Je Remercie

Paul Cazeaux En Deuxième Année

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Keywords: discontinuous coecients, cell dynamics AMS 65M08, 35L50, 35R05, 92C37

Je remercie mes directrices de thèse, Marie Postel et Frédérique Clément, pour leur conance en mon travail et mes idées, avec qui j'ai beaucoup appris, et qui m'ont toujours bien encadré.

Je remercie Marie d'avoir partagé avec moi sa grande expérience en numérique, de m'avoir poussé à être plus rigoureux, de m'avoir appris à mettre proprement en forme mes idées, avec notamment beaucoup d'astuces de rédaction. Merci aussi de m'avoir soutenu pour mes candidatures, en thèse puis en postdoc. Merci enn pour votre patience, et toutes nos séances de travail au tableau, qui m'ont été très protables.

Je remercie Frédérique, scientique polyvalente, à la fois Mathématicienne et Biologiste, avec qui j'ai aussi beaucoup appris. J'apprécie (et j'admire) votre capacité à prendre du recul sur un projet et en avoir une vision à long terme, mais aussi à en gérer l'évolution pas à pas. Je remercie Danielle Monniaux, de l'INRA Tours-Nouzilly, pour toutes nos réunions d'échanges et de confrontations de résultats, qui ont été très constructives, et même indispensables à l'évolution de notre projet. Merci aussi de nous avoir invités à plusieurs reprises à l'INRA, où nous avons eu l'immense privilège de visiter les installations du centre.

Merci à mes anciens professeurs qui ont eu un rôle décisif dans ma scolarité. Merci à Laurent Mertz, moniteur en L3, qui m'a fait découvrir l'analyse numérique des EDOs, et m'a poussé à continuer, au moment où j'allais arrêter mes études. Merci à Martin Vohralik, qui m'a initié en M1 à l'analyse numérique des EDPs. Un grand merci à Christophe Chalons, avec qui j'ai débuté la résolution numérique de systèmes de lois de conservation, et qui m'a fait l'honneur d'accepter de faire partie de mon jury de thèse.

Un immense merci à Frédéric Coquel, qui a été mon enseignant en M2, m'a initié aux problèmes hyperboliques, et m'en a transmis sa passion. Merci de m'avoir contacté pour me mettre en relation avec Marie, avec qui j'ai fait mon stage, qui a débouché sur cette thèse. Cela a été un honneur, et un plaisir, de travailler avec toi, car en plus d'être une encyclopédie vivante des problèmes hyperboliques, tous plus intéressants les uns que les autres, j'ai beaucoup apprécié tes qualités humaines.

Numerical study of multiscale non conservative transport equations modeling cell kinetics Abstract

The thesis focuses on the numerical simulation of a biomathematical, multiscale model explaining the phenomenon of selection within the population of ovarian follicles, and grounded on a cellular basis. The PDE model consists of a large dimension hyperbolic quasilinear system governing the evolution of density functions of cells (microscopic scale) for N f follicles (in practice N f is on the order of twenty).

These equations make use of two structural variables, the age and maturity, which play the role of the space variables. The equations are coupled in a nonlocal way by control terms involving moments of the solution with respect to maturity, dened on either the mesoscopic (follicle) or the macroscopic (ovaries) scale.

Three chapters of the thesis, presented in the form of articles [START_REF] Aymard | Numerical simulation of the selection process of the ovarian follicles[END_REF][START_REF]A numerical method for transport equations with discontinuous ux functions : Application to mathematical modeling of cell dynamics[END_REF][START_REF] Aymard | Adaptive mesh renement strategy for a non conservative transport problem[END_REF], develop the method used to simulate the model numerically. The numerical code is designed to be implemented on a parallel architecture (MPI). PDEs are discretized with a Finite Volume scheme on an adaptive mesh driven by a multiresolution analysis.

Flux discontinuities, at the interfaces between dierent cellular states, require a specic treatment to be compatible with, on the one hand, the high order numerical scheme and, on the other hand, the mesh renement.

A chapter of the thesis is devoted to the calibration method, which translates the biological knowledge into constraints on the parameters and the model outputs.

The multiscale character is crucial here again in the sense that parameters are used at the microscopic level in the equations governing the evolution of the density of cells within each follicle, whereas quantitative biological data are available at the mesoscopic and macroscopic levels.

The development of tests with a signicant biological meaning requires intensive simulations, and the last chapter of the thesis focuses on the analysis of computational performances of the parallel code, based on statistical methods inspired from the eld of uncertainty quantication.

Simulation numérique d'un modèle multi-échelle de cinétique cellulaire formulé à partir d'équations de transport non conservatives Résumé La thèse porte sur la calibration d'un modèle biomathématique multi-échelle expliquant le phénomène de sélection des follicules ovariens à partir du niveau cellulaire.

Le modèle EDP consiste en un système hyperbolique quasi linéaire de grande taille gouvernant l'évolution des fonctions de densité cellulaire pour N f follicules (en pratique N f est de l'ordre d'une vingtaine). Ces équations d'évolution utilisent deux variables structurantes, l'âge et la maturité, qui jouent le rôle de variables d'espace.

Les équations sont couplées de manière non locale par l'intermédiaire de termes de contrôle faisant intervenir les moments en maturité de la solution, intégrée à l'échelle mésoscopique (du follicule) et macroscopique (de l'ovaire).

Trois chapitres de la thèse présentent, sous forme d'articles publiés [START_REF] Aymard | Numerical simulation of the selection process of the ovarian follicles[END_REF][START_REF]A numerical method for transport equations with discontinuous ux functions : Application to mathematical modeling of cell dynamics[END_REF][START_REF] Aymard | Adaptive mesh renement strategy for a non conservative transport problem[END_REF], la méthode développée pour simuler numériquement ce modèle. Elle est conçue pour être implémentée sur une architecture parallèle (MPI). Les EDP sont discrétisées avec un schéma Volumes Finis sur un maillage adaptatif piloté par une analyse multirésolution. Le modèle présente des discontinuités de ux aux interfaces entre les diérents états cellulaires, qui nécessitent la mise en ÷uvre d'un couplage spécique pour être compatible avec, d'une part, le schéma d'ordre élevé et, d'autre part, le ranement de maillage.

Un chapitre de la thèse est dévolu à la méthode de calibration, qui consiste à traduire les connaissances biologiques en contraintes sur les paramètres et sur les sorties du modèle. Le caractère multi-échelle est là encore crucial au sens où les paramètres interviennent au niveau microscopique dans les équations gouvernant l'évolution des densités de cellules au sein de chaque follicule, alors que les données biologiques quantitatives sont disponibles au niveau mésoscopique et macroscopique sur les sorties agrégées en espace.

La mise au point de cas tests signicatifs du point de vue biologique est très consommatrice de temps de calcul, et le dernier chapitre de la thèse porte sur l'analyse des performances numériques du code parallèle, avec une méthodologie empruntée aux méthodes statistiques de propagation d'incertitudes.

A Plume. Les follicules ovariens sont des tissus de géométrie grossièrement sphérique, servant d'abri aux ovocytes (cellules germinales femelles), dont ils accompagnent la maturation, et assurant en partie la fonction endocrine de l'ovaire (sécrétion d'hormones, en particulier l'÷stradiol). Le processus de sélection des follicules ovulatoires, qui a lieu au cours de la phase terminale du développement folliculaire, peut être considéré comme un phénomène de compétition indirecte entre follicules à l'issue duquel un ou plusieurs follicules dominants sont sélectionnés pour l'ovulation. Les autres follicules subissent un processus de dégénérescence et deviennent atrétiques. La sélection est opérée via un contrôle hormonal, la ressource principale nécessaire au développement terminal étant l'hormone hypophysaire FSH (follicle-stimulating-hormone) dont les niveaux sont en retour modulés par les hormones ovariennes produites par les follicules ovariens.

Le processus de sélection s'étend donc sur plusieurs échelles spatiales (voir Figure 1.1), allant de l'échelle microscopique (intracellulaire et cellulaire) à l'échelle macroscopique (organique : niveau ovarien et systémique : axe hypothalamo-hypophysogonadique), en passant par le niveau mésoscopique (tissulaire : follicule ovarien). [START_REF] Aymard | Numerical simulation of the selection process of the ovarian follicles[END_REF]). A gauche : boucle de rétrocontrôle entre les ovaires et l'hypophyse modulant les niveaux de FSH. Au milieu : coupe schématique d'un follicule ovarien. La granulosa, dont certaines cellules entourent l'ovocyte, est délimitée par une cavité liquidienne, l'antrum et par une autre couronne cellulaire, la thèque. A droite : diérents états cellulaires rencontrés par les cellules de granulosa.

Le modèle que nous considérons dans ce manuscrit étudie la dynamique de densités cellulaires évoluant dans un espace fonctionnel âge cytologique × maturité ; il permet de distinguer diérents états cellulaires (cellules en prolifération, cellules diérenciées, cellules en apoptose), mais aussi diérentes phases au sein du cycle cellulaire, et de considérer des termes de vitesse et de second membre spéciques et représentant l'action diérentielle du contrôle hormonal qui s'exerce sur les cellules de granulosa en fonction de leur position dans le domaine. question a ensuite été étudiée rigoureusement dans [START_REF] Shang | Cauchy problem for multiscale conservation laws : Application to structured cell populations[END_REF], en utilisant un argument de type point xe combiné à la méthode des caractéristiques. Ce résultat assure, dans un cadre général, l'existence et l'unicité de la solution du problème, sans hypothèses simplicatrices. Un problème de contrôle optimal a ensuite été étudié sur une expression un peu simpliée du modèle (contrôle s'exerçant en boucle ouverte sur les termes de vitesse, omission du second membre) [START_REF] Clément | Optimal control of cell mass and maturity in a model of follicular ovulation[END_REF]. Les résultats montrent qu'il existe au moins une stratégie de contrôle de type bang-bang, appliquée sur les termes de vitesse, qui permette de maximiser la maturité folliculaire (moment d'ordre un en maturité) en un temps nal xé. Ces résultats, qui généralisent en quelque sorte ceux qui avaient été obtenus sur le modèle compartimental [START_REF] Clément | Optimal control of the cell dynamics in the granulosa of ovulatory follicles[END_REF], sont établis pour une formulation du modèle exprimée sous la forme d'une somme (possiblement innie) de masses de Dirac, et restent à l'état de conjecture pour la formulation EDP. Une autre approche s'est focalisée sur le phénomène de compétition entre follicules, envisagé sous l'angle de la théorie des jeux [START_REF] Michel | Multiscale modeling of follicular ovulation as a mass and maturity dynamical system[END_REF]. En se basant sur des résultats de convergence asymptotique de la maturité, et en formulant quelques hypothèses simplicatrices (une seule phase pour le cycle cellulaire et mitose distribuée), l'auteur parvient à réduire le modèle originel, d'abord de 2D en 1D, puis comme un système d'ODEs couplées et fortement non linéaires. L'étude de ce système permet d'étudier la pression des autres follicules sur une trajectoire folliculaire et de classer les follicules dans diérentes catégories (saved, sensitive et doomed).

Ce modèle a aussi fait l'objet d'études numériques exposées dans [START_REF]Echenim Modélisation et contrôle multi-échelles du processus de sélection des follicules ovulatoires[END_REF] et [START_REF]Hombourger Rapport de stage de 3 ème année. Modélisation multiéchelle du développement folliculaire[END_REF], qui se basent sur des schémas numériques de type volumes nis, implémentés jusqu'ici dans l'environnement académique Bearclaw (Boundary Embedded Adative Renement Conservation LAW package) [START_REF] Leveque | Wave-propagation methods and software for complex applications[END_REF]. Non seulement ces approches ont dû proposer des solutions alternatives pour éviter de traiter directement les conditions de transmission discontinues (par exemple en les régularisant), mais elles ont été confrontées à un problème de temps de calcul pénalisant, qui constituait un obstacle majeur à la réalisation de simulations en grand nombre, ou de grande dimension (c'est-à-dire avec des tailles de cohorte folliculaire réalistes), pourtant indispensables à l'exploitation complète du modèle en terme d'interprétation biologique, que ce soit pour calibrer les valeurs de paramètres en fonction des connaissances disponibles ou pour tester diérents scénarios de sélection.

Ce sont ces dés numériques que nous proposons de relever dans cette thèse, en développant une stratégie combinant l'élaboration d'un schéma numérique dédié, la construction d'un maillage adaptatif compatible avec ce schéma et leur mise en ÷uvre dans un cadre de calcul intensif sur une machine parallèle.

Modèle multiéchelles

Schéma fonctionnel (Block Diagram). Le modèle EDP de processus de sélection des follicules ovariens peut être représenté sous la forme générale d'un schéma avec N f le nombre de follicules.

Système de lois de conservations

Dynamique d'un follicule. A chaque follicule f est associé une dynamique, donnée par : une loi de conservation avec ux contrôlés

∂φ f ∂t + ∂g f (u f )φ f ∂a + ∂h f (u f )φ f ∂γ = -λ(U )φ f (a, γ, t),
une condition initiale :

φ f (a, γ, 0) = φ 0 f (a, γ), des conditions aux limites : condition de périodicité sur l'axe des abscisses φ f (0, γ, t) = φ f (N c , γ, t) ; u f contrôle propre au follicule f , dit mésoscopique ; U contrôle commun à tous les follicules, dit global ou macroscopique ;

des conditions de transmission entre les sous domaines.

Conditions de transmission. Les fonctions de vieillissement g f et de maturation h f sont dénies par sous-domaines, et sont en général discontinues aux interfaces séparant ces derniers. Pour que le problème soit mathématiquement bien posé, il faut imposer des conditions de transmission (voir [START_REF] Shang | Cauchy problem for multiscale conservation laws : Application to structured cell populations[END_REF]).

Ces conditions, qui peuvent s'exprimer en 1D sans perte de généralité, prennent la forme suivante (voir Figure 1.4)

f (φ(x + s , t)) = ψ(f (φ(x - s , t))) frontière en x s .

Le ux à droite de l'interface positionnée en x s , f (φ(x + s , t)), est une fonction ψ du ux à gauche de l'interface, f (φ(x - s , t)). [START_REF] Aymard | Adaptive mesh renement strategy for a non conservative transport problem[END_REF]). Le champ de vitesse est régulier sur chaque sous-domaine Ω 1 , Ω 2 , Ω 3 , mais est potentiellement discontinu aux interfaces séparant ces derniers (en x s et en y s ). An de fermer mathématiquement le problème, il est nécessaire d'imposer des conditions de transmission.

Précisons les conditions de transmission dans notre cas. continuité de ux entre les phases G1 et SM : ψ(F ) = Id(F ), doublement de ux (mitose en cinétique cellulaire) entre les phases SM et G1 :

ψ(F ) = 2F , paroi étanche entre SM et D : ψ(F ) = 0.
Vitesse de propagation selon l'axe des abscisses. La fonction g f dénit la vitesse de vieillissement (propagation selon l'axe des abscisses). En zone G1, où les g f (a, γ, u f ) = g 1 u f + g 2 en phase G1, 1 en phase SM ∪ D.

Sur la Figure 1.5, on représente une coupe horizontale (à gauche) du domaine de calcul 2D (à droite), pour diérentes valeurs de contrôle mésoscopique u f . Plus le contrôle est élevé, plus la vitesse est élevée, favorisant la production de cellules de granulosa.

