j 2 j 2 j 2 k

 5722222222 étant décroissante, nous avonsδ opt ≥ δ * . Comme E (V δ ) = m m-n+1 || F || 2 2 , alors E (V δ ) → || F || 2 2 δ→0. En supposant S ≥ || F ||2 2 , il vient alors P (V δ > S) → 0 δ→0 Nous énonçons un lemme technique que nous utiliserons par la suite.∀k ≥ l, ∀α ∈ 0, 1 2 , 1 -(2α) k-l ≥ 0. Donc Q l ≥ e 2α . ♦Nous disposons d'un résultat sur la monotonie de δ → P (V δ ≥ S). Supposons que l'on se restreigne à des valeurs entières del := m-n+1 Supposons S ≥ || F || 2 2 . Alors la fonction δ → P (V δ ≥ S) est croissante. Preuve. Posons α := . S ≥ || F || 2 2 implique 0 ≤ α ≤ 1 2 . Comme V δ ∼ m|| F || 2 2 1 c(m-n+1) , alors P (V δ ≥ S) = F c(m-n+1) (2mα), où F c(m-n+1) désigne la fonction de répartition d'une χ 2 à mn + 1 degrés de liberté. Nous avons F c(m-n+1) (2mα) = γ( , où γ(•, •) est la fonction Gamma incomplète inférieure, et Γ (•) est la fonction Gamma. On a F c(m-n+1) , où Γ (•, •) est la fonction Gamma incomplète supérieure. Lorsque l = m-n+1 2 est entier, nous avons l'expression :F c(m-n+1) (2mα) = 1e -(2l+n-1)α l-1 i=0 ((2l + n -1) α) i i! . (B.31) Il nous reste à montrer que m → F c(m-n+1) (2mα) est décroissante (car m = 1+ς δ 2 ), ou, de façon équivalente, que l → f α l := e -(2l+n-1)α l-1 i=0 ((2l+n-1)α) i i!est croissante pour tout α ∈ 0, 1 2 . Nous avonsf α l+1f α l = e -(2l+n+1)α (2l + n + 1) l α l n + 1) ie 2α (2l + n -1) i . (B.32)La fonction l → f α l est croissante si et seulement si∆f α l := (2l + n + 1) l α l n -1) i ≥ 0. (B.33)En utilisant le Lemme B.6, il vient∆f α l ≥ (2l + n + 1) l α l (2l + n -1) k-j j! (kj)! α k -(2l + n -1) k-j j! (kj)! α kl (2l + n -1) i i!k! α i+k-l (2l + n -1) k-j j! (kj)! α k -(2l + n -1) j-k (jk)!k! α j .(B.34) Dans la dernière double somme, j ≥ k, donc en intervertissant les deux sommations, cette quantité devient l-1 j=0 j k=02 k (2l-n+1) j-k k!(j-k)!α j , et la dernière ligne dans l'expression précédente se simplifie. ∀l ∈ N * , e -l-1 k=0 1 k! ≥ 0 et ∀i ∈ N, -α i ≥ -1 2 i , donc nous pouvons régler la dépendance en α :

l+ x-1 2 .l+ x-1 2 .

 22 ∀l ∈ N * , ∀i ∈ N * tel que i < l, la fonction x → g(x) := (2l+x-1) i (2l+x+1) l est décroissante sur R + . En effet, g ′ (x) = (2l+x-1) i (2l+x+1) l i 2l+n-1 -l 2l+n+1 , donc g ′ (x) ≤ 0 ⇔ i+1 l -1 l ≤ 1 -1 Or, d'après nos hypothèses, i+1 l ≤ 1 et -1 l ≤ -1 En particulier, ∀l ∈ N * , ∀i ∈ N * tel que i < l, ∀n ∈ N * , -(2l+n-1) i (2l+n+1) l ≥ -2 i-l l i (l+1) i ,et nous avons réglé la dépendance en n en écrivant l! (2l + n + 1) l

Corollaire B. 8 Corollaire B. 9

 89 l -1) , (B.[START_REF] Dubois | Lorentz transform and staggered finite differences for advective acoustics[END_REF] où I l := 1 0 e -t 1 + t l l dt. Il nous reste à voir que la suite (I l ) l est bien majorée par 1 pour que(A l ) l le soit également. Posons g(t) = ln e -t 1 + 1 l l . On a g ′ (t) = -t l 1 + t l -1 ≤ 0 pour t ∈ R + . Donc e g(t) ≤ e g(0) = 1. Ainsi, I l ≤ 1, et la Proposition B.7 est démontrée. ♦ La fonction δ → P (V δ ≥ S)étant croissante, l'ensemble des états admissibles du problème d'optimisation (B.19) est un intervalle de R + . La fonction de coût étant décroissante, δ opt est donné par la solution unique de P V δopt > S = ǫ. Nous pouvons désormais imaginer un algorithme de résolution de P V δopt > S = ǫ initialisé par δ * donné par le théorème B.4. Tout modèle de Wishart pour le système mécanique avec la même matrice moyenne et avec un nombre de degrés de liberté supérieur à m opt := m(δ opt ) verra sa solution vérifier la contrainte du problème d'optimisation (B.19). La figure B.4 représente la fonction δ → P (V δ < S). La solution du problème d'optimisation sous contrainte en probabilité pour ǫ = 5% est alors δ opt ≈ 0.36.