Vitesse de propagation selon l'axe des ordonnées. La vitesse de maturation des cellules de granulosa peut être dénie comme l'évolution de la réponse d'une cellule à la FSH en terme de synthèse d'AMP cyclique. En phase SM, en notant γ la maturité d'une cellule de granulosa, on a forcément dγ dt = 0, elle se prépare à la mitose. Les cellules sont insensibles à la FSH, et n'évoluent pas en maturation. En phase G1 et D, le niveau évolue selon la sensibilité de la cellule à FSH. Dans [START_REF] Clément | A mathematical model of FSH-induced cAMP production in ovarian follicles[END_REF], la dynamique du couplage entre les récepteurs à FSH et la synthèse de l'AMP cyclique a été modélisée par un système d'EDO non linéaires. L'étude de l'état d'équilibre stable de ce système a conduit, moyennant quelques simplications, à la fonction de maturation suivante Figure 1.6 Fonction de maturation h f sur une coupe verticale (à gauche) du domaine de calcul (à droite), pour diérentes valeurs de contrôle mésoscopique u f , croissante de u 1 à u 5 . Le contrôle mésoscopique règle la position du zéro de h f , qui dénit une maturité asymptotique.

h f (a, γ, u f ) = τ f (-γ 2 + (c 1 γ + c 2 )(1 -exp( -u f ū )
)) en phases G1 ∪ D, 0 en phase SM.

Sur la Figure 1.6, on a représenté la fonction h f sur une coupe verticale (à gauche) du domaine de calcul (panel de droite), pour diérentes valeurs de contrôle. A γ constant, h f est une fonction croissante de u f , et bornée. Un contrôle élevé favorise une maturation rapide.

Terme source : perte par apoptose. La perte de cellules par apoptose est modélisée par un terme source de la forme -Λ(a, γ, U )φ f (a, γ, t),

avec Λ(a, γ, U ) = K exp - (γ -γ s ) 2 γ (U max -U ) U min .
Il est composé de plusieurs termes : le paramètre K règle l'intensité de l'apoptose, le terme en exponentielle assure la nullité du terme source loin de l'interface G1-D, et le terme de contrôle modélise la réponse au contrôle global. Lorsque le contrôle global est maximal, la fonction Λ est nulle, ce qui traduit un environnement favorable à la croissance cellulaire (et donc folliculaire), car il n'y a pas de perte par apoptose.

Inversement, lorsque le contrôle est minimal, la fonction Λ atteint un maximum, ce qui traduit un environnement défavorable pour l'accroissement cellulaire, avec beaucoup de pertes par apoptose.

Sur la Figure 1.7, on a représenté la fonction Λ sur une coupe (à gauche) du domaine de calcul (à droite), pour diérentes valeurs de contrôle global U . Jusqu'à présent, nous avons parlé de dynamique pour chaque follicule séparément, ainsi que de contrôle, mais nous n'avons pas précisé comment s'opère le couplage entre les follicules. C'est l'objet du paragraphe suivant.

Contrôle non local

Couplage des follicules par les moments de la solution. Le couplage des follicules se fait via les termes de contrôle, dénis par les moments des solutions. On dénit les moments d'ordre 1 en γ pour chaque follicule, que l'on appelle maturités folliculaires (échelle mésoscopique),

             m 1 1 (t) = [0,Nc]×[0,1] γφ 1 (t, a, γ)dγda, ... m N f 1 (t) = [0,Nc]×[0,1]
γφ N f (t, a, γ)dγda.

Mathématiquement, la maturité folliculaire va représenter la position verticale moyenne du nuage de densité. Biologiquement, elle quantie la production d'÷stradiol et d'inhibin de chaque follicule, ainsi que la sensibilité à la FSH et à la LH.

A partir des maturités folliculaires, on peut dénir la maturité ovarienne (échelle macroscopique)

M 1 (t) = N f f =1 m f 1 (t).
Cette quantité permet de modéliser la quantité d'÷stradiol et d'inhibin sécrétée par l'ovaire, information reçue par l'axe hypothalamus-hypophyse. Contrôle macroscopique : taux plasmatique de FSH. A partir de la maturité ovarienne, on dénit le terme de contrôle global

U (t) = U max -U min 1 + exp(c(M 1 (t) -M )) + U min ,
Ceci dénit un premier couplage entre les follicules, via le terme source de perte par apoptose.

Sur la Figure 1.8, on représente le niveau plasmatique de FSH (terme U ) en fonction de la maturité globale M 1 . Au départ, le contrôle est maximal, ce qui favorise la croissance folliculaire, et entraîne donc un accroissement de la valeur de M 1 . une fois une valeur seuil atteinte, le niveau plasmatique de FSH décroît vers une valeur minimale, activant le terme de perte par apoptose (avec une intensité de ce terme commune à tous les follicules).

Contrôle mésoscopique : vascularisation et accès du follicule à la FSH.

Une fois le contrôle global déni, on peut dénir le contrôle mésoscopique, propre à chaque follicule. Ce dernier représente la vascularisation, la capacité d'accès à la ressource FSH. On dénit pour cela une fonction de vascularisation b f :

b f (m f ) = min b 1 + e b 2 m 1 (f,t) b 3 , 1
,
puis les termes de contrôle mésoscopiques, qui sont des fractions du terme de contrôle macroscopique Figure 1.9 Fonction de vascularisation b f . Fonction croissante de la maturité folliculaire, jusqu'à l'atteinte de la valeur 1, signe d'un accès sans contrainte au niveau plasmatique de FSH.

u f (t) = b f (m f 1 )U (t) Sur la Figure 1.9, on représente la vascularisation b f en fonction de la maturité folliculaire m f 1 . La vascularisation est une fonction croissante de la maturité folliculaire, jusqu'à atteindre la valeur 1, signiant que le niveau de FSH mésoscopique est égal au niveau macroscopique de FSH.

Temps d'arrêt. Le processus prend n lorsque le niveau de maturité globale, M 1 (t), dépasse un seuil, que l'on note M o . Cela dénit un temps d'arrêt

t f inal = inf{t, M 1 (t) ≥ M o }.
Les follicules sont alors triés en deux classes, suivant leur niveau de maturité folliculaire. Ceux dont la maturité dépasse un seuil, noté M f , sont dits ovulatoires, les autres étant atrétiques.

Apports de la thèse

Dans cette section, nous passons en revue les résultats obtenus dans le cadre de la thèse.

Numerical simulation of the selection process of the ovarian follicles

La première étape de la thèse fut l'écriture et l'implémentation d'un schéma numérique dédié au problème. Le point de départ était le suivant : le modèle est bien posé, et des simulations numériques faites précédemment ( [START_REF]Echenim Modélisation et contrôle multi-échelles du processus de sélection des follicules ovulatoires[END_REF], [START_REF]Hombourger Rapport de stage de 3 ème année. Modélisation multiéchelle du développement folliculaire[END_REF]) ont permis d'identier des valeurs de plages de paramètres donnant des sorties plausibles. Ces simulations étaient réalisées avec le logiciel BearClaw ( [START_REF] Leveque | Wave-propagation methods and software for complex applications[END_REF]), utilisant divers schémas volumes nis en maillage adaptatif : CTU ( [START_REF] Colella | Multidimensional Upwind Methods for Hyperbolic Conservation Laws[END_REF]), et F-Wave ( [START_REF] Bale | A wave propagation method for conservation laws and balance laws with spatially varying ux functions[END_REF]).

Le principal écueil de cette stratégie numérique était la prise en compte des conditions de couplage entre les sous-domaines, qui était contourné en régularisant les discontinuités inhérentes au modèle dans les coecients de vitesse et du terme source.

Cette approximation introduisait de l'imprécision numérique et pénalisait énormément les performances en temps de calcul, car elle nécessitait un ranement excessif du maillage dans les zones de régularisation.

A partir de ce constat, un des buts de la thèse était de dénir puis implémenter une méthode numérique dédiée pour ce type de modèle incluant des équations couplées de manière non locale et posées sur un domaine de calcul avec des conditions de transmissions non triviales aux interfaces entre les sous-domaines.

Nous avons pensé dès le début à l'implémentation de la méthode numérique sur une architecture parallèle, en nous inspirant du schéma fonctionnel représenté sur la Figure 6.1, qui met en relief le comportement "en parallèle" des follicules au sein de l'ovaire.

Nous sommes repartis des équations du modèle. Ces dernières sont des équations de transport scalaires couplées de manières non locales, dénies sur une géométrie simple. Nous avons adopté dans un premier temps un schéma volumes nis upwind sur des mailles cartésiennes uniformes.

En tirant parti du couplage non local des équations, via les moments de la solution, nous avons proposé un algorithme de type SIMD Single Instruction, Multiple Data). On peut citer deux principaux résultats obtenus (voir Figure 1.10). Le schéma proposé est bien convergent (voir Figure 1.10, panel de gauche), et la méthode de parallélisation donne de très bons résultats en test de scalabilité faible (voir Figure 1.10, panel de droite). Il est important de noter que, dans le cadre du test de scalabilité faible eectué pour cette gure, la distribution des paramètres est la même pour tous les processeurs.

Ces résultats ont fait l'objet d'une publication dans ESAIM Proceedings ( [START_REF] Aymard | Numerical simulation of the selection process of the ovarian follicles[END_REF]), et constituent le chapitre 1.

A numerical method for transport equations with discontinuous ux functions : application to mathematical modeling of cell dynamics

Une fois obtenue une méthode de calcul multiéchelle à l'ordre 1, nous avons cherché à augmenter la précision du schéma. Pour cela, nous avons utilisé des techniques de limitation de ux, pour monter en précision en espace, et des méthodes de type Runge-Kutta, pour monter en ordre en temps.

Les méthodes de limitation de ux consistent à écrire les ux numériques aux interfaces séparant les mailles sous la forme [START_REF] Aymard | Numerical simulation of the selection process of the ovarian follicles[END_REF]. Test de convergence en zone G1 (à gauche), et test de scalabilité faible (à droite). Le schéma est bien convergeant, loin des interfaces, et la parallélisation est excellente. avec F L un ux numérique d'ordre 1, F H un ux numérique d'ordre élevé, l une fonction limiteur et r un rapport, indicateur de régularité locale. Il est connu que les schémas d'ordre 1 sont diusifs, et ont tendance à trop lisser les solutions. Les schémas d'ordre élevé n'ont pas ce problème, mais ils génèrent des oscillations aux voisinages des discontinuités. La limitation de ux (ou de pente), permet de pallier ce problème. L'indicateur de régularité est proche de 1 dans les régions où la solution varie peu, et proche de 0 ou de l'inni dans les régions à fort gradient. La fonction limiteur vaut 1 quand r = 1 et tend vers 0 en 0 et en l'inni. Le schéma ainsi construit est hybride. Il est à l'ordre 1 proche des discontinuités, à l'ordre élevé dans les régions où la solution est régulière.

F = F L + l(r)(F H -F L )
Cette méthode ne fonctionne cependant pas aux interfaces non conservatives de type mitose dans notre modèle. Ces interfaces créent localement des discontinuités, ce qui dégrade l'ordre du schéma à l'ordre 1 (vérié numériquement). Nous avons proposé une solution pour éviter ce problème. Tout d'abord, on ne calcule pas un, mais deux ux par interface (un ux à gauche, et un ux à droite). Selon le signe de la vitesse, on calcule d'abord l'un des deux ux, puis le second, en prenant en compte la condition de transmission ψ. Pour que le ux soit calculé à l'ordre élevé, on injecte dans le limiteur non plus la solution, mais la solution via un changement de variable, qui permet de manipuler une quantité conservée. Typiquement, à gauche de l'interface, le schéma numérique ne voit pas la condition de transmission (on donne aux limiteurs des quantités non doublées à l'interface). Pour avoir pleinement l'ordre 3, dans notre cas, il faut passer au limiteur des quantités conservées, sur les interfaces de doublement, mais aussi sur les interfaces directement voisines, à gauche et à droite.

An de valider notre méthode, nous avons créé des cas tests directement inspirés du modèle EDP de processus de sélection des follicules ovariens. Nous avons isolés quatre cas représentatifs. Une fois ces problèmes posés, nous avons proposé des solutions exactes, Pour ce faire, l'idée est d'adapter la méthode des caractéristiques (voir Figure 1 .11). Loin des interfaces, on peut remonter les caractéristiques pour atteindre l'axe t = 0 Lorsqu'une caractéristique atteint une interface, on relève le temps d'atteinte, puis on utilise la condition de transmission pour passer de l'autre côté de l'interface, et enn on remonte la nouvelle caractéristique jusqu'à l'axe t = 0.

x t Figure 1.11 Caractéristiques correspondant à deux équations de transports linéaires à coecients constant avec un contraste de vitesse en x = 0, avec v > 1 pour x < 0 et v < 1 pour x > 0. La solution est constante sur les droites caractéristiques.

Sur la Figure 1.12, on compare la solution exacte d'un problème d'équation de transport avec doublement de ux à l'interface avec la solution approchée utilisant notre méthode, et ce pour diérentes valeur de contraste de vitesse (accélération, vitesse constante ou ralentissement de la densité au passage de l'interface). A chaque fois, on représente à gauche du domaine la densité au temps initial (il s'agit d'une gaussienne). On note que les courbes se superposent, ce qui semble indiquer que notre méthode fonctionne qualitativement.

Ce résultat est conrmé sur la Figure 1.13, panel b), où l'on fait une courbe d'erreur au temps nal en fonction du pas de discrétisation. Il y a plusieurs courbes, indexées par g R , qui représente la vitesse à droite de l'interface, en supposant que le vitesse à gauche est de 1. On note que l'ordre numérique de convergence est supérieur à 2. Sur le panel a), on trace la masse (moment d'ordre 0 de la densité) au cours du temps. On note qu'elle a bien doublée.

On pourra consulter le chapitre 2 pour plus de détails. Figure 1.12 Cas test 1D (Figures extraites de [START_REF]A numerical method for transport equations with discontinuous ux functions : Application to mathematical modeling of cell dynamics[END_REF]) : doublement de ux au passage de l'interface x s = 0.5. Prols de la densité φ(x, t) au temps initial et lorsqu'elle passe l'interface séparant les zones G1 et SM (t = 0.2).

Ces

∆x g R = 0.5 g R = 1 g R = 2 g R = 3
b) Erreur en norme L 1 en fonction de la taille de maille ∆x vitesses de vieillissement et de maturation g f et h f étant précédemment établis, nous avons déni la notion de perte de masse instantanée, et celle de perte cumulée, en s'inspirant de la notion de taux de risque (hazard rate, on pourra consulter [START_REF] Rodriguez | Princeton University Lecture Notes[END_REF] pour plus de détails), pour mieux calibrer le terme de perte par apoptose (voir Figure 1. q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0 10 20 Abstract This paper presents the design and implementation of a numerical method to simulate a multiscale model describing the selection process in ovarian follicles. The PDE model consists in a quasi-linear hyperbolic system of large size, namely N f × N f , ruling the time evolution of the cell density functions of N f follicles (in practice N f is of the order of a few to twenty). These equations are weakly coupled through the sum of the rst order moments of the density functions. The time-dependent equations make use of two structuring variables, age and maturity, which play the roles of space variables. The problem is naturally set over a compact domain of R 2 . The formulation of the time-dependent controlled transport Chapitre 2. Numerical simulation of the selection process of the ovarian follicles coecients accounts for available biological knowledge on follicular cell kinetics. We introduce a dedicated numerical scheme that is amenable to parallelization, by taking advantage of the weak coupling. Numerical illustrations assess th e relevance of the proposed method both in term of accuracy and HPC achievements.

t n φ f f a γ
∂ t φ + ∂ x φ = 0 x φ = 0.05 N m = 10 L = 4
U M 1 m f 1 u f ζ γ m f 0 5 [0, 1] [0.195, 0.225] Chapitre 1. Introduction calcul.