46 )

 46 Supposons maintenant queA i ∼ W n 1 m i D i + ω 2 I , m i et posons δ i = 1 m i 1 + (tr(Di+ω 2 I)) 2tr(D i +ω 2 I)

  argmin

δ 2 (

 2 C (δ 1 , δ 2 )) , tq P (V δ 1 ,δ 2 < S) ≥ 1ǫ. (B.49) Contrairement à la Proposition B.1 dans cas d'une seule corde, nous ne sommes pas en mesure de caractériser simplement la quantité d'intérêt en termes de variables aléatoires classiques. Un résultat équivalent à la Proposition B.7 serait : si S ≥ M = lim δ 2 →0 E (V δ 1 ,δ 2 ), alors la fonction δ 2 → P (V δ 1 ,δ 2 ≥ S) est croissante. Nous vérifions si cette conjecture est raisonnable en représentant à la figure B.5 la fonction δ 2 → P (V δ 1 ,δ 2 < S) avec δ 1 = 0.2, S = 2.2, N 1 = N 2 = 100 et n 1 = n 2 = 14. La monotonie de la fonction est cohérente avec le résultat conjecturé.

Table des matières 1 Introduction générale

  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Table 1 . 1 .

 11 une approximation suffisamment bonne. L'idée des algorithmes gloutons est de choisir de façon itérative les éléments g i du dictionnaire. Nous supposons que pour tout u ∈ H, il existe g ∈ D tel que g ∈ argmax Nous pouvons distinguer deux principaux types d'algorithmes gloutons : le Pure Greedy Algorithm (PGA) et l'Orthogonal Greedy Algorithm (OGA), présentés dans la Table 1.1. Présentation des algorithmes PGA et OGA. Ces algorithmes convergent dans le sens suivant. Nous considérons les algorithmes PGA et OGA présentés dans la Table 1.1 avec ǫ = 0. Pour tout dictionnaire D et tout u ∈ H,

	•• ,dn u H ,	(1.37)

où P d 1 ,••• ,dn est la projection orthogonale sur Span (d 1 , • • • , d n ) pour le produit scalaire •, • H . Trouver la meilleure approximation est un problème difficile et sujet au fléau de la dimension ; nous cherchons i

, 1 ≤ i ≤ n + 1}, u OGA n+1 := P H OGA n+1 (u) et r OGA n+1 := u -P H OGA n+1 (u), remplacer n ← n + 1. uu PGA

  8. end for 9. Initialiser N = 1 10. Choisir µ1 ∈ P trial aléatoirement et initialiser P select = {µ1} 11. Calculer uµ 1 12. Initialiser V1 = Span{uµ 1 } 13. for all k in {1, ..., d b } do

	14.

Calculer et sauvegarder b k (uµ 1 ) [Premier coefficient du membre de droite de (1.47)] 15. end for 16. for all k in {1, ..., d a } do 17. Calculer et sauvegarder a k (uµ 1 , uµ 1 ) [Premier coefficient du membre de gauche de (1.47)] 18.

  [START_REF] Langou | Solving large linear systems with multiple right-hand sides[END_REF] où d x est la dimension de l'espace d'arrivée des fonctions de V. Nous prouvons maintenant ce résultat. En utilisant le lemme de Céa (1.55), il vient

  [START_REF] Amestoy | Multifrontal parallel distributed symmetric and unsymmetric solvers[END_REF]

  3) is not sufficient to determine the unknown potentials [γ 0 u] and [γ 1 u]. It is merely a necessary relation satisfied by any radiating Helmholtz solution. Actually, no boundary condition has been enforced yet, and the system (2.3) is still valid for Dirichlet, Neumann or Robin boundary conditions on Γ . We will see that injecting a Robin boundary condition enables to recover the injectivity property.

Remark 2.5 (Symmetry vs Hermitian symmetry) Consider two normed spaces X and Y . The spaces X

  for all x h ∈ H h .