Histogram of x

Résumé Ce document présente la conception et l'implémentation d'une méthode numérique servant à simuler un modèle multiéchelle décrivant le processus de sélection des follicules ovariens. Le modèle EDP consiste en un système hyperbolique quasi linéaire de grande taille, typiquement N f × N f , gouvernant l'évolution des fonctions de densité cellulaire pour N f follicules (en pratique N f est de l'ordre de quelque uns à une vingtaine). Ces équations d'évolution utilisent deux variables structurantes, l'âge et la maturité, qui jouent le rôle de variables d'espace. Le problème est naturellement posé sur un domaine compact de R 2 . La formulation du

Introduction

This work is motivated by a mathematical modelling approach of a complex physiological system, the development of ovarian follicles. The model describes both the cell dynamics within each follicle and the competition process within the population of follicles. The resulting model ( [START_REF] Echenim | Multi-scale modeling of the follicle selection process in the ovary[END_REF]) is a large scale system of weakly coupled quasilinear transport equations, where integro-dierential terms occur both in the velocity and source term. The coupling terms account for the endocrine-based dependence of one follicle dynamics on all other developing follicles. The well-posedness of the Cauchy problem is established in [START_REF] Shang | Cauchy problem for multiscale conservation laws : Application to structured cell populations[END_REF]. Existence and uniqueness of weak solutions is proved for bounded initial conditions. The competition process was investigated in a game theory approach after reducing the PDE model to coupled ODE systems [START_REF] Michel | Multiscale modeling of follicular ovulation as a mass and maturity dynamical system[END_REF].

Control problems associated with this model are investigated in [START_REF] Echenim | Sorine Multiscale modeling of follicular ovulation as a reachability problem[END_REF] (computation of backwards reachability sets) and [START_REF] Clément | Optimal control of cell mass and maturity in a model of follicular ovulation[END_REF] (optimal control in minimal time).

We are specically interested in the numerical issues raised by this multiscale model.

In previous works [START_REF]Multiscale modeling of endocrine systems : new insight on the gonadotrope axis[END_REF][START_REF]Echenim Modélisation et contrôle multi-échelles du processus de sélection des follicules ovulatoires[END_REF], a CTU numerical scheme has been implemented in the Bearclaw 1 environment, which is based on adaptive mesh renement using wave- propagation algorithms [START_REF] Leveque | Wave propagation algorithms for multidimensional hyperbolic systems[END_REF][START_REF] Leveque | Adaptive mesh renement using wavepropagation algorithms for hyperbolic systems[END_REF]. Here, we develop a dedicated code, which allows us (i) to handle the conservative form of the equations, (ii) to deal with the discontinuous coecients and (iii) to use high performance computing (HPC) techniques in order to speed up the computing and be able to simulate as many as twenty follicles. ovarian follicles The paper is organized as follows. In section 2.1, we introduce the biological background and the multiscale model. In section 2.2, we describe the numerical scheme in detail. In section 2.3, we illustrate the simulation outputs and assess the algorithm robustness, accuracy and scalability on parallel architectures. The commitment of a follicle to either ovulation or atresia is driven by the changes occurring in the follicular cell population.

Macroscopic scale : competition process and endocrine feedback loop

In some sense, the terminal part of follicle development can be considered as a com- and estradiol, whose plasma levels are determined by the summed contributions of all maturing follicles (gure 2.1). This ovarian hormonal feedback induces a drop in FSH levels that will penalize the follicles, except those (in the poly-ovulating situations) or that (in the mono-ovulating ones) that are suciently mature to survive in a FSH poor environment. The rising levels of estradiol nally triggers the ovulatory surge that leads to the ovulation of the surviving follicles. At this time they are exposed to a great risk of apoptosis, and may die if the FSH environment is not favorable. After exiting the cell cycle, the cells stop proliferating denitively, but their maturity still increases, so that they contribute more and more to hormone (and especially estradiol) secretion.

Ω = {(a, γ), 0 ≤ a ≤ N c × D a , 0 ≤ γ ≤ 1}
where N c is the number of cell cycles and D a is the duration of one cycle. The dierent cell states described in paragraph 2.1.1.3 take place in three subdomains as illustrated in Figure 2.3. The dierentiation phase D corresponds to the area of the domain where the maturity overcomes the cellular maturity threshold γ s

           G1 = {(a, γ) ∈ Ω, pD a ≤ a ≤ (p + 1/2)D a , p = 0, . . . , N c -1, 0 ≤ γ ≤ γ s }, SM = {(a, γ) ∈ Ω, (p + 1 2 )D a ≤ a ≤ (p + 1)D a , p = 0, . . . , N c -1, 0 ≤ γ ≤ γ s }, D = {(a, γ) ∈ Ω, γ s ≤ γ}.

Hyperbolic system

If we consider, for each follicle, a conservation law for the granulosa cell population density, we obtain the following hyperbolic system, satised by Φ : The initial condition is given by the distribution of the granulosa cell populations at the initial time

                                         ∂φ 1 (a, γ, t) ∂t + ∂(g(a, γ, u 1 (t))φ 1 (a, γ, t)) ∂a + ∂(h(a, γ, u 1 (t))φ 1 (a, γ, t)) ∂γ = -λ(a, γ, U (t))φ 1 (a, γ, t) . . . ∂φ f (a, γ, t) ∂t + ∂(g(a, γ, u f (t))φ f (a, γ, t)) ∂a + ∂(h(a, γ, u f (t))φ f (a, γ, t)) ∂γ = -λ(a, γ, U (t))φ f (a, γ, t) . . . ∂φ N f (a, γ, t) ∂t + ∂(g(a, γ, u N f (t))φ N f (a, γ, t)) ∂a + ∂(h(a, γ, u N f (t))φ N f (a, γ, t)) ∂γ = -λ(a, γ, U (t))φ N f (a, γ, t) (2.

1) ovarian follicles

Φ(a, γ, 0) = Φ 0 (a, γ)
where Φ 0 is compactly supported in ]0, N c D a [×]0, 1[. Since the functional domain Ω is compact, we need to express boundary conditions. Considering some qualitative properties of the maturation function h, dened below, the horizontal top boundary γ = 1 is never reached. Similarly, we can x the number of cell cycles N c in accordance with the ageing function g, so that the right vertical boundary a = N c D a is never reached. For sake of computing simplicity, we therefore assume spatial periodicity on the outer boundaries of Ω in the numerical simulations. The ageing function g appearing in (4.38) is dened by

g(a, γ, u) = g 1 u + g 2 for (a, γ) ∈ G1 1 for (a, γ) ∈ SM ∪ D (2.2)
where g 1 , g 2 are real positive constants. The maturation function h is dened by

h(a, γ, u) = τ h (-γ 2 + (c 1 γ + c 2 )(1 -exp( -u ū ))) for (a, γ) ∈ G1 ∪ D 0 for (a, γ) ∈ SM (2.3)
where τ h , c 1 , c 2 and ū are real positive constants. The source term, that represents cell loss through apoptosis, is dened by

λ(a, γ, U ) =    K exp(-( (γ -γ s ) 2 γ )) × (1 -U ) for (a, γ) ∈ G1 ∪ D 0 for (a, γ) ∈ SM (2.4)
Chapitre 2. Numerical simulation of the selection process of the ovarian follicles where K, γ s and γ are real positive constants.

Remark. The precise values of the constants depends on species and breeds ; they will be xed later.

m 0 (f, t) = 1 0 NcDa 0 φ f (a, γ, t)dadγ, 1 ≤ f ≤ N f (2.5)
This moment is an interesting quantity since it corresponds to an observable variable (the total cell number in one follicle) even if it does not come as such into play in the equations.

follicular maturity (local, one by follicle)

m(f, t) = 1 0 NcDa 0 γφ f (a, γ, t)dadγ, 1 ≤ f ≤ N f (2.6)
ovarian maturity (global, shared by all follicles)

M (t) = N f f =1 m(f, t), (2.7)
plasma FSH level (global, shared by all follicles)

U (t) = U min + 1 -U min 1 + exp(c(M (t) -M )) , (2.8)
where U min , c and M are real positive constants locally bioavailable FSH level

u f (t) = min b 1 + e b 2 m(f,t) b 3 , 1 U (t), (2.9)
where b 1 , b 2 and b 3 are real positive constants.

It is important to notice that the ageing function (5.2) is discontinuous (in general) on the interfaces G1-SM , and that the maturation function (4.40) is discontinuous (in general) on the interfaces SM -D (Figure 2.3). It is necessary to introduce transmission conditions in order to overcome possible failure of uniqueness due to the discontinuities in coecients (see [START_REF] Godlewski | Raviart The numerical interface coupling of nonlinear hyperbolic systems of conservation laws. I. The scalar case[END_REF] or [START_REF] Bouchut | One-dimensional transport equations with discontinuous coecients[END_REF] for instance). The precise denition of the required transmission conditions has been addressed in the paper by Peipei Chapitre 2. Numerical simulation of the selection process of the ovarian follicles Shang ([71]). We suppose that, for each cycle p = 1, . . . , N c and each follicle f = 1, . . . , N f , the ux on the a-axis is continuous between the phases G1 and SM

φ f (t, a + , γ) = (g 1 u f + g 2 )φ f (t, a -, γ), a = (p -1/2)D a , 0 ≤ γ ≤ γ s , (2.10)
and that the ux is doubling on the interfaces SM -G1, which accounts for the birth of new cells at the end of each cell cycle

(g 1 u f + g 2 )φ f (t, a + , γ) = 2φ f (t, a -, γ), a = pD a , 0 ≤ γ ≤ γ s .
(2.11)

Finally, we suppose homogeneous Dirichlet condition to the north of the interface

SM -D φ f (t, a, γ + s ) = 0, (p -1/2)D a ≤ a ≤ pD a .
(2.12)

The time dependent ovarian maturity M (t) dened in (2.7) is compared to the ovarian maturity threshold, denoted by M o , to dene the nal time, beyond which the competition is over

t f inal = inf{t, M (t) ≥ M o }.
(2.13)

The follicles are then sorted into two classes : ovulatory if the maturity is higher than the follicle maturity threshold, denoted by M f , atretic otherwise.

Numerical Method

Discretization

In this paragraph the cycle duration is set to D a = 1. We denote by ∆a (respectively ∆γ ) the space step in the age (respectively maturity) direction. In practice we choose ∆a = ∆γ. The discretization step ∆γ and the cellular maturity threshold γ s must be chosen so that the interfaces where the speed coecients (5.2,4.40) are discontinuous fall on grid points covering the computational domain of size [N c , 1]. Denoting by N m the number of grid cells by half granulosa cell cycle, we set N γ = 2N m and ∆a = ∆γ = 1/N γ and we introduce the dedicated2 notations for grid points (a k , γ l ) and mesh centers (a k+1/2 , γ l+1/2 )

a k = k∆a, a k+1/2 = (k + 1/2)∆a, for k = 0, . . . , N c × N γ , (2.14)
γ l = l∆γ, γ l+1/2 = (l + 1/2)∆γ, for l = 0, . . . , N γ .

(2.15)

Considering that the time step ∆t n may change at every iteration, in order to preserve stability, the time discretization is dened by

t 0 = 0, t n+1 = t n + ∆t n , for n = 0, . . . , N t (2.16)
with N t such that t Nt = t nal . The unknowns are the approximate mean values of the density vector in each grid mesh

Φ n k,l ≈ 1 ∆a∆γ a k+1 a k γ l+1 γ l Φ(a, γ, t n )dγda, for k = 0, ..., N c N γ -1,
and l = 0, ..., N γ -1

whose components φ n f,k,l are the discrete density values for each follicle.

Chapitre 2. Numerical simulation of the selection process of the ovarian follicles

Macroscopic scale : piecewise constant approximation of the hormonal control

We dene the approximation of the control terms (2.5 -4.44) at each time step n = 0, . . . , N t m n 0,f = ∆a∆γ

Nγ -1 l=0 NcNγ -1 k=0 φ n f,k,l , for f = 1, . . . , N f , (2.17)
m n f = ∆a∆γ Nγ -1 l=0 γ l+1/2 NcNγ -1 k=0 φ n f,k,l , for f = 1, . . . , N f , (2.18)
M n = N f f =1 m n f , (2.19)
U n = U min + 1 -U min 1 + exp(c(M n -M )) , (2.20)
u n f = min(b 1 + e b 2 m n f b 3 , 1)U n , for f = 1, . . . , N f .
(2.21)

Mesoscopic scale : nite volume scheme

We use a splitting strategy in order to compute the solution of the PDE system (4.38), which amounts to a convective equation

∂ t φ f (a, γ, t) + ∂ a (g(a, γ, u f (t))φ f (a, γ, t)) +∂ γ (h(a, γ, u f (t))φ f (a, γ, t)) = 0 (convective part),
combined with a source equation

∂ t φ f (a, γ, t) = -λ(a, γ, U (t))φ f (a, γ, t) (source part),
for each follicle f = 1, . . . , N f .

Convective part

The convection part (2.2.3) is treated with a classical nite volume method. The approximate mean values of the solution at t = 0 are initialized using a midpoint formula, accurate at the order 2 in space Φ 0 k,l = Φ 0 (a k+1/2 , γ l+1/2 ).

Using the integral form of the conservation law

     t n+1 t n a k+1 a k γ l+1 γ l (∂ t φ f (a, γ, t) + ∂ a (g(a, γ, u f (t))φ f (a, γ, t)) +∂ γ (h(a, γ, u f (t))φ f (a, γ, t)))dγdadt = 0,
Chapitre 2. Numerical simulation of the selection process of the ovarian follicles we obtain a recursion on the approximate density of each follicle, where we drop the index f for clarity sake

φ n+1 k,l = φ n k,l - ∆t ∆a G k+1,l+ 1 2 (Φ n ) -G k,l+ 1 2 (Φ n ) - ∆t ∆γ H k+ 1 2 ,l+1 (Φ n ) -H k+ 1 2 ,l (Φ n ) (2.22)
where G k,l+ 1 2

(respectively H k+ 1 2 ,l ) is the numerical ux across the vertical edge [(a k , γ l ), (a k , γ l+1 )] (respectively the horizontal edge [(a k , γ l ), (a k+1 , γ l )]) dened by

         G k,l+ 1 2 (Φ n ) ≈ 1 ∆t n ∆γ t n+1 t n γ l+1 γ l g(a k , γ, u f (t))φ(a k , γ, t)dγdt, H k+ 1 2 ,l (Φ n ) ≈ 1 ∆t n ∆a t n+1 t n a k+1 a k h(a, γ l , u f (t))φ(a, γ l , t)dadt. (2.23)
The dependence of the ux functions on all the follicle densities through the control term u f is emphasized by the Φ n argument. It appears in the explicit in time approximations of the speeds (5.2), and (4.40) at the center of the mesh

   g n k,l = g(a k+ 1 2 , γ l+ 1 2 , u n f ), h n k,l = h(a k+ 1 2 , γ l+ 1 2 , u n f ).
Note that the transmission conditions (4.45) and (4.46) on the interfaces where these speed coecients (5.2) and (4.40) are discontinuous are exactly treated with the method described in the article of Godlewski and Raviart ([43]).

The numerical uxes (2.23) are designed using a limiter strategy. Indeed, it is well known that rst order schemes, like the Godunov scheme, are diusive, and that second order schemes, like Lax Wendro scheme, generate oscillations in the neighborhood of discontinuities. In order to get a stable as well as precise scheme, we take a weighting of a low order scheme and a high order scheme, and we dene limited numerical uxes

G = G Low + (r G )(G High -G Low ), H = H Low + (r H )(H High -H Low ), (2.24)
where is a limiter function, for example van Leer function

(r) = r + |r| 1 + r and r G , r H are r G =                g k-1,l φ k-1,l -g k-2,l φ k-2,l g k,l φ k,l -g k-1,l φ k-1,l if g k-2
,l ≥ 0 and g k-1,l ≥ 0 and g k,l ≥ 0,

g k+1,l φ k+1,l -g k,l φ k,l g k,l φ k,l -g k-1,l φ k-1,l if g k-1
,l ≤ 0, and g k,l ≤ 0 and g k+1,l ≤ 0, 0 otherwise, ovarian follicles

r H =                h k,l-1 φ k,l-1 -h k,l-2 φ k,l-2 h k,l φ k,l -h k,l-1 φ k,l-1 if h k,l-2 ≥ 0 and h k,l-1 ≥ 0 and h k,l ≥ 0, h k,l+1 φ k,l+1 -h k,l φ k,l h k,l φ k,l -h k,l-1 φ k,l-1 if h k,l-1 ≤ 0 and h k,l ≤ 0 and h k,l+1 ≤ 0, 0 otherwise.
These ratios are good indicators of the regularity of the function in each direction (see [START_REF] Sweby | High resolution schemes using ux limiters for hyperbolic conservation laws[END_REF]). In fact, a steep gradient or a discontinuity gives a ratio far from 1, whereas a smooth function gives a ratio close to 1.

The rst order uxes entering equation (3.13) are the Godunov uxes

     G Low k,l+ 1 2 (Φ n ) = (g n k-1,l ) + φ n k-1,l + (g n k,l ) -φ n k,l , H Low k+ 1 2 ,l (Φ n ) = (h n k,l-1 ) + φ n k,l-1 + (h n k,l ) -φ n k,l ,
and the high order uxes are the Lax Wendro ones

       G High k,l+ 1 2 (Φ n ) = 1 2 (g n k-1,l φ n k-1,l + g n k,l φ n k,l ), H High k+ 1 2 ,l (Φ n ) = 1 2 (h n k,l-1 φ n k,l-1 + h n k,l φ n k,l ).
Consequently, since ∆a = ∆γ, the CFL condition which guarantees the stability is

∆t n f,F V S ≤ CF L ∆γ max k,l (|g k,l |, |h k,l |) (2.25) with CF L ≤ 1 2 .

Source part

The source part (2.2.3) of the PDE system is explicitly dealt with

φ n+1 k,l = φ n k,l -∆tλ(a k+ 1 2 , γ l+ 1 2 , U n )φ n k,l , (2.26)
which implies a stability condition on the time step, which we strengthen to enforce positivity Denoting by E the evolution operator, that consists in steps (2) and (3), the second order in time is achieved by a second order Runge-Kutta method (Heun)

∆t n f,source ≤ 1 max k,l |λ(a k+ 1 2 , γ l+ 1 2 , U n )| . ( 2
φ * = E(φ n ), φ * * = E(φ * ), φ n+1 = 1 2 (φ n + φ * * ).

Numerical simulations

The code was developed in C++ using the parallel library MPI. The tests were performed on the Jacques-Louis Lions Laboratory super calculator SGI Altix UV 100. The current conguration is 64-core 2 GHz. The visualizations were made with OpenGL and gnuplot.

Initial condition

In the rst test case, we used, for sake of comparison with anterior results, a piecewise constant function similar to the initial condition used in [START_REF]Echenim Modélisation et contrôle multi-échelles du processus de sélection des follicules ovulatoires[END_REF][START_REF] Echenim | Sorine Multiscale modeling of follicular ovulation as a reachability problem[END_REF]. Its support in age ovarian follicles covers the rst cycle while its amplitude is a step-decreasing function of the maturity.

φ 0 (a, γ) =              8 if 0.05 ≤ γ < 0.1 7 if 0.1 ≤ γ < 0.15 5 if 0.15 ≤ γ < 0.2 0 otherwise.        if 0 ≤ a ≤ 1, 0 otherwise.
(2.30)

For the other test cases we used as initial condition a gaussian function centered in

C = (C a , C γ ) φ 0 (a, γ) = 1 2πσ exp 1 2 (a -C a ) 2 σ 2 + (γ -C γ ) 2 σ 2 .
(2.31)

The variance σ 2 can be chosen so that φ 0 be smooth enough to be used for testing the convergence rate, which is checked in the paragraph 3.5.

Sets of default parameters

Since one of our goals was to improve the computational method, we used the same set of parameters as in [START_REF] Echenim | Sorine Multiscale modeling of follicular ovulation as a reachability problem[END_REF] to be able to compare at least qualitatively our results to previous simulation outputs. We distinguished two sets of parameters, one, in On the rst snapshot 2.4 a), we can observe the piecewise constant initial condition (2.30). The cell mass is equal to 1.

On the second snapshot 2.4 b), at time 0.91, we can see that the density has moved to the second cycle. The cell mass is equal to 2.17. At the interface a = 2.5 between zone G1 and SM the ageing function decreases, which results in a local increase in the density, whose prole becomes narrower. The cell density is starting to double at the end of the second cell cycle, on the SM -G1 interface at a = 3.

On the third snapshot 2.4 c), at time 1.57, the cell density has reached the third cycle. It is splitting into a fully dierentiated subpopulation and a still proliferating one. It is worth noticing that, consistently with the model, there is no crossing of the SM -D interface. The cell mass is equal to 3.7.

On the last snapshot 2.4 d), around time t = 5, towards the end of the simulation, the density is concentrated in the D phase above the seventh cell cycle. Even if all cells have exited the cell cycle, we can distinguish three dierent density clouds, each of which being issued from one of the previous cycles. The cell mass is equal to 14.7.

In Figure 2.5 the time evolution of the cell mass (panel A) and follicular maturity (panel B) are displayed as green curves. While the cell mass reaches a constant value as soon as all cells have entered the dierentiating phase, the maturity goes on increasing. The blue curves correspond to a similar simulation with a slower maturation function, where τ h = 1 instead of 1.2. The cells spend more time in the proliferating phase and they enter the dierentiating phase at a later time. Since more mitosis have happened, the cell mass reaches a higher value, around 25 instead of 14.7.

In the rst case, with the faster maturity function, the nal time condition dened by (2.13) is not reached and the simulation is stopped at t max = 7 after 1967 time steps. This is more than the time required to cover the eight cell cycles. Yet at that time, the follicular maturity, displayed on panel B of Figure 2.5, is around 11. far below the follicular maturity threshold of M f = 16.5. In the second case the follicular maturity overcomes the follicular maturity threshold at time t = 6.85 and is around 17. at the end of the simulation. This behavior of the numerical solution is in accordance with the previous simulations presented in [START_REF] Echenim | Sorine Multiscale modeling of follicular ovulation as a reachability problem[END_REF] and [START_REF]Echenim Modélisation et contrôle multi-échelles du processus de sélection des follicules ovulatoires[END_REF].

Competition between ten follicles

The second test consists in a simulation of a competition process between ten follicles. The parameters dening the plasma FSH level are set to U min = 0.9 and c = 10 and the intensity of the source term is set to K = 1. The follicles are distinguished by the parameter dening their ageing rate (5.2) at the origin. We consider a range of g 2 values running from 0.5 to 0.95, with a 0.05 increment from one follicle to the other g 2 = 0.5, 0.55, . . . , 0.95. This setup will be used in future work to mimic realistic congurations where ten to twenty follicles can interact together. However the calibration of the model is still in its preliminary stage and the parameters used in this simulation do not meet exactly the biological specications such as the number of cell cycles performed before ovulation or the nal cell number in ovulatory follicles.

Convergence rate test

We now turn to the validation of the code, which consists in numerically verifying the asymptotic order of convergence when the time step ∆t and space discretization step ∆γ go to zero. We use the parameters in Tables 2.1 and 2.2, except for the gaussian function, whose variance is set to σ 2 = 0.002, so that it is very smooth. The simulation is stopped at t max = 0.05, which allows us to reduce the number of cell cycles to N c = 1 and to discretize the solution on a square grid N γ × N γ . We compute the solution for six dierent levels discretizations N γ = 80, 160, 320, 640, 1280, 2560

with the time discretization provided by the stability condition (2.28).

Convergence of the numerical scheme for the linear transport

We rst study the convergence of the numerical scheme when the initial condition is centred in the SM phase C = (0.7, 0.15) and the nal time is small enough for the density to remain in this phase. The maturity function (4.40) and the source term (4.41) are both null, and the ageing function (5.2) is constant and equal to one.

For such a constant linear transport, we can compute the exact solution at the nal time t Nt = t max using the characteristic method as well as the error in L 1 -norm of the numerical solution

E(∆γ) = ∆γ 2 Nγ -1 k=0 Nγ -1 l=0 φ Nt k,l -φ 0 (a k+ 1 2 -t Nt , γ l+ 1 2
) .

The error curve in log scale is superposed with the theoretical order O(∆γ 2 ) in Figure 2.7.

Full model convergence test

We now set the initial condition near the center C = (0.2, 0.15) of the G1 phase, still with a maximum time t max = 0.05 ensuring that the cell density remains in this zone. In that case the source term (4.41) and the age and maturity speeds are no more constant, and we no longer know the exact solution of the PDE. The error is computed using the solution with N γ = 5120 as reference solution, that is compared to the solution on the other meshes. Since we have used a dyadic renement, the size N ref of the reference mesh in one direction is always a power of two times the ovarian follicles size N ∆γ of the current mesh. Denoting by P = N ref N ∆γ the ratio between the current discretization and the reference nest one, we estimate the discretization error by

E(∆γ) = ∆γ 2 N ∆γ -1 k=0 N ∆γ -1 l=0 φ Nt k,l - 1 P 2 P p=0 P m=0 φ ref kP +p,lP +m
The asymptotic order O(∆γ 1.95 ) which best ts the behavior of E(∆γ) is displayed in Figure 2.8.

HPC test

The improvement in terms of computing time provided by the parallelization is tested through a set of simulations involving an increasing number of identical follicles.

The grid for these simulations consists in one cycle of 200 × 200 cells, and the simulation goes on for 900 times steps. In this experiment, we disposed of enough processors to do the computing with one follicle by processor and since the time command is used to monitor the computing time, the output of the program has been commented out. Figure 2.9 a) shows the real computing time, which is basically the time elapsed from the beginning of the computation. Figure 2.9 b) shows the user computing time, which cumulates all the processors computing time. The computing time cannot be shorter than that required for a computation involving only one, this constitutes the so-called theoretical limit. The fact that the real computing time remains close to the computing time for one follicle and that the user computing time grows linearly with the number of follicles is very encourageing.

The communications between processors appear not to aect signicantly the gain in computing time due to the parallel computing. A small increase in real computing time can be noticed beyond eight follicles. This is due to the super calculator architecture, where the processors are pooled in eight processor nodes. The communications within one node are faster than across dierent nodes. This leads to a threshold eect observed as soon as more than eight processors are needed, therefore involving more than one single node.

Conclusion

This paper summarizes preliminary works in the development of a dedicated software to illustrate numerically the development of follicles. The uniform grid numerical method have been successfully tested in terms of robustness, accuracy and scalability on parallel architecture. As part of a challenging project involving biologists mathematicians and computer scientists, the dierent situations engendered by the model (mono-ovulation, poly-ovulation or anovulation) from given combinations of parameters will now be systematically and intensively tested. In order to achieve this goal within realistic delay the HPC aspect of the method must be enriched using adaptive mesh renement. The multiresolution method developed in [START_REF] Cohen | Fully adaptive multiresolution nite volume schemes for conservation laws[END_REF] Chapitre Abstract In this work, we propose a numerical method to handle discontinuous uxes arising in transport-like equations. More precisely, we study hyperbolic PDEs with ux transmission conditions at interfaces between subdomains where coecients are discontinuous. A dedicated nite volume scheme with a limited high order enhancement is adapted to treat the discontinuities arising at interfaces. The validation of the method is done on 1D and 2D toy problems for which exact solutions are available, allowing us to do a thorough convergence study. We then apply the method to a biological model focusing on complex cell dynamics, that initially motivated this study, and illustrates the full potentialities of the scheme.

Introduction

This paper is dedicated to the study of rst order equations with discontinuous ux functions in the space variable. Our generic setup is the broad eld of coupling problems which has received considerable attention over the past decade.

Let Ω ⊂ R n , n ≥ 1 be a simply connected open set, {Ω i } i=1,...,N a non overlapping collection of subsets of open subsets of Ω, with ∪ N i=1 Ω i = Ω. Then consider the phase space K ⊂ R F , with integer F ≥ 1, a convex open set. For a given T > 0, the problem reads : nd φ : [0, T ] × Ω → K solution of ∂ t φ + div(f i (x, φ)) = S i (x, φ), on (0, T ) × Ω i , for all i = 1, . . . , N , φ(0, x) = φ 0 (x), on Ω, φ(t, x) = 0, on (0, T ) × ∂Ω, ψ i,j (φ(t, x -)) = φ(t, x + ) on (0, T ) × Ω i ∩ Ω j , where f i : Ω i × K → R n×F and S i : Ω i × K → R F are smooth functions.

The last equation models the transmission condition between any pair of adjacent open sets

Ω i and Ω j , hence such that Ω i ∩ Ω j = ∅, for (i, j) ∈ {1, . . . , N } 2 . The transmission condition is prescribed thanks to the smooth function ψ i,j : K → K which allows to dene the right hand trace of φ(t, x + ) at Ω i ∩ Ω j from the left hand trace φ(t, x -) . This setup covers a wide range of applications besides mathematical modeling of cell proliferation which is our current interest here and will be detailed in the next paragraph : multiphase ow in porous media [START_REF] Boutin | Coupling techniques for nonlinear hyperbolic equations. III. well-balanced approximation of thick interfaces[END_REF], trac ow with discontinuous road surface conditions [START_REF] Bürger | Towers A conservation law with discontinuous ux modeling trac ow with abruptly changing road surface conditions , in Hyperbolic problems : theory, numerics and applications[END_REF], sedimentation in thickener-clarier units [START_REF] Bürger | Risebro A relaxation scheme for continuous sedimentation in ideal clarier-thickener units[END_REF] for instance, which have generated a lot of interest and mathematical work in recent years. In a dierent domain of application, hydrodynamic limit for particle systems also lead to hyperbolic conservation laws with discontinuous uxes with f (x, φ) = λ(x)h(φ) as analyzed in [START_REF] Chen | Hyperbolic conservation laws with discontinuous uxes and hydrodynamic limit for particle systems[END_REF].