	♦

* := L(X, C) and Y * := L(Y, C) are respectively the dual spaces of X and Y . In what follows, (•, •) X,X * denotes the duality pairing between X and X * . For instance, if u ∈ X and g ∈ X * , (u, g) X,X * = g(u). Consider an operator A : X → Y . Its adjoint operator A

2.5.2 Kernel of boundary integral operators Proposition 2.6 For all

	.30)
	where the exponent N stands for the Neumann boundary condition enforced on Γ and where γ -1 stands for the interior Neumann trace on Γ . We denote {λ N n } the set of eigenvalues and {φ N n } the set of eigenfunctions.
	We observe that there exist geometries for which {λ D n }∩{λ N n } = ∅. Indeed, fix a dimension d ∈ N * and consider Ω -= [0, L] d , with L > 0. Considering d j=1 sin( 2πx j L ) as an eigenfunction of (2.29) and d j=1 cos( 2πx j L ) as an eigenfunction of (2.30), we see that 4dπ 2 L 2 ∈ {λ D n } ∩ {λ N n }.
	k 2 ∈ {λ D n }, the interior Neumann traces of the eigenfunctions of the Dirichlet eigenvalue problem (2.29) are nonzero elements of Ker(S).

a nontrivial solution of Sλ = 0. ♦ Corollary 2.7 For all k 2 ∈ {λ

	Proof. Let λ ∈ Ker(S) and define w := Sλ. From [80, Section 3.2.1], w is a radiating Helmholtz solution. In particular, ∆w -+ k 2 w -= 0. The interior Dirichlet trace of w is γ -0 w -= Sλ = 0. Therefore, w -solves (2.29). Moreover, γ -1 w -= ( D + 1 2 I)λ = λ from Proposition 2.8. If k ∞ ∈ {λ D n }, from Corollary 2.7, we can suppose λ = 0. Hence, w -= 0, and w -is an eigenvalue of problem (2.29). If k ∞ / ∈ {λ D n }, problem (2.29) admits only zero as solution. Hence w -= 0, and λ = γ -1 w -= 0. ♦
	Corollary 2.10 The nonzero elements of Ker(S) are exactly the interior Neumann traces of
	the eigenfunctions of the Dirichlet eigenvalue problem (2.29).
	Remark 2.11 We can show in the same fashion that Ker(N ) = Ker(D + 1 2 I) = {0}, and that the nonzero elements of Ker(N ) are exactly the interior Dirichlet traces of the eigenfunctions of
	the Neumann eigenvalue problem (2.30).
	Remark 2.12 For all k 2 ∈ {λ D n }, D -1 2 I is noninvertible, as the transpose (or dual) of the noninvertible operator D -1 2 I. Indeed, (Im(D -1 2
	D n }, Ker(S) = {0}.
	Proposition 2.8 For all k > 0, Ker(S) = Ker( D -1 2 I).
	Proof. Let λ ∈ H -1 2 (Γ ) and define w := Sλ. From [80, Section 3.2.1], w is a radiating Helmholtz solution. The exterior Dirichlet and Neumann traces of w are γ + 0 w + = Sλ and γ + 1 w + = ( D -1 2 I)λ. Suppose λ ∈ Ker(S). Hence, γ + 0 w + = 0, and by uniqueness of the solution to the exterior Helmholtz problem with Dirichlet boundary condition, w + = 0, and hence ( D -1 2 I)λ = 0. Therefore Ker(S) ⊂ Ker( D -1 2 I). Suppose now that λ ∈ Ker( D -1 2 I). Hence, γ + 1 w + = 0, and by uniqueness of the solution to the exterior Helmholtz problem with Neumann boundary condition, w + = 0, and hence Sλ = 0. Therefore Ker(S) ⊃ Ker( D -1 2 I). ♦
	Proposition 2.9 For all k 2 ∈ {λ D n }, the nonzero elements of Ker(S) are interior Neumann traces of eigenfunctions of the problem (2.29); for all k 2 / ∈ {λ D n }, Ker(S) = {0}.

  the surface unknown.

	{λ D n } ∩ {λ N n } is nonempty, let k 2 ∈ {λ D n } ∩ {λ N n } and consider the unique solution u of (2.31) at this wave number. Then, ∀v * ∈ Ker(S), S(γ + 1 u+γ -1 f inc +v * ) = -γ 0 f inc and ∀w * ∈ Ker( 1 2 I + D), 1 2 I + D (γ + 1 + γ -1 f inc + w * ) = -γ 1 f inc .
	Consider a geometry for which {λ D n } ∩ {λ N n } is nonempty, and take k 2 ∈ {λ D n } ∩ {λ N n }. Then, the normalized Dirichlet eigenfunction at k 2 is linealy independent of the normalized Neumann
	eigenfunction at k 2 .
	Proposition 2.14 For all k > 0, there holds Ker(S) ∩ Ker 1 2 I + D = {0}.
	Proof. Let λ ∈ Ker(S) ∩ Ker 1 2 I + D \{0} and set u = Sλ. Then, u is a radiating Helmholtz solution, such that γ + 0 u + = Sλ = 0 and γ + 1 u + = -1 2 I + D λ = -λ. Therefore, u + solves an exterior Helmholtz equation with a homogeneous Dirichlet boundary condition at Γ . By
	uniqueness of the solution to this problem, u + ≡ 0, and hence λ = -γ + 1 u + = 0.	♦
	Remark 2.15 After introducing the exterior Helmholtz problem with a Neumann boundary con-
	dition, we can show in the same fashion that for all k > 0, Ker(N) ∩ Ker(D -1 2
	Remark 2.13 Even if the exterior Helmholtz problem (2.31) has a unique solution at all fre-
	quencies, the integral equations (2.32), written as necessary relations verified by its solution,
	admit an infinity of solutions. This phenomenon is related to the eigenvalues and eigenfunc-
	tions of the complementary interior problem, and has no physical meaning. In (2.32), the
	first equation is not invertible for k 2 ∈ {λ D n }, whereas the second equation is not invertible for k 2 ∈ {λ N n }. Furthermore, there exist geometries and wave numbers for which both equa-tions in (2.32) have infinitely many solutions. More precisely, consider a geometry for which