In all these models, the mathematical diculty lies in proving the well posedness of the initial boundary value problems (IBVP), as it is well summarized in the review [START_REF] Bürger | Conservation laws with discontinuous ux : a short introduction[END_REF]. Already in the scalar case, namely with F = 1, the L 1 contraction principle cannot be inferred from the Kruzkov's analysis [START_REF]Kruºkov First order quasilinear equations with several independent variables[END_REF], since it would require the Lipschitz continuity of the ux in the space variable. Besides classical waves, discontinuous solutions involve additional (standing) discontinuities. On the one hand, those additional waves may prevent the existence of an a priori BV bound for the solutions [START_REF] Adimurthi | Optimal entropy solutions for conservation laws with discontinuous ux-functions[END_REF]. On the other hand, they also prevent classical entropy requirements from restoring uniqueness of the solution, even for the Riemann problem. Several distinct strengthened entropy conditions have been introduced in the literature and Chapitre 3. A numerical method for transport equations with discontinuous ux functions : application to mathematical modeling of cell dynamics we refer for instance the reader to [START_REF] Adimurthi | Optimal entropy solutions for conservation laws with discontinuous ux-functions[END_REF], [5], [START_REF] Chen | Hyperbolic conservation laws with discontinuous uxes and hydrodynamic limit for particle systems[END_REF] and [START_REF] Bürger | Conservation laws with discontinuous ux : a short introduction[END_REF] for a review and exhaustive references. Distinct compactness frameworks based on these entropy conditions have then been proposed to prove well posedness of the problem (see again [START_REF] Adimurthi | Optimal entropy solutions for conservation laws with discontinuous ux-functions[END_REF]5,[START_REF] Chen | Hyperbolic conservation laws with discontinuous uxes and hydrodynamic limit for particle systems[END_REF][START_REF] Bürger | Conservation laws with discontinuous ux : a short introduction[END_REF]).

Here we deal with a non conservative coupling setting where the velocities are nonlocal. As highlighted below, these two new mathematical features arise naturally in a biological context.

We address the model developed in [START_REF] Echenim | Multi-scale modeling of the follicle selection process in the ovary[END_REF] in the general context of cell dynamics, describing the time evolution of a density function depending on age and maturity variables (n = 2). This unknown is governed by a kinetic like equation involving velocities that are function of integro-moments of the unknown and the age and maturity variables. Closure equations for these velocities are naturally discontinuous in the age and maturity variables, precisely at biological checkpoints which correspond to the interfaces between the biological phases [START_REF] Tyson | Temporal organization of cell cycle[END_REF]. These discontinuities require additional information which are handled as local double IBVP, where inner boundary conditions are formulated to express the biological switch.

The conservation hypothesis must clearly be abandoned : cell proliferation (through mitosis 1 ) is modeled by a non conservative coupling and cell death (through apop- tosis) also alters the global mass of the system. An additional specicity of the equation lies in its quasi linear integro dierential nature. The ux functions f i are of the form

f i (x, φ) = u i (x, φ)φ, for i = 1, . . . , N .
They arise from the weak dependence of the velocity eld u on the solution φ through a single moment m f (t)

u i (x, φ) = v i (x, {m f } f =1,...,F ), with m f (t) = Ω yφ f (t, x)dx, where x = (x, y).
More details on the biological background of the model will be provided in Section 4.5 which is devoted to its numerical simulation.

The theoretical study of this model has been done in [START_REF] Shang | Cauchy problem for multiscale conservation laws : Application to structured cell populations[END_REF]. The main diculties come from the nonlocal velocity, the coupling between boundary conditions and the vector nature of the coupling problem (F > 1). Roughly speaking, the specic moment m f (t) of the solution entering the model is shown to exist as the x point of a map from the space of continuous functions. This leads to the existence and uniqueness of rst a local weak solution, from which a global weak solution is then built. Various tools are at hand to compute the numerical approximation of this solution. Let us rst mention that the general case of transport equations with discontinuous equations was rst studied in a series of work by Bouchut and James [START_REF] Bouchut | One-dimensional transport equations with discontinuous coecients[END_REF] using duality arguments. Another standpoint is to address those discontinuities in terms of a coupling problem, as pioneered by Godlewski and Raviart [START_REF] Godlewski | Raviart The numerical interface coupling of nonlinear hyperbolic systems of conservation laws. I. The scalar case[END_REF]. We will adopt this later setting and show that it is well suited to our purpose. We will use techniques proposed in [START_REF] Godlewski | Raviart The numerical interface coupling of nonlinear hyperbolic systems of conservation laws. I. The scalar case[END_REF] and subsequent studies [3,[START_REF] Boutin | Coupling techniques for nonlinear hyperbolic equations. III. well-balanced approximation of thick interfaces[END_REF]. Here we pay special Chapitre 3. A numerical method for transport equations with discontinuous ux functions : application to mathematical modeling of cell dynamics attention to the extension of these algorithms to higher order accuracy, which up to our knowledge, have not yet been addressed in 2D simulations for non conservative coupling and also barely in 2D in general. Although the well posedness issue is not handled with the same mathematical tools as in [START_REF] Shang | Cauchy problem for multiscale conservation laws : Application to structured cell populations[END_REF], the numerical methods that we have designed in this paper can be used in the application contexts cited above, like for instance trac models [START_REF] Seguin | Vovelle Analysis and approximation of a scalar conservation law with a ux function with discontinuous coecients[END_REF].

Outline

From the numerical point of view we are focusing in this work on the discontinuous non conservative uxes. The remaining of the paper is therefore organized in four sections. In Section 2, we describe toy models consisting of linear hyperbolic PDE with piecewise constant speeds and linear source term, in 1D and 2D. In Section 3, we present the numerical scheme, rst recalling the Finite Volume scheme and then specically dealing with the transmission conditions. For piecewise constant speed linear transport with or without linear source term, we design a third order scheme by combining a limited reconstruction in space and a Runge Kutta time quadrature.

We describe in details the numerical scheme at the interfaces where the speed and source coecients are discontinuous, with or without discontinuity of the ux, in order to maintain the precision of the overall computation.

The numerical validation is performed in Section 4 on several 1D and 2D test cases and based on a thorough convergence study. Whenever it is possible, the exact solutions are used as references (the derivation of the analytical solutions is postponed to the Appendix). Finally, we present in Section 5 the biological model describing the cell dynamics in developing ovarian follicles [START_REF] Echenim | Multi-scale modeling of the follicle selection process in the ovary[END_REF] which has motivated our work and perform simulations that validate the numerical method for this application. If this model has already been subject to numerical simulations based on nite volumes (see [START_REF]Echenim Modélisation et contrôle multi-échelles du processus de sélection des follicules ovulatoires[END_REF][START_REF] Echenim | Multi-scale modeling of the follicle selection process in the ovary[END_REF] for instance), the rst rigorous numerical test was performed recently in a preliminary study [START_REF] Aymard | Numerical simulation of the selection process of the ovarian follicles[END_REF], that only dealt with low-order treatment of discontinuities.

The novelty here is the high order enhancement at discontinuous interfaces, which is a particularly challenging question.

Toy models

In this section we present generic toy models with piecewise constant coecients for which exact solutions are available, and which we will use to validate the numerical method. The weak nonlinearity arising in biological models from the dependence of the speeds and source term on the moments of the solution is left out. It is therefore Chapitre 3. A numerical method for transport equations with discontinuous ux functions : application to mathematical modeling of cell dynamics sucient to consider a simplied scalar model dened for (x, y)

∈ Ω = R 2 , t > 0.              ∂φ(x, y, t) ∂t + ∂(g(x, y)φ(x, y, t)) ∂x + ∂(h(x, y)φ(x, y, t)) ∂y = -Λ(x, y)φ(x, y, t) φ(x, y, 0) = φ 0 (x, y) (initial condition) ψ L (g(x - s , y)φ(x - s , y, t)) = g(x + s , y)φ(x + s , y, t) (ux cond. on Ω 1 ∩ Ω 2 ) ψ B (h(x, y - s )φ(x, y - s , t)) = h(x, y + s )φ(x, y + s , t) (ux cond. on Ω i ∩ Ω 3 , for i = 1, 2) (3.1)
with the piecewise constant velocities g(x, y) and h(x, y) and source term Λ(x, y) dened in the subregions depicted on Figure 3.

1, namely Ω 1 = {(x, y) ∈ R 2 , x < x s , y < y s }, Ω 2 = {(x, y) ∈ R 2 , x > x s , y < y s } and Ω 3 = {(x, y) ∈ R 2 , y > y s }.
To x ideas, we can already assume that φ(x, y) represents a density of cells, and that its integral over Ω is the total mass of the system. Furthermore, thanks to the simplicity of the geometry, we can use a cartesian grid discretization and a numerical scheme deduced from the 1D problem by tensorization. We will therefore describe the numerical method on the following 1D toy problem

O Ω 1 Ω 2 x xs ys y Ω 3 g h Figure 3
.1 Spatial set-up for the 2D toy model. g and h respectively denote the velocities in the x and y directions.

       ∂ t φ(x, t) + ∂ x g(x)φ(x, t) = -Λ(x)φ(x, t) for x ∈ R, t > 0 φ(x, 0) = φ 0 (x) (initial condition) ψ L (g(x - s )φ(x - s , t)) = g(x + s )φ(x + s , t) (ux condition) (3.2) with g(x) = g L , for x < x s , g R elsewhere, Λ(x) = Λ L , for x < x s , Λ R elsewhere. (3.3)
In order to deal with all the situations encountered in the real-life problem to be addressed in Section 4.5, the transmission conditions in (3.2) will be in turn ψ L (z) = z (ux continuity between the left and right zones)

(3.4)
ψ L (z) = 2z (doubling of ux between the left and right zones)

(3.5)
ψ L (z) = 0 (waterproof interface between the left and right zones) (3.6) and the source term will be in turn dened by

Λ L = Λ R = 0 (no source) (3.7) Λ L = 1, Λ R = 0 (source in the left zone only) (3.8)
Λ L = 0, Λ R = 1 (source in the right zone only) (3.9) Chapitre 3. A numerical method for transport equations with discontinuous ux functions : application to mathematical modeling of cell dynamics

Numerical scheme

We now turn to the numerical method. We rst recall the nite volume scheme in the continuous regions, and then present the method that we have designed to deal with the transmission conditions.

Discretization

In this paragraph we describe the numerical scheme for the simplied problems (3.1) and (3.2). To solve (3.1) we restrict the computational domain to [0, L x ] × [0, L y ], enforcing periodical boundary conditions on the outer boundaries. We set the domain dimensions as L x = L y = 1 and we select the initial condition such that, at all times, the outer boundaries are far enough from the region where the solution is not zero. We can therefore compare the numerical solution obtained when applying periodic boundary conditions with the exact solutions described in the Appendix.

We denote by N x the number of grid meshes in each direction and by ∆x = L x /N x the space step. The step needs to be chosen so that the locations of the interfaces

x s and y s , where the speed coecients g(x, y) and h(x, y) are discontinuous, fall on grid points. From now on we focus on the 1D nite volume scheme to solve (3.2), since the 2D scheme can be obtained by tensorization. Along the spatial grid

x k = k∆x, x k+1/2 = (k + 1/2)∆x, for k = 0, . . . , N x , (3.10)
the time discretization is dened by

t 0 = 0, t n+1 = t n + ∆t n , for n = 0, . . . , N (3.11)
with N such that t N = t nal , and time steps ∆t n that may change at each iteration, in order to preserve stability. The unknowns are the approximate mean values of the solution in each grid mesh [x k , x k+1 ]

φ n k ≈ 1 ∆x x k+1 x k φ(x, t n )dx. for k = 0, . . . , N x -1. Equation (3.2) is then explicitly discretized to obtain a recursion formula φ n+1 k = φ n k - ∆t n ∆x (F k+1 (φ n ) -F k (φ n )) -∆t n Λ(x k+ 1 2 )φ n k (3.12)
where F k is the numerical ux across x k , designed using a limiter strategy. Indeed, it is well known that rst order schemes, like the Godunov scheme, are diusive, and that second order schemes, like the Lax-Wendro scheme, generate oscillations in the neighborhood of discontinuities. In order to get a stable as well as precise scheme, we take a weighting of a low order scheme and high order scheme, and we dene the limited numerical ux

F k = F Low k + (r k )(F High k -F Low k ), (3.13)
Chapitre 3. A numerical method for transport equations with discontinuous ux functions : application to mathematical modeling of cell dynamics where is the limiter function designed by Koren [START_REF] Koren | A robust upwind discretisation method for advection, diusion and source terms[END_REF] Koren (r) = max(0, min(2r,

2 + r 3 , 2
)),

and r k is dened by                r k = R ((g k+i φ k+i ) i=-2,...,1 ; (g k+i ) i=-2,...,1 ) R ((z k+i ) i=-2,...,1 ; (g k+i ) i=-2,...,1 ) =            z k-1 -z k-2 z k -z k-1 if g k+i ≥ 0 ∀i = -2, . . . , 1 z k+1 -z k z k -z k-1 if g k+i ≤ 0 ∀i = -2, . . . , 1 0 otherwise, (3.14)
with the notation g k = g(x k ). This ratio is a good indicator of the regularity of the function (see [START_REF] Sweby | High resolution schemes using ux limiters for hyperbolic conservation laws[END_REF]). In fact, a steep gradient or a discontinuity gives a ratio far from 1, whereas a smooth function gives a ratio close to 1.

The rst order ux entering equation (3.13) is the Godunov ux

F Low k (φ n ) = (g n k-1 ) + φ n k-1 + (g n k ) -φ n k , (3.15)
and the high order ux is the Lax-Wendro one, which is of second order in space wherever the function g(x) is continuous.

F High k (φ n ) = 1 2 (g n k-1 φ n k-1 + g n k φ n k ), (3.16)
The CFL stability condition is

∆t n ≤ min CF L ∆x max k |g k | , 1 max k |Λ k+ 1 2 | (3.17)
with CF L ≤ 1 2 and the notation Λ k+1/2 = Λ(x k+1/2 ).

Choosing Koren limiter in (3.13) provides third order in space for the convective part of the equation in each domain where it is continuously dened. The source term being discretized by a center point quadrature is at most second order in each domain. Second order in time (third order when Λ = 0) is achieved by a third order Runge-Kutta method ( [START_REF] Gottlieb | Total Variation Diminishing Runge Kutta schemes[END_REF])

B 1 k = F k+1 (φ n ) -F k (φ n ) + ∆xΛ k+1/2 φ n k , φ * k = φ n k - ∆t n ∆x B 1 k , B 2 k = F k+1 (φ * ) -F k (φ * ) + ∆xΛ k+1/2 φ * k , φ * * k = φ n k - 1 4 ∆t n ∆x B 1 k - 1 4 ∆t n ∆x B 2 k B 3 k = F k+1 (φ * * ) -F k (φ * * ) + ∆xΛ k+1/2 φ * * k , φ n+1 k = φ n k - 1 6 ∆t n ∆x B 1 k - 1 6 ∆t n ∆x B 2 k - 2 3 ∆t n ∆x B 3 k ,
We will now focus on the interface between two subdomains where function g can be discontinuous, neglecting the source term for the time being.

55 Chapitre 3. A numerical method for transport equations with discontinuous ux functions : application to mathematical modeling of cell dynamics

Treatment of discontinuous coecients

The domain is discretized in such a manner that the interface between two subregions where g is continuous is placed on an edge between two grid meshes. Let us consider that the interface position x s = x K is at the interface between meshes [x K-1 , x K ] and [x K , x K+1 ] as depicted on Figure 3.2. To adapt the nite volume scheme designed in

F R K x K+1 x K x K-1 φ K+1 φ K φ K-1 φ K-2 F L K Figure 3.

Left and right ux surrounding the interface between meshes

[x K-1 , x K ] and [x K , x K+1 ].
the previous paragraph so that it can handle the transmission condition, we dene for each interface at grid point x k two uxes : the left ux F L k and the right ux F R k (Figure 3.2). Then we rewrite equation (3.12) as

φ n+1 k = φ n k - ∆t n ∆x (F L k+1 -F R k )
If there is no transmission condition on the ux, we have (3.13). In contrast, if there is a transmission condition such as (3.5), we set

F L k = F R k = F k dened by
F R K = ψ L (F L K ). (3.18)
The value of the numerical uxes dened by (3.15) and (3.16) are derived for smooth coecients g(x) and continuous uxes g(x)φ(x). They have therefore to account for the transmission condition ψ L , seen from the left side of x s . We dene, at the interface K, the rst order ux

F Low,L K (φ n ) = (g n K-1 ) + φ n K-1 + 1 2 (1 -sign(g n K ))ψ -1 L (g n K φ n K ), (3.19)
and the second order ux

F High,L K (φ n ) = 1 2 g n K-1 φ n K-1 + ψ -1 L (g n K φ n K ) . (3.20)
Furthermore, the high order enhancement using a limited combination such as (3.13) is aected by the presence of an interface not only at x K but also in its vicinity. Indeed, if the velocity g(x) is positive, the limiter sees interface K at neighboring interfaces k = K -1, . . . , K + 1 through the ratio r k . The transmission condition (3.18) causes the ratio r k (3.14) to depart from 1 on these interfaces and consequently induces a loss of accuracy in the numerical scheme. This can be avoided Chapitre 3. A numerical method for transport equations with discontinuous ux functions : application to mathematical modeling of cell dynamics by computing the ratio r k from the continuous quantities seen from x k . Namely, at interfaces K -1,K,K + 1, the limiter is computed using

       r K-1 = R g K-3 φ K-3 , g K-2 φ K-2 , g K-1 φ K-1 , ψ -1 L (g K φ K ); (g K+i ) i=-3,...,0 r K = R (ψ L (g K-2 φ K-2 ), ψ L (g K-1 φ K-1 ), g k φ K , g K+1 φ K+1 ; (g K+i ) i=-2,...,1 ) r K+1 = R (ψ L (g K-1 φ K-1 ), g K φ K , g K+1 φ K+1 , g K+2 φ K+2 ; (g K+i ) i=-1,...,2 ) (3.21) instead of (3.14).
To x ideas, we apply this method in cases (3.4) and (3.5) encountered in our application model.

First application : Flux continuity with discontinuous speed

In this case the transmission condition is

ψ L = Id. At interface K, the condition is (gφ) R K = (gφ) L K with g K-1 = g K .
The transmission condition (3.18) provides the ux from the right side of the interface

F R K = F L K .
The scheme works well without modication, both at rst and third order.

Second application : Doubling ux

In this case the transmission condition (3.18) is

ψ L (F) = 2F. At interface K, the condition is (gφ) R K = 2(gφ) L K , with g K-1 = g K .
The ux on the left of the interface is computed using continuous values in the denitions of (3.15) and (3.16). From the left side, the continuous values are gφ for x < x K and ψ L (gφ) for x > x K . Therefore we write for the rst order ux

F low,L K = (g K-1 ) + φ K-1 + (g K ) -φ K 2 ,
Chapitre 3. A numerical method for transport equations with discontinuous ux functions : application to mathematical modeling of cell dynamics and for the second order ux

F high,L K = g K-1 φ K-1 + g K φ K 2 2
.

Concerning the limiter, it has to take into account the doubling condition on three meshes in the vicinity of the interface. At interfaces K -1,K,K +1 the limiter (3.21) is computed from the continuous quantities

         r K-1 = R g K-3 φ K-3 , g K-2 φ K-2 , g K-1 φ K-1 , g K φ K 2 ; (g K+i ) i=-3,...,0 , r K = R (2g K-2 φ K-2 , 2g K-1 φ K-1 , g K φ K , g K+1 φ K+1 ; (g K+i ) i=-2,...,1 ) , r K+1 = R (2g K-1 φ K-1 , g K φ K , g K+1 φ K+1 , g K+2 φ K+2 ; (g K+i ) i=-1,...,2 ) .
The transmission condition (3.18) provides the ux from the right side of the interface

F R K = 2F L K .

Numerical validation

In the sequel, we present several numerical simulations to substantiate the validation of our method. Two 1D situations are handled, one with a ux continuity (3.4), another with a doubling at the interface (3.5), with or without a source term (3.7),

(3.8), (3.9). We then present two 2D test cases : a shear phenomenon which is encountered between zones Ω 1 and Ω 3 , and a waterproof condition (3.6), between zones Ω 2 and Ω 3 .

1D test cases

For this set of test cases we can compare the numerical solution at nal time t N with the exact solution described in Appendix 3.6.1 and compute the L1-norm relative error

E ∆x = N x k=0 φ N k -φN k N x k=0 φN k where φN k is the mean value of the exact solution at nal time on mesh [x k , x k+1 ].
This mean value is itself estimated with a quadrature formula the second order central point formula. This is actually justied for the 1D test cases where the discontinuity of the solution coincides with a grid point. In order to recover the expected behavior the initial condition is a smooth gaussian function of total mass equal to 1 and centered on c x =0.3.

φ 0 (x) = 1 2πσ exp 1 2 (x -c x ) 2 σ 2 .
(3.22) Chapitre 3. A numerical method for transport equations with discontinuous ux functions : application to mathematical modeling of cell dynamics test description x s = 0.5. Snapshot of the density φ(x, t) at initial time and when it crosses the interface between zones Ω 1 and Ω 2 (t = t N ).

ψ L (F) g L g R Λ L Λ R p E.10

Test case 1. Continuous ux, speed discontinuity on Ω 1 -Ω 2 interface, no source

This tests mimics what happens at the transition (3.4) between zones Ω 1 and Ω 2 . The speed is equal to 1 in the second half of the domain, and takes values 0.5, 1 (for reference), 2 or 3 in the rst half of the domain. As described in Appendix 3.6.1 the exact solution in that case is

φ exact (x, t) =        φ 0 (x -g L t) for x < x s , g L φ 0 (x s -g L (t -x -x s )) for x s -t ≤ x -t < x s , φ 0 (x -t) for x -t ≥ x s ,
In Figure 3.3, we plot the density φ at a time t N = 0.2/g L , chosen so that half the mass has already crossed the interface between zones Ω 1 and Ω 2 . Since the ux is continuous at x s = 0.5 and the speed is discontinuous, the solution presents a discontinuity (except for the special case g L = 1). We can notice the good quality of the approximation, compared to the exact solution. There are no oscillations in the neighborhood of the discontinuity, neither numerical diusion. Remark that if the speed decreases much on the right side of the interface, the density shape sharpens. This is qualitatively similar to what happens for a trac ow when there is a sudden speed limitation. This phenomenon penalizes the accuracy. In panel b) of Figure 3.4, the L1-norm error with respect to the space step ∆x is displayed in logarithmic scale, which allows us to compute the convergence rate. Depending on the precise value of g L , the convergence rate varies between 2.26 and 2.4. More precisely, it gets worse as the value of g L increases. This can be explained by looking at the contrast between g L and g R = 1. By analogy with the trac ow, the interface between a high speed on the left and a lower one on the right hand side corresponds to a sudden speed decrease which generates an accumulation of trac. The density prole becomes narrower and it becomes harder to make accurate simulations. In contrast, when the speed is higher on the right, the density prole becomes smoother Chapitre 3. A numerical method for transport equations with discontinuous ux functions : application to mathematical modeling of cell dynamics and it becomes easier to approximate the solution. In any case, the best order of convergence 2.4 is achieved when g L = g R = 1. In panel a) of Figure 3.4, the total mass m 0 (t) = Lx 0 φ(x, t)dx is displayed with respect to the time to check whether the scheme is conservative.

We also perform a 2D generalization of this test in the case g L = 2, with a smooth The convergence rates (around 2.4) are gathered in Table 3.4.1.

initial condition

φ 0 (x, y) = 1 2πσ 2 exp( 1 2 (x -c x ) 2 σ 2 + (y -c y ) 2 σ 2
). The density bump lies in zone Ω 2 . x s = 0.5. Snapshot of the density φ(x, t) at initial time and when it crosses the interface between zones Ω 1 and Ω 2 (t = 0.2).

Test case 2. Doubling ux, speed discontinuity on Ω 1 -Ω 2 interface, no source

The second test case addresses ux discontinuity described in paragraph 3.3.2.2.

The speed is equal to g L = 1 for x < x s and takes successively the values g R = 0.5,1,2 or 3 for x > x s . This time the exact solution is

φ exact (x, t) =        φ 0 (x -t) for x < x s , 2 g R (φ 0 (x s -t -(x-xs) g R )) for (x s -g R t) ≤ (x -g R t) < x s , φ 0 (x -g R t) for (x -g R t) ≥ x s ,
In Figure 3.6, we plot the density φ at time t = 0.2. This corresponds to the time at which half the mass has already crossed the interface between zones Ω 2 and Ω 1 . The solution is continuous when g R = 2, but exhibits a discontinuity in slope. We can draw the same conclusions as for the rst test case. In particular, there are no oscillations in the neighborhood of the discontinuity, even if there is a discontinuity in the solution.

In panel b) of Figure 3.7 we plot in a logarithmic scale the L1-norm error with respect to the space step ∆x. The convergence rates, gathered in Table 3.4.1, are similar to the rst test case, which means that the transition condition has not reduced the precision of the scheme. We can notice that the lower the g R , the worse the error. This is a similar situation to the rst test case which can be explained by the same analogy with the trac ow. In panel a) of Figure 3.7 we plot the mass with respect to time. Once all the cells have been transported in the right subdomain, the mass is doubled. As expected, since the speed is always g L = 1 in the left subdomain, in all cases the cells reach the interface at the same time and the mass proles are identical.

∆x g R = 0.5 g R = 1 g R = 2 g R = 3
b) L 1 -norm error with respect to ∆x Test case 3. Continuous ux, speed discontinuity on Ω 1 -Ω 2 interface, linear source on the left

The third test case is the same as the rst one except that we add a linear source term Λ L = 1 in the left part of the domain. The exact solution, detailed in Appendix, is

φ exact (x, t) =        φ 0 (x -g L t) exp(-t) for x < x s , g L φ 0 (x s -g L (t -x -x s )) exp(-t + x -x s ) for x s -t ≤ x -t < x s , φ 0 (x -t) for x -t ≥ x s ,
As in the rst test case, we plot the density at time t N , when half the mass has already crossed the interface between zones Ω 1 and Ω 2 . Two cases g L = 0. .9 1D test case 3 : ux continuity condition and velocity jump at interface x s = 0.5, with source on the left. Snapshot of the density at initial time and when it crosses the interface between zones Ω 1 and Ω 2 (t = t N ).

on the left subdomain, there is a loss in mass until the density bump reaches the interface (see panel a) of Figure 3.10). In the case where g L = 3, the mass stabilizes after the density has passed through the interface. In the case where g L = 0.5 the density bump has not entirely passed through the interface at t = 0.25 so that the mass is not yet stabilized.

Test case 4. Doubling ux, speed discontinuity on Ω 1 -Ω 2 interface, linear source

The fourth test case is the same as the second one except that we add a linear source term Λ R = 1 in the right part of the domain. The exact solution is then

φ exact (x, t) =        φ 0 (x -t) for x < x s , 2 g R (φ 0 (x s -t -(x-xs) g R )) exp( xs-x g R ) for (x s -g R t) ≤ (x -g R t) < x s , φ 0 (x -g R t) exp(-t) for (x -g R t) ≥ x s ,
As in the second test case, we plot the density at time t = 0.2, for two values g R = 0.5 and g R = 3 respectively on panels a) and b) of Figure 3.11. Here also the asymptotic order of convergence for the error is roughly 2.4 (see panel b) of Figure 3.12). The source term is active in the right subdomain. The mass, observed in panel a) of Figure 3.12, rst doubles as the density bump goes through the interface, then diminishes as soon as the loss term becomes active. The drop in mass begins at the same time in both cases g R = 0.5 and g R = 3 since the interface is reached with the same left speed.

2D test cases

We now turn to the validation of the method in 2D. As shown on Figure 3.1, the computing domain includes three internal boundaries : one vertical between zones Ω 1 and Ω 2 , and two horizontal ones, between zones Ω 1 and Ω 3 , and Ω 2 and Ω 3 . The initial condition is centered on (c x , c y ) = (0.3, 0.15). x s = 0.5, linear source on the right. Snapshot of the density φ(x, t) at t = 0 and t = 0.2, when it crosses the interface between zones Ω 1 and Ω 2 .

Error Computing

In 2D the relative error due to discretization is computed with

E(∆x) = N x-1 k=0 N y-1 l=0 |φ N k,l -φN k,l | N x-1 k=0 N y-1 l=0 | φN k,l | (3.24)
where φN k,l is the mean value of the exact solution on the grid mesh [x k , x k+1 ] × [y l , y l+1 ] at time t N . This value is computed using the 2nd order quadrature formula.

2D test case 1 : shear interface

For this test case the speed is vertical g(x, y) = 0, h(x, y) = 1 in the bottom part of the computing domain (Ω 1 ∪ Ω 2 ). In the top part (Ω 3 ), the speed is oblique Chapitre 3. A numerical method for transport equations with discontinuous ux functions : application to mathematical modeling of cell dynamics 0.8 0.9

∆x g R = 0.5 g R = 1 g R = 2 g R = 3
b) L1 norm error Figure 3.12 1D test case 4 : doubling ux condition and velocity jump at interface x s = 0.5, linear source on the right. The mass increases just after the interface, then decreases when the density has passed the interface. The convergence rates (around 2.4) are displayed in Table 3.4.1.

g(x, y) = 1 and h(x, y) = 1. The discontinuity in speed is located on the horizontal internal boundary between Ω 1 and Ω 3 at y s = 0.3. On the snapshots in Figure 3.13, we can observe a shear phenomenon that could not occur in 1D. We also notice that the change of direction of the speed transforms the shape of the density bump, from circular to elliptic. The total mass remains constant. Due to the shear, the asymptotic rate of convergence of the error drops to 2.1 (see panel a) of Figure 3.

14)

.