Remark 2.21 (The particular case of the scattering by an impedant surface) The

	without resorting to any regularizing operator, since 1 iη S + D is
	a compact perturbation of 1 2 I.
			CFIE
	consists in a complex linear combination of an equation posed in H	1 2 (Γ ) and an equation posed in
	H -1 2 (Γ ). For the scattering by an impedant surface, such a combination is implicitely contained
	in the Robin boundary condition (2.6), which directly links χ, the surface unknown in H	1 2 (Γ ),

  Hiptmair and Meury[START_REF] Hiptmair | Stable FEM-BEM Coupling for Helmholtz Transmission Problems[END_REF] derived abstract trace transformation operators and generalized Calderón projectors for the Helmholtz transmission problem. By construction, integral operators written using these projectors enjoy the uniqueness property whatever the value of k∞ . The map DtN stab corresponds to a particular choice of the trace transformation operator in the

	Proof. See Section 3.7.	♦
	Theorem 3.10 Problem (3.45) is well-posed at all frequencies.
	Proof. See Section 3.7.	♦
	Remark 3.11

t H 1 (Γ∞) . Proposition 3.9 If f solves

(3.21)

, then (f -, γ 1 f, 0) solves

(3.45)

. Conversely, if (Φ, λ, p) solves

(3.45)

, then R(Φ, λ) solves (3.21) and p = 0, where R is defined in Proposition 3.5. general setting of

[START_REF] Hiptmair | Stable FEM-BEM Coupling for Helmholtz Transmission Problems[END_REF] Section 9]

. Hence, Theorem 3.10 is an extension of

[START_REF] Hiptmair | Stable FEM-BEM Coupling for Helmholtz Transmission Problems[END_REF] Theorem 7.3]

to the case where a flow exists and is nonuniform in a bounded domain, for a particular choice of the trace transformation operator.

The formulation

(3.45)

corresponds to the stabilization of the formulation (3.36), using a well-posed CFIE technique taken from the literature.

  has two contributions: one sparse and nonsymmetric and one dense and symmetric, therefore this block is dense and nonsymmetric. The blocks A unstab

					.55)
	All the blocks are complex-valued. The blocks A unstab 1,1	, A unstab 1,2	and A unstab 2,1	are sparse. The block
	A unstab 1,1	is not symmetric, and the block A unstab 1,2	is neither the transpose nor the hermitian trans-
	pose of block A unstab 2,1	. The block A unstab 2,2
					2,3

Table 3 .

 3

		Mesh 1 Mesh 2 Mesh 3 Mesh 4
	number of volumic dofs Φ	1796	687	194	79
	number of surfacic P0 dofs λ	808	510	270	148
	number of surfacic P1 dofs p	406	257	137	76
	proportion of dofs p in the total number of dofs 11.9% 15.0% 18.6% 20.0%
	smallest edge (mm)	7.09	8.78	15.71 19.18
	mean edge (mm)	22.64 32.20 49.78 66.46
	largest edge (mm)	56.87 70.62 103.59 112.71

1. Characteristics of the four considered meshes.

  93, Theorem 2.6.11]), it is deduced from (3.71) that (3.68b) holds, and from (3.71) and (3.45b) respectively that

  .[START_REF] Mises | Praktische verfahren der gleichungsauflösung[END_REF] Problem(3.45) can then be written: Find (Φ, λ, p) ∈ H, such that ∀ Φ t , λ t , p t ∈ H, a (Φ, λ, p) , Φ t , λ t , p t = b Φ t , λ t , p t ,(3.78)

Table 4 .

 4 1. Relative difference on the scattered pressure in Euclidian norm, between the formulation (3.36) and the reference version of ACTIPOLE.

  Let k ∈ {1, 2}, let µ ∈ P select and suppose that the stopping criterion (5.19) is satisfied. Then, ûµ = u µ , but u µ does not exactly solve E µ . First, by definition of the

		.19)
	then the formulae E 1 and E 2 are valid for computing the error bound with tolerance tol if β-1 min δξ ≤ tol.	(5.20)
	Proof.

13 (Goal-oriented case, approximate reduced basis functions

  Since the • * V ′ norm is hard to compute, the stopping criterion(5.19) uses in practice the Hermitian norm in C N or the V-norm of the corresponding functions in V.

	1 min δξ.
	Hence, if β-1
	Remark 5.

min δξ ≤ tol, the validity of E 1 and E 2 follows from Definition 5.6. ♦ ) The formulae E go 1 and E go 2 are valid for computing the error bound with tolerance tol if βd min -1 δγξ 2 ≤ tol.

  are reached, the formulae E 1 and E 2 are valid for computing the error bound with tolerance tol if,

	respectively, and the formulae E go 1 and E go for E 1 , for E 2 , 2 are valid for computing the error bound with tolerance tol if, 2 β-1 min δ max (ξ, ǫ) ≤ tol, 2 β-1 min δ max ξ, √ (5.21) ǫ ≤ tol, respectively, for E go 1 , 2 βd min -1 δγ max ξ 2 , ǫ 2 ≤ tol, for E go 2 , min 2 βd -1

Table 5 . 1 .

 51 Comparison between stabilized Gram-Schmidt and stabilized EIM.

	2 stab v2, ..., δ σ stab vσ) orthogonal basis of Span(v1, ..., vσ)	(I σ v)(µ)

Table 5 .

 5

		E1	E2	E3	E4
	Online efficient	No	Yes	Yes	Yes
	Unconditionally well-posed	Yes	Yes	No	Yes
	Dependence on ǫ of the observed accuracy ǫ	√	ǫ	ǫ, if well-posed	ǫ
	Equals E1 in exact arithmetics	-	Yes	Yes	Yes, if σ = σ No, if σ < σ
	Complexity of the offline stage	-(d N + 1)Nsol	σNsol	σ4 σM + σNsol with classical EIM σ5 σM + σN sol with stabilized EIM
	Complexity of the online stage	-	N 3 + σ	N 3 + σ 3	N 3 + σ3

2.

Comparison of the considered formulae for computing the error bound.

  .18)

	Proposition 7.3 (Interpolation with S2) The approximation procedures S2O1 and S2O2
	are interpolant with respect to the second variable: for all µ ∈ P,

  où le partitionnement des matrices K i se fait sur les degrés de liberté intérieurs, puis de frontière. En injectant (B.41) dans (B.40) pour les 2 sous-domaines, et en multipliant à gauche par

	K i	1,1) K i(1,2) (1,2) K i(2,2) t	,	(B.42)
	B i S i t t 0 I	,		(B.43)
	nous obtenons

1.2 L'équation d'Helmholtz convectée

Acoustic scattering by an impedant object

A coupled FEM/BEM for the convected Helmholtz equation with non-uniform flow in a bounded domain

A nonintrusive EIM to approximate linear systems with nonlinear parameter dependence

A nonintrusive Reduced Basis Method applied to aeroacoustic simulations

A multiscale problem in thermal science

Part I

Two aeroacoustic problems solved by integral equations and (3.21f). Finally, taking the exterior traces of R(Φ, λ), the transmission conditions (3.21d) and (3.21e) are directly obtained from (3.68c), (3.68d) and λγ - 1 Φ ∈ Ker(S) = Ker( D -1 2 I), in the same fashion as in the proof of Proposition 3.9. ♦ Proof. (of Theorem 3.7) Let (Φ, λ) solve (3.36) with γ 0 f inc = 0 and γ 1 f inc = 0. From Proposition 3.5, R(Φ, λ) solves (3.21). As seen in the proof of Proposition 3.18, R(Φ, λ) ∈ H 1 loc (Ω). It is then deduced from Proposition 3.16 that R(Φ, λ) = 0 in Ω. As a consequence, Φ = R(Φ, λ)| Ω -= 0. From the proof of Proposition 3.5, λ-γ - 1 Φ ∈ Ker(S). Suppose -k2 ∞ / ∈ Λ. Then, Ker(S) = {0}, leading to λ = γ - 1 Φ = 0, so that problem (3.36) has at most one solution; well-posedness is then obtained using the Fredholm alternative by proceeding similarly to the proof of Theorem 3.10. Suppose -k2 ∞ ∈ Λ. Let λ * ∈ Ker(S) = Ker( D -1 2 I), and f be the solution to (3.21). From Proposition 3.6, (f -, γ 1 f + λ * ) solves (3.36).

♦

Part II

The Reduced Basis Method

Extension to more general parameter dependence

The goal of this section is to show how to extend the nonintrusive procedure described in Section 6.4 to more complex parameter dependence. We illustrate the procedure on an industrial test case, namely a frequency-dependent three-dimensional aeroacoustic scattering problem.