2D test case 2 : waterproof interface

In this test case, the speed is diagonal g(x, y) = 1, h(x, y) = 1 in zone Ω 1 ∪ Ω 3 , and horizontal g(x, y) = 1, h(x, y) = 0 in zone Ω 2 . On the snapshots in Figure 3.15, we can observe a phenomenon of waterproof interface that could not occur in 1D.

The fraction of the mass which crosses the vertical interface between zones Ω 1 and Ω 2 remains trapped in zone Ω 2 and can only move horizontally. The convergence rate drops to 1 (panel b) of Figure 3.14). In order to understand better the drop in precision for the 2D test case 2, we have studied other values for h(x, y) = h B in zone Ω 2 , ranging from the reference case with no speed variation (h B = 1) to the waterproof test case (h B = 0). The density bump lies in zone Ω 3 . Here the space step for the reference solution is ∆x = 0.0005, corresponding to N f ine x = 2000. The dierent error curves and the variation of the error order with respect to h B are displayed respectively on panel a) and b) of Figure 3.17. This numerical experiment conrms that there is a regular drop in precision as the contrast between the vertical speeds on both sides of the interface between zones Ω 2 and Figure 3.15 2D test case 2 : waterproof interface. Density at nal time (First and third order computations and exact solution). Some numerical diusion aects the shape of the density, due to the splitting of the initial density bump into two separate clouds. Ω 3 increases. It is worth noting that the asymptotic order for the waterproof case (h B = 0) computed using the converged solution is 1.25 instead of the value 1 obtained when using the exact solution in formula (4.35). On the other hand, the asymptotic order for the case with constant diagonal speed (h B = 1) is 2.25, which is less than the asymptotic order 2.4 obtained in the 1D test case 1 for g L = g R = 1.

This small drop in precision is generally encountered with 2D schemes on cartesian grids obtained by tensorization of a 1D scheme.

a) h B = 0.9 b) h B = 0.5 c) h B = 0.001

Application to cell dynamics

In this last section we turn to the numerical simulation of a biological model dedicated to the cell dynamics in ovarian follicular development described in Echenim equations presented in Chapter 3 of [START_REF]Perthame Transport equations in biology[END_REF]). This is however a rough description of the Chapitre 3. A numerical method for transport equations with discontinuous ux functions : application to mathematical modeling of cell dynamics mitosis event since it is distributed over all cell ages. An alternative way to represent mitosis is to consider appropriate boundary conditions coupling the dynamics of the population of proliferating cells with another population of cells that have exited the cell cycles. Instances of corresponding models can be found in the context of hematopoiesis, the process by which blood cells are produced (see e.g. [START_REF] Adimy | A mathematical study of the hematopoiesis process with applications to chronic myelogenous leukemia[END_REF]).

Here, we consider a model where one cannot get rid of discontinuity problems, since not only the mitosis event, but also the distinction between dierent phases of the cell cycle are embedded in the cell population dynamics. Due to the phase-dependent sensitivity of cells to the extracellular signals that make them progress along the cell cycle, discontinuities on the velocities have to be dealt with in addition to the mitosis-induced discontinuity. More precisely, we account for both the Start transition from phase G1 to S and the Exit transition after mitosis completion [START_REF] Tyson | Temporal organization of cell cycle[END_REF].

Besides the mitosis process which increases the total cell mass, the follicular development model takes into account apoptosis, and therefore total mass loss, via a source term which is active only locally in a narrow zone delimited by a skewed gaussian law and centered on the boundary between the rst phase G1 of the cycle and the dierentiation phase. The coecients of the cell loss term depend on the rst moment of the solution. An additional diculty compared to the toy models is the weak coupling between dierent follicles through the generalized moments of the cell density. The HPC issues in the case of realistic numbers of follicles -around 20-have already been addressed in [START_REF] Aymard | Numerical simulation of the selection process of the ovarian follicles[END_REF]. Here we will rst focus on the case of one follicle to study the precision of the scheme, and then illustrate the vector case with a competition between two follicles. Let us denote by F the number of follicles in the general case. The density Chapitre 3. A numerical method for transport equations with discontinuous ux functions : application to mathematical modeling of cell dynamics Φ = (φ f ) f =1,...,F satises the following system of equations, for f = 1, . . . , F :

   ∂φ f (x, y, t) ∂t + ∂(g f (x, y, u f (t))φ f (x, y, t)) ∂x + ∂(h f (x, y, u f (t))φ f (x, y, t)) ∂y = -Λ(x, y, U (t))φ f (x, y, t), (3.26) set in the computing domain Ω in the (x, y) plane, Ω = {(x, y), 0 ≤ x ≤ N c × D c , 0 ≤ y ≤ 1}
                 Ω p 1 = {(x, y) ∈ Ω, pD c ≤ x ≤ (p + 1/2)D c , 0 ≤ y ≤ y s }, p = 0, . . . , N c -1, Θ 1 = ∪ Nc p=1 Ω p 1 Ω p 2 = {(x, y) ∈ Ω, (p + 1/2)D c ≤ x ≤ (p + 1)D c , 0 ≤ y ≤ y s }, p = 0, . . . , N c -1, Θ 2 = ∪ Nc p=1 Ω p 2 Ω 3 = {(x, y) ∈ Ω, y s ≤ y}.
Each cell cycle consists of the Ω p 1 ∪ Ω p 2 subdomain and Θ i for i = 1, 2 denotes the disconnected union of the N c corresponding phases Ω i p , for p = 1, . . . , N c . The aging function g f appearing in (4.38) is dened by

g f (x, y, u) = γ 1 u + γ 2 for (x, y) ∈ Θ 1 1 for (x, y) ∈ Θ 2 ∪ Ω 3 (3.27)
where γ 1 , γ 2 are real positive constants that may depend on the follicle f . The maturation function h f is dened by

h f (x, y, u) = τ h (-y 2 + (c 1 y + c 2 )(1 -exp( -u ū ))) for (x, y) ∈ Θ 1 ∪ Ω 3 0 for (x, y) ∈ Θ 2 (3.28)
where τ h , c 1 , c 2 and ū are real positive constants that may depend on the follicle f . The source term, that represents cell loss through apoptosis, is dened by

Λ(x, y, U ) =    Λ exp(-( (y -y s ) 2 γ )) × (1 -U ) for (x, y) ∈ Θ 1 ∪ Ω 3 0 for (x, y) ∈ Θ 2 (3.29)
where Λ, y s and γ are real positive constants.

M (t) = F f =1 m f (t) (3.31)
Note that in the case of one single follicle m f (t) and M (t) are identical. The plasma FSH level U (t) showing up in the arguments of the source term in (4.38) is dened by

U (t) = U min + 1 -U min 1 + exp(c(M (t) -M )) , (3.32)
where U min , c and M are real positive constants. The locally bioavailable FSH level u f (t) showing up in the arguments of the speeds in (4.38) is dened by The precise denition of the required transmission conditions along the successive cell cycles of the domain has been formulated in [START_REF] Shang | Cauchy problem for multiscale conservation laws : Application to structured cell populations[END_REF]. For each cycle p = 1, . . . , N c , the ux on the x-axis is continuous on the interface between Ω p 1 and Ω p 2 The ux is doubling on the interface between Ω p 2 and Ω p+1

u f (t) = min b 1 + exp(b 2 m f (t)) b 3 , 1 U (t), for f = 1, . . . , F (3.33
φ f (t, x + , y) = (γ 1 u f + γ 2 )φ f (t, x -, y), x = (p -1/2)D c , 0 ≤ y ≤ y s . ( 3
1 , which accounts for the birth of two daughter cells from one mother cell at the end of each cell cycle

(γ 1 u f + γ 2 )φ f (t, x + , y) = 2φ f (t, x -, y), x = pD c , 0 ≤ y ≤ y s . (3.35)
A homogeneous Dirichlet condition holds to the north of the interface between

Ω p 2 and Ω 3 φ f (t, x, y + s ) = 0, (p -1/2)D c ≤ x ≤ pD c . (3.36)
We refer the reader to [START_REF] Aymard | Numerical simulation of the selection process of the ovarian follicles[END_REF] and the references therein for more details on the model, but nevertheless stress out that, as mentioned in the introduction, an important feature, which is inherent to cell dynamics, is the mitosis event occurring at the end of each cell cycle, materialized in our setup by the interfaces between phases Ω p 2 and Ω p+1

1 . In most transport equations dedicated to cell dynamics, it is not accounted for as such but rather by a gain term distributed over the cell population [START_REF]Perthame Transport equations in biology[END_REF]. In [START_REF] Shang | Cauchy problem for multiscale conservation laws : Application to structured cell populations[END_REF], the model with discontinuous coecients was shown to have a unique weak solution, provided that the discontinuities in the velocities were taken into account by ux continuity at the interfaces between phases Ω p 1 and Ω p 2 and ux doubling at the end of each cycle. The Ω 2 phase being a transport in age direction only, null ux at the interface with the dierentiating phase Ω 3 ensures the well posedness. Chapitre 3. A numerical method for transport equations with discontinuous ux functions : application to mathematical modeling of cell dynamics

Numerical simulation

We rst perform a convergence study on the full model. For this time consuming validation, we restrict ourselves to only one follicle F = 1, and we stop the simulation after a duration t N = 1, which is short enough so that the computations on very ne grids remain tractable and would correspond to the transit time across the rst cell cycle if there were no control of the ageing velocity (i.e. if the cell age evolved as time in Ω 1 1 ). Even if, for the chosen values of the biological parameters, gathered in Table 4.2, the rst cell cycle duration is slightly less than one, all of the three interesting transitions :

Ω 1 1 -Ω 1 2 , Ω 1 2 -Ω 2 1 and Ω 2 1 -Ω 3 do

Conclusion

The numerical method presented in Sections 2 and 3 and validated by the numerical convergence study in Section 4 is applied to a biological model for follicular 74 Chapitre 3. A numerical method for transport equations with discontinuous ux functions : application to mathematical modeling of cell dynamics development. Thanks to the original treatment of interface conditions it shows very promising results in terms of accuracy. This numerical tool will enable us to calibrate the model with respect to quantitative biological specications. More extensive simulations with realistic numbers of follicles in competition will be performed to better understand the phenomena under study, like for instance multiple ovulations.

Appendix : Exact solutions

We now give the details of the exact solution computations for the 1D and 2D test cases. In all cases, the solutions will exhibit well dened traces on the left and right hand sides of the discontinuity. Those limits will be explicit in the solution.

Exact solutions of the 1D problem with piecewise constant speeds

Problem

                           ∂ t φ + ∂ x gφ = Λφ on R + × (-∞, x s [∪]x s , +∞) g(x) = g L if x < x s (discontinuous speed) g R if x ≥ x s Λ(x) = Λ L if x < x s (discontinuous speed) Λ R if x ≥ x s φ(x, 0) = φ 0 (x) (initial condition) ψ L (g L φ(x - s , t)) = g R φ(x + s , t) (ux condition) (3.37)
Case without source term We rst solve this problem when Λ L = Λ R = 0, using the method of characteristics. The case when g R < g L is displayed in Figure 3.22. If there were no transmission conditions, the characteristics would be straight lines passing through the vertical axis. The analysis is detailed in the case where g L and g R speeds are positive which is the most usual situation in problems arising from biology.

Solution For x < x s , considering the characteristics of

∂ t φ + g L ∂ x φ = 0, φ(x, 0) = φ 0 (x),
the solution is φ(x, t) = φ 0 (x-g L t) for x < x s . Using the same argument, considering the characteristics of

∂ t φ + g R ∂ x φ = 0, φ(x, 0) = φ 0 (x),
we nd that φ(x, t) = φ 0 (xg R t) for xg R t > x s . The solution for x sg R t <

xg R t < x s is obtained thanks to the transmission condition ψ L (g L φ(x - s , t)) = Chapitre 3. A numerical method for transport equations with discontinuous ux functions : application to mathematical modeling of cell dynamics

g R φ(x + s , t), combined with the fact that φ(x - s , t) = φ 0 (x s -g L t), which leads to g R φ(x + s , t) = ψ L (g L φ(x - s , t)) = ψ L (g L φ 0 (x s -g L t)). Dening a trace function T r(t) = 1 g R ψ L (g L φ 0 (x s -g L t)), (3.38)
which acts as a boundary condition, we follow the characteristics

ẋ(t) = g R , x(t s ) = x s ,
given by x(t) = x s + (tt s )g R , to reach x s at time

t s = t - (x -x s ) g R . (3.39)
Finally, the solution is

φ(x, t) =        φ 0 (x -g L t) for x < x s , T r(t s ) for x s -g R t < x -g R t < x s , φ 0 (x -g R t) for x -g R t < x s ,
where T r is dened by condition (4.27) and t s is dened by (4.28).

Case with source term It can be deduced from the homogeneous case by changing the unknowns. In the subregion where

Λ is constant, if φ(x, t) is solution of ∂ t φ + ∂ x gφ = -Λφ then φ(x, t) = exp(Λt)φ(x, t) is solution of ∂ t φ + ∂ x g φ = 0. We distinguish two cases :
With a source term in the left subdomain Λ L > 0 and Λ R = 0, the solution is

φ(x, t) =        φ 0 (x -g L t) exp(-Λt) for x < x s , T r(t s ) exp(-Λt s ) for x s -g R t < x -g R t < x s , φ 0 (x -g R t) for x -g R t > x s .
where T r is dened by the transmission condition (4.27) and t s is dened by (4.28).

With a source term in the right subdomain

Λ L = 0 and Λ R > 0, the solution is φ(x, t) =        φ 0 (x -g L t) for x < x s , T r(t s ) exp(-Λ(t -t s )) for x s -g R t < x -g R t < x s , φ 0 (x -g R t) exp(-Λt) for x -g R t > x s .
where T r is dened by the transmission condition (4.27) and t s is dened by (4.28). Chapitre 3. A numerical method for transport equations with discontinuous ux functions : application to mathematical modeling of cell dynamics 3.6.2 Exact solution of the 2D problems with piecewise constant speeds

3.6.2.1 Horizontal speed Problem                  ∂ t φ + ∂ x gφ = 0 on R + × (-∞, x s [∪]x s , +∞) × R g(x, y) = g L if x < x s (discontinuous speed) g R if x > x s φ(x, y, 0) = φ 0 (x, y) (initial condition) ψ L (φ(x - s , y, t)) = φ(x + s , y, t) (ux condition) (3.40) Solution The solution is φ(x, y, t) =        φ 0 (x -g L t, y) for x < x s , T r(t s , y) for x s -g R t < x -g R t < x s , φ 0 (x -g R t, y) for x -g R t > x s ,
where T r is dened by the transmission condition T r(t, y) = ψ L (φ 0 (x sg L t, y)).

(3.41) and t s is dened by (4.28).

Shear

Problem

       ∂ t φ + ∂ y φ = 0 on R + × R × (-∞, y s [ ∂ t φ + ∂ x φ + ∂ y φ = 0 on R + × R×]y s , +∞) φ(x, y, 0) = φ 0 (x, y) (initial condition) (3.42)
Solution The characteristics are of the form

x(t) = x 0 + t D y(t) = y 0 + t s + t D
with t D = yy s the time spent in the upper zone. We can then dene the solution piecewise φ(x, y, t) = φ 0 (x, yt) for (x, y) ∈ R × (-∞, y s [ φ 0 (xt D , yt) for (x, y) ∈ R×]y s , +∞) Chapitre 3. A numerical method for transport equations with discontinuous ux functions : application to mathematical modeling of cell dynamics

3.6.2.3 Waterproof Problem        ∂ t φ + ∂ x φ = 0 on R + ×]x s , +∞) × (-∞, y s [ ∂ t φ + ∂ x φ + ∂ y φ = 0 on R + × (∞, x s [×(-∞, y s [ and R + × R×]y s , +∞) φ(x, y, 0) = φ 0 (x, y) (initial condition) (3.43)
In order to close this problem, we have to add a homogeneous Dirichlet condition on the north of the limit between the 2 zones φ(x, y + s , t) = 0 for x ∈]x s , +∞)

Solution As in the precedent case the characteristics are of the form

x(t) = x 0 + t D , y(t) = y 0 + t s + t D .
We can then dene the solution piecewise

φ(x, y, t) =           
φ 0 (xt, yt) for (x, y) ∈ R×]y s , +∞) and xt D < x s 0 for (x, y) ∈ R×]y s , +∞) and xt D > x s φ 0 (xt, yt) for (x, y) ∈ Ω 1 φ 0 (xt, yt s ) for (x, y) ∈ Ω 2 Chapitre 3. A numerical method for transport equations with discontinuous ux functions : application to mathematical modeling of cell dynamics

a) t = 0.66, cell cycles 1 -2 transition Ω 1 2 → Ω 2 1 . φ max = 140 b) t = 2.58, cell cycles 3 -4 transition Ω 3 2 → Ω 4 1 and Ω 4 1 Ω 3 , φ max = 430 c) t = 3.

Introduction

When studying phenomena over long periods in the context of transport equations, one has to make simulations on large computational domains so as to follow the interesting part of the solution throughout the duration of the experiment. On the other hand it can happen that the solutions of the problem occupy only a bounded and time varying portion of the domain. Such set-ups can be very costly to simulate numerically with uniform grids. Time-adaptive mesh renement methods enable one to minimize the size of the mesh, hence both the memory requirement and CPU time, by restricting the use of ne discretization to the areas of interest, or more generally to the regions where the solution varies rapidly.

Among these methods, the Multiresolution (MR) method is specially interesting for hyperbolic systems of PDEs. Firstly, the underlying wavelet theory enables one to control the numerical error induced by the adaptive scheme with a small parameter ε that is explicitly used as a threshold in the algorithm. Secondly, MR provides one with a strategy to monitor the evolution in time of the adaptive grid, relying on the hyperbolicity of the equations to predict the displacement of the singularities. We refer the reader to [START_REF] Cohen | Fully adaptive multiresolution nite volume schemes for conservation laws[END_REF][START_REF] Muller | Adaptive multiscale schemes for conservation laws[END_REF] for a thorough analysis of the MR method applied to a scalar hyperbolic equation and to [START_REF] Cohen | Postel Adaptive multiresolution for nite volume solutions of gas dynamics[END_REF][START_REF] Coquel | Entropysatisfying relaxation method with large time-steps for Euler IBVPs[END_REF] and references therein for applications to more realistic models and set-ups. Although originally designed for and theoretically studied in the case of hyperbolic equations, Multiresolution is currently used to simulate reaction-diusion and parabolic equations with interesting computational performances, as illustrated in [START_REF] Coquel | Tran Multiresolution and adaptive methods for convection-dominated problems[END_REF][START_REF] Louvet | Massot Summer school on multiresolution and adaptive mesh renement methods[END_REF].

In the current work we deal with non conservative transport equations frequently arising in biological models involving cell dynamics. In these models, which are most of the time 1D, the space variable plays the role of age and marks the progression along the cell cycle. Mitosis is the endpoint of the cell cycle, that gives birth to two daughter cells from one mother cell. From the modeling viewpoint, mitosis can be distributed over the cell cycle thanks to a positive linear source term, amounting to doubling the cell density in a time equivalent to the duration of the cell cycle (see for instance [START_REF]Perthame Transport equations in biology[END_REF]). Another approach treats the mitosis phenomenon more directly, as a localized event at the end of each cell cycle, corresponding to a doubling transmission condition which must be satised by the cell density at the interface between cell cycles (see for instance [START_REF] Shang | Cauchy problem for multiscale conservation laws : Application to structured cell populations[END_REF]). These doubling transmission conditions are handled thanks to discontinuous ux conditions at the corresponding interfaces. transport problem Such conditions are tricky to discretize numerically, even on a uniform grid, and require a specic scheme in order to achieve high order of accuracy. We refer the reader to [START_REF]A numerical method for transport equations with discontinuous ux functions : Application to mathematical modeling of cell dynamics[END_REF] for a complete study of a Finite Volume (FV) scheme well suited to this purpose. We validate our adaptive strategy on both the localized and distributed mitosis models. The specic application we have in mind is the numerical calibration of terminal follicular development modeled in [START_REF] Echenim | Multi-scale modeling of the follicle selection process in the ovary[END_REF]. Realistic simulations involve solving simultaneously about twenty coupled PDEs in two space dimensions, during about one dozen of cell cycles, with an initial cell population concentrated in the rst cell cycle. The weak structure of the coupling is suited to an ecient implementation on parallel architecture, already described in [START_REF] Aymard | Numerical simulation of the selection process of the ovarian follicles[END_REF]. Nevertheless one simulation takes several minutes to run and the extensive simulations required to calibrate the model therefore motivate our search for additional reduction of computational costs by using a MR adaptive nite volume strategy. In this paper we address the problem arising from coupling the MR strategy with the FV method designed in [START_REF]A numerical method for transport equations with discontinuous ux functions : Application to mathematical modeling of cell dynamics[END_REF] in the scalar case and with a single processor architecture, leaving the vector case with parallel implementation to future works. Actually, the main diculty arising from this coupling is already present in the scalar case, where the resulting adaptive scheme (FVMR) unfortunately exhibits unstable numerical noise. The mesh adaptation is tuned by thresholding the details of the MR analysis, which introduces oscillations. In the standard conservative set-up, this spurious noise is bounded by a small parameter, and remains so at future times. In our non conservative situation it is amplied by the discontinuous ux conditions and becomes apparent only after long time simulations -let us say more than ten cell cycles. This corresponds to a proliferation ratio of 2 10 which explains how any numerical error, well within any given tolerance at the beginning of the simulation, can end up having macroscopic consequences on the solution mesh or even its shape (see Figure 4.14). It is a quite specic set-up, obviously not encountered in all PDEs with discontinuous uxes.

Hopefully they do not all require any particular adaptation of the MR method. This is for instance the case for Multiresolution applied to a trac ow model [START_REF] Bürger | Sepúlveda Fully adaptive multiresolution schemes for strongly degenerate parabolic equations with discontinuous ux[END_REF] with only one (xed and a priori known) ux discontinuity. If the discontinuity is well captured by the initial graded tree, it will remain that way, which explains why no specic strategy is needed. We will show that the non conservative nature of the equation is indeed the factor responsible for the noise amplication.

The main object of this article is therefore twofold : rst to fully document the unstable noise appearance when using the standard MR method on such an equation with doubling conditions. This is done in section 4.2 where we introduce a simple 1D model and detail the elementary numerical steps, showing how they lead to spurious numerical noise. Secondly, we propose a numerical strategy to stop the noise appearance, and then validate it by extensive numerical tests. This numerical validation, exposed in section 4.3, uses the FVMR method that we also use in the latest section 4.4 to solve more realistic biological models. Eventually we show that our method correctly handles both types of models -localized or distributed mitosiswith CPU and memory gain between 10 and 20 for an overall relative error remaining below 10 -3 .

Investigation of numerical noise appearance

In this section we investigate the circumstances under which numerical noise can appear. For this we consider a simplied 1D problem with piecewise constant speeds.

We compute the approximated solution with a simple rst order FV scheme. This allows us to perform one time step of the algorithm by hand and completely describe the interaction of the doubling condition with a two-level MR algorithm. We illustrate this example by numerical simulations performed with a method coupling the 2D FV scheme with a mean value MR adaptive strategy.

Description of the simplied set-up

We study the simple case of a 1D transport model with piecewise constant speeds and discontinuous ux conditions

∂ t φ(t, x) + ∂ x (g(x)φ(t, x)) = 0, for x ∈ R, t > 0 φ(0, x) = φ 0 (x), for x ∈ R, (4.1)
with N c interfaces numbered from 1 to N c spaced by x s . The speeds are constant in each subdomain

g(x) = g p-1 for (p -1)x s ≤ x ≤ px s , p = 1, . . . , N c + 1.
Transmission conditions are dened by

g p φ(t, px + s ) = k p g p-1 φ(t, (p -1)x - s ) (4.2)
where k p is equal to 2 to model a doubling ux interface, or to 1 to model a continuous ux interface.

For the sake of simplicity we rst choose g p = 1 and k p = 2 for all p = 0, . . . , N c and we apply a doubling condition on the outer boundary

φ(t, 0 + ) = 2φ(t, N c x - s ).
This model is a simple prototype for transport equations modeling cell proliferation with localized mitosis at the end of each cell cycle, which is relevant for the granulosa cells in ovarian follicles [START_REF] Echenim | Multi-scale modeling of the follicle selection process in the ovary[END_REF]. We will also consider an alternative model frequently encountered in the literature (see for instance [START_REF]Perthame Transport equations in biology[END_REF] chapter 3), which consists in distributing the mitosis over the cell cycle duration, with a linear source term

∂ t φ(t, x) + ∂ x (g(x)φ(t, x)) = Bφ(t, x), for x ∈ R, t > 0, φ(0, x) = φ 0 (x), for x ∈ R. (4.
3)

The coecient B in the source term must be such that the averaged total mass is doubled at the end of a cell cycle, which leads to

exp(BT ) = 2,
where T is the duration of the cycle.

D 0,K-1 D 0,K D 1,2K-1 D 1,2K ∆x 0 = 2∆x D 1,k = [k∆x, (k + 1)∆x], D 0,k = D 1,2k ∪ D 1,2k+1 = [k∆x 0 , (k + 1)∆x 0 ]. , k = 0 = 1 x s = 2K∆x = K∆x 0 th D 0,pK-1 D 0,pK D 1,2pK-1 D 1,2pK p = 1 φ n 1,k ≈ 1 ∆x D 1,k φ(n∆t, x)dx, φ n 0,k = 1 2 φ n 1,2k + φ n 1,2k+1 . φn 1,2k = φn 1,2k+1 = φ n 0,k , φn 1,2k = 1 8 φ n 0,k-1 + φ n 0,k - 1 8 φ n 0,k+1 , φn 1,2k+1 = - 1 8 φ n 0,k-1 + φ n 0,k + 1 8 φ n 0,k+1 .
Chapitre 4. Adaptive mesh renement strategy for a non conservative transport problem

We use a FV scheme to compute the numerical solution as

φ n+1 j = φ n j - ∆t ∆x j F n j+1/2 -F n j-1/2 , (4.7)
where F n j+1/2 is the numerical ux at time t n through the edge between meshes D j and D j+1 , and ∆x j the size of mesh D j . A simple upwind rst order scheme on a uniform grid consists, in this case with unit speed, in taking

F n j+1/2 = φ n j if j = 0, pK -1 and 1, 2pK -1, 2φ n j otherwise, (4.8)
where the factor 2 in bold takes into account the doubling ux condition (4.2) when k p = 2, for p = 1, . . . , N c.

We can therefore compute the numerical solution on the ner level as

φ n+1 1,k = (1 -λ)φ n 1,k + λφ n 1,k-1 if k = 2pK, (1 -λ)φ n 1,k + 2λφ n 1,k-1 if k = 2pK, for p = 1, . . . , N c, (4.9)
with λ = ∆t ∆x , which must satisfy 0 < λ ≤ 1 in order to ensure the stability of the solution. If the solution is smooth enough -with a criterion that we will precise below -we can compute it instead on the coarser level, using λ/2 instead of λ

φ n+1 0,k = (1 -λ 2 )φ n 0,k + λ 2 φ n 0,k-1 if k = pK, (1 -λ 2 )φ n 0,k + λ 2 2φ n 0,k-1 if k = pK, for p = 1, . . . , N c.
Note that the numerical solution becomes locally small but nonzero, even when starting from a piecewise constant solution : due to the numerical diusion of the numerical scheme, the solution rapidly becomes nonzero upstream of the initial condition. For instance for the simple scheme (4.9) we have

φ n 1,k = (1 -λ)φ n-1 1,k + λφ n-1 1,k-1 , = (1 -λ) 2 φ n-2 1,k + 2(1 -λ)λφ n-2 1,k-1 + λ 2 φ n-1 1,k-2 , . . . = . . . = (1 -λ) n φ 0 1,k + . . . + λ n φ 0 1,k-n .
(4.10)

In the general case with an arbitrary number of levels and arbitrary high dimensions, one time step of the FVMR scheme consists in a loop over levels starting from the nest. For each level, a rst loop computes the numerical uxes between meshes active at the current level using (4.8), possibly requiring local reconstruction if neighboring meshes entering the ux estimation are coarser than the current level.

Then a second loop updates the solution on the meshes of the current level using (4.7).

Having in mind this algorithm is necessary to correctly handle the cases of neighboring meshes of dierent sizes if the left hand side neighbor of mesh D 1,2k is the coarser mesh D 0,k-1 , then the solution φn 1,2k-1 at time n on mesh D 1,2k-1 must be reconstructed, using (4.5) or (4.6), in order to compute F n (1,2k)-1/2 if the left hand side neighbor of mesh D 0,k is the ner mesh D 1,2k-1 , then

F n (0,k)-1/2 = F n (1,2k-1)+1/2 = φ n 1,2k-1
has already been computed during the loop on the ner level if the right hand side neighbor of mesh D 1,2k+1 is the coarser mesh D 0,k+1 , no reconstruction is required to compute F n

(1,2k+1)+1/2 for the simple rst order scheme (4.8). In the case of a higher order scheme, φn

1,2k+2 would have to be reconstructed if the right hand side neighbor of mesh D 0,k is the ner mesh D 1,2k+2 , then

F n (0,k)+1/2 = F n (1,2k+2)-1/2 = φn 1,2k+1
has already been computed during the loop on the ner level, using reconstructed value φn

1,2k+1

The choice between the two levels of discretization on a given mesh D 0,k in the domain is made by comparing the solution on its subdivisions D 1,2k ∪ D 1,2k+1 with the solution reconstructed on the same mesh using the values on the coarser grid with formulas (4.5) or (4.6). The local smoothness of the solution is indeed measured by the detail, dened by

d n 1,k = φ n 1,2k -φn 1,2k (4.11)
In the Haar case, a null detail corresponds to a locally constant function

d n 1,k = φ n 1,2k -φ n 0,k = 1 2 (φ n 1,2k -φ n 1,2k+1
).

(4.12)

In the quadratic basis, a null detail corresponds to a locally quadratic function

d n 1,k = φ n 1,2k - 1 8 φ n 0,k-1 + φ n 0,k - 1 8 φ n 0,k+1 = - 1 16 (φ n 1,2k-2 + φ n 1,2k-1 ) + 1 2 (φ n 1,2