Generalization of the nonintrusive procedure

Recall the general form of the matrix A µ to approximate:

where A ̺ µ are matrices that require to integrate some functions g ̺ (µ, x) over Ω, ψ s are given functions of µ and A s are µ-independent matrices resulting from some integration over Ω. EIM g is applied independently to each g ̺ (µ, x), for all 1 ≤ ̺ ≤ R, where the number of interpolation points, respectively (d g ) ̺ , may differ from one EIM g to the other. These procedures lead to the construction of the functions (λ g m ) ̺ (µ), for all 1 ≤ ̺ ≤ R, all 1 ≤ m ≤ (d g ) ̺ , and all µ ∈ P trial , using (7.8). Then, define the functions (z p (µ)) 1≤p≤dmax with d max := R ̺=1 (d g ) ̺ + S such that Even if the inf-sup constant depends on the parameters, its values are not expected to exhibit significant variations since the considered formulations do not feature any resonant frequency. In Section 7.4.4, the test case has much more unknowns that those from the two previous sections, and we do not compute the inf-sup constant so as to control the offline computational cost. The power iteration method (see [START_REF] Mises | Praktische verfahren der gleichungsauflösung[END_REF]) associated with the inverse matrix can be used to approximately compute the smallest eigenvalue. This would imply to solve many eigenvalue problems associated with the inverse operator, and therefore does not appear reasonable for industrial test cases. Dealing further with this issue goes beyond the present scope. Consider the object whose mesh is represented in the left panel of Figure 7.1. The surface of this object, denoted by Γ , consists of three zones denoted by Γ 1 , Γ 2 and Γ 3 respectively. The surface Γ 2 is represented in the right panel of Figure 7.1. On each of these zones, a Robin boundary condition is enforced with a specific impedance coefficient µ i for i ∈ {1, 2, 3}. Thus, the impedance coefficient on Γ , denoted by µ Γ , is piecewise constant and takes the form

An optimization problem for an impedant object in the air at rest

, for all x ∈ Γ , where 1 1 Γ i , i ∈ {1, 2, 3}, are characteristic functions. The source is a plane wave whose wave vector is supported by the axis of symmetry of the object, creating an incident acoustic pressure field denoted by p inc . The variational formulation of the problem is as follows:

where (•, •) Γ denotes the extension of the L 2 (Γ )-inner product to the duality pairing on H

, with ω the pulsation of the source and c the speed of sound in the air at rest and where ůγ 0 and γ 1 respectively denote the Dirichlet and Neumann traces on Γ . The operators N µ , D µ , Dµ , and S µ are boundary integral operators, expressed in terms of the Green kernel G µ (x, y) = exp(iµ|x-y|) 4π|x-y| associated with the Helmholtz equation at wave number µ. The pressure field around the object is then obtained by applying a representation formula to (χ, λ), the solution to (7.42). We refer to Chapter 2 for more details on this formulation and its wellposedness. The considered finite-dimensional approximation of (7.42) has 2240 unknowns.

The parameters of the problem are the frequency of the source f r = ω 2π , and the impedance coefficient of each of the three zones composing the surface of the object. The frequency varies from 487 to 1082 Hz, and each impedance coefficient varies from 1 to 5. The quantity of interest is the far-field acoustic pressure along the axis of symmetry of the object, but in the opposite direction of the source. A goal-oriented RBM is carried out to select a basis of n = 20 truth solutions using the nonintrusive formula (7.41) to approximate the matrix, the right-hand side of the direct problem, and the right-hand side of the adjoint problem needed to evaluate the quantity of interest. For the matrix, the approximation procedure S1O1 is applied to

and the procedure S2O2(z) is applied to

For the approximation formula of the right-hand side of the direct and dual problems, the procedure S1O1 is applied to

where d is respectively the direction of the incoming plane wave and the direction of measure of the far-field; and the procedure S2O2(z) is applied to

The EIM algorithms are carried out with d = 13 and d z = 20 for the matrix, and d = 13 and d z = 13 for right-hand side of the direct and dual problems. Over the considered parameter values, the relative error for the three nonintrusive formulae is of the order of 10 -12 (in Frobenius norm for the matrix and Euclidian norm for the vectors). The maximum error bound (over a 8.3 A reduced basis approach for the electronic component problem

Review of the method

The reduced basis (RB) method aims at reducing the computation time in a precise context: running many times the same calculation with a small change of a set of parameters. The idea is somehow close to modal decomposition in mechanical vibrations: the solution should be well represented by a small set of precomputed solutions (the reduced basis). A wide variety of problems can be tackled by this method (see [START_REF] Machiels | Reliable real-time solution of parametrized partial differential equations: Reduced-basis output bound methods[END_REF], [START_REF] Boyaval | A reduced basis approach for variational problems with stochastic parameters: Application to heat conduction with variable Robin coefficient[END_REF]). We will present it in the context of the present study: the steady decoupled incompressible Navier-Stokes / heat model (8.3) applied to the electronic component. The reduced basis method has been applied to a heat conduction problem by Sen (see [START_REF] Sen | Reduced-basis approximation and a posteriori error estimation for many-parameter heat conduction problems[END_REF]), and recently to the unsteady Boussinesq equations by Knezevic, Nguyen and Patera (see [START_REF] Knezevic | Reduced Basis Approximation and A Posteriori Error Estimation for the Parametrized Unsteady Boussinesq Equations[END_REF]). Consider that the fluid equations have been solved once for all, the RB method will be applied to the set of equations:

where u NS is the result of the preliminary incompressible Navier-Stokes computation, and Q(x) = q1 x∈Ω IC (Ω IC stands for integrated circuits and is defined in Figure 8.7). u NS is extended by 0 in the integrated circuits and in the board.

Equation (8.12) can be rewritten as

with µ a set of n parameters (for instance c p IC , κ board , etc...) in a n-dimensional subspace D of R n . Intervals of variation for the parameters are specified in section 8.3.3. Here, A µ is the linear operator of the problem, f the source term and T µ represents the temperature, solution of equation (8.12) with the parameters set µ.

Let us first check that the RB approach is reasonable in the present case, by selecting by hand a finite set of parameter values, computing the corresponding solutions T i , selecting a problem-related scalar product -(T, Θ) = Ω ∇T • ∇Θ, and checking the variation of the eigenvalues of the matrix M ij = (T i , T j ). This is close to finding principal components of the energy operator (like modal basis). On Figure 8.9, we see that the eigenvalues of the gramian matrix M of the problem lose 8 orders of magnitude with a reduced basis of size 81 chosen randomly, and the decreasing is exponential. This indicates that the solutions, for µ ∈ D, can be efficiently represented by a linear combination of a small number of functions.

We are interested in the variational formulation of equation (8.13): Find

where M h is a finite dimensional subspace of X := H 1 per,0 (Ω) = {T ∈ H 1 (Ω) |T Γ in = 0, T periodic at Γ per }, (we can make the Dirichlet condition homogeneous, considering a lifting of T in ). We use the Lagrange's P 1 finite elements for this heat equation (see section 8.2.3). Since we have mixed boundary condition with a homogeneous Dirichlet condition, ||.|| H 1 0 (Ω) is a norm on H 1 per,0 (Ω) . The considered output is proportional to the mean temperature in the integrated circuits. This specific output allows efficient mathematical analysis.

The RB consists in two steps:

-A computationaly heavy offline stage: construct a low dimensional basis, which is a good basis for the high dimensional problem (8.12) for every parameter µ in D, -Fast online stages: solve light low dimensional problems. The challenge of this approach is to guaranty that the approximate solution of equation (8.12) is a good one. This is possible thanks to the efficient computation of an a posteriori error estimate for our specific quantity of interest.

Goal-oriented a posteriori error estimate : certified RB

Consider T FE

µ the solution of (8.14) and T RB µ the current RB approximation of (8.14) (the proper construction of T RB µ is explained in section 5.2.5). These quantities verify a µ T FE µ , Θ = l(Θ), ∀Θ ∈ M h and a µ T RB µ , Θ = l(Θ), ∀Θ ∈ M RB , where M RB is the space spanned by the current reduced basis, and has much lower dimension than M h .

The quantity of interest is s FE µ , which is computed using the expensive finite element solution, is approximated by s RB µ , which is computed using reduced basis approximation. We need an accurate and fastly computable a posteriori error estimate ∆ µ for the approximation error between the RB and the FE solution : |s RB µs FE µ |. By "fast", we mean that the evaluation of ∆ µ should not require the computation of the FE solution.

The bilinear form a µ in Equation (8.14) is coercive and non symmetric: given u NS ∈ V 2 h (Ω air ) we seek T ∈ M h such that for all Θ ∈ M h :

where the last integral is the Temam term.

The non-symmetry is induced by the convective terme u NS • ∇T . Recall that the precomputed velocity field u NS is incompressible and cancels on ∂Ω:

; the Temam term ensures coercivity for this bilinear form defined on M 2 h (T ∈ P 1 , therefore T 2 / ∈ P 1 and Ω div (u NS )T 2 = 0). This lower bound is quite pessimistic, especially when κ µ varies a lot in the computational domain. For simplicity, we take this one in the present work. Methods exist to determine sharper constant for the discrete problem, based on solving eigenvalue problems (see [START_REF]Mathematical modelling and numerical simulation in materials science[END_REF] remark 16 p.114 for symmetric problems, see [START_REF] Huynh | A successive constraint linear optimization method for lower bounds of parametric coercivity and inf-sup stability constants[END_REF] for a presentation of the successive constraint method used in nonsymmetric problems).

We introduce the adjoint problem of (8.14

(8.16)

We have to construct a reduced basis for this problem as well: define Ψ * RB µ the current RB approximation of (8.16) (verifying a µ v, Ψ * RB µ = -l(v), ∀v ∈ M * RB , the space spanned by the current reduced basis of (8.16)).

Consider the residual for the direct and the adjoint problems: for all w, v ∈ M h ,

(8.17) Make use of the version of Riesz-Fréchet representation theorem applied to continuous bilinear forms: there exists a unique application

(see [START_REF]Mathematical modelling and numerical simulation in materials science[END_REF] section 4-I-B-c). This specific output has been chosen to ensure the posteriori estimate given in the proposition 8.1. Note that the FE corresponding quantity is l T

only contains approximation errors introduced by the two reduced basis. Make use of proposition 23 p.115 of Boyaval's PhD thesis [START_REF]Mathematical modelling and numerical simulation in materials science[END_REF] (see also [START_REF] Boyaval | Reduced basis techniques for stochastic problems[END_REF] eq.22):

where α LB,µ is a computable lower bound for the coercivity constant α µ of a µ (., .) (recall that a bound for the continuous case is α LB,µ = min (κ µ )).

Proof. Using elements of proof from Boyaval's PhD thesis (propositions 18 and 23):

Using (8.16),

|, by definition (8.17). Using Cauchy-Schwarz inequality, we can write:

Then by coercivity, (8.16), (8.17) and the Cauchy-Schwarz inequality:

Pluging this in (8. [START_REF] Boyaval | A reduced basis approach for variational problems with stochastic parameters: Application to heat conduction with variable Robin coefficient[END_REF], we get the inequality (8.1). ♦ Remark 8.2 Note that the error estimate ∆ µ does not require the evaluation of the FE solutions, enabling fast computation.

The inexpensive a posteriori error estimate (8.1) is useful to check rapidly if an online call approximates well the FE reference and to construct iteratively the reduced basis using a greedy algorithm (see section 5.2.5). This algorithm was proposed by Patera, Prud'homme, Rovas and Veroy in [START_REF] Patera | A posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations[END_REF].

Defining the vectors (α) j = α j , and (F ) j = l (b j ), one then just has to solve the ddimensional linear system A µ α = F, and then compute the RB solution u RB µ = d i=1 α i b i . Note that the α i contain the dependence of the solution on µ.

In practice, when the size of the reduced basis increases, the matrix A µ may be close to singular. One can improve this by orthonormalizing the basis using modified or simple Gram-Schmidt (see [START_REF]Mathematical modelling and numerical simulation in materials science[END_REF], p.109).

Then, an online call should also contain an evaluation of the a posteriori error estimate ∆ µ , and if it is larger than ǫ, one can compute the FE solution and enrich the basis with this function. This way, the error estimate is guaranteed, but we may need to compute the expensive solution sometimes.

We also use precomputed quantities to get an evaluation of ∆ µ in a complexity independent of the FE problem size.

where the affine dependence has been used in the second line and the RB decomposition in the third line. All the scalar products involved have been precomputed and stored during the offline stage. Therefore

, which corresponds to a fast computation compared to the FE problem. The faster the estimator is computed, the larger D trial we can explore in the greedy algorithm for a given computation time.

Numerical results

The greedy algorithm has been carried out to compute 10 basis functions, with a parameter set of 20, 000 points. The finite elements problem has 9012 degrees of freedom.

We took as parameters: κ IC , κ board , κ air the integrated circuit, board and air heat conductivity and c p air the air thermal capacity, varying in the following intervals (international system of units): Consider now the evolution of the maximum error estimate ∆ µ for µ ∈ D trial and the corresponding value of the error s RB, * RB µ s FE µ (see Figure 8.12). The error estimate and the error are not strictly decreasing. Maybe this is due to the fact that the RB is certified for a scalar quantity different from the projection error represented in Figure 8.11. Recall also that we did not compute a satisfying lower bound for the coercivity constant of the bilinear form a µ . The error estimate for a random parameter is not strictly decreasing with the size of the basis, but globally tends to zero. Remark that an error of 8 × 10 -5 correspond to a mean temperature error of 1 K over the integrated circuits.

Once the basis constructed, we tried an online call taking a random parameter in D (see Figure 8.13):

-A posteriori error estimate : 7.24 × 10 -6 -Error : 1.20 × 10 -6

Annexe

A

A well conditioned kernel interpolation

A.1 Kernel interpolation

Consider a unknown function µ → f (µ), µ ∈ P. We want to derive an interpolation formula for f : given interpolation points µ i ∈ P, 1 ≤ i ≤ d where we dispose of the value of f (µ i ), how can we derive an approximation f (µ) of f (µ). Consider a given kernel K(µ, µ ′ ), the Kernel Ridge Regression with kernel K and parameter λ > 0 is defined by f0 (µ) := with y i = f (µ i ), 1 ≤ i ≤ d and Kij = K(µ i , µ j ), 1 ≤ i, j ≤ d. In (A.1) is used a mapping between the set P and an reproducing kernel Hilbert space, determined by the kernel K, without having the compute explicitly (we only compute evaluation of the kernel). For these reasons, (A.1) is often referred as kernel trick).

The following proposition is readily seen. The parameter λ allows the linear system (A.2) to be numerically solvable, since in practice, the matrix K can have a very large condition number, especially when some learning points are very close to each other. However, with λ = 0, Proposition A.1 is lost in the general case. In what follows, we use the Empirical Interpolation Method (EIM) to improve the numerical behavior of the linear system (A.2) when λ = 0.

A.2 Empirical interpolation method

Consider a discrete subset of P denoted P trial . Suppose that the following EIM approximation of the kernel has been computed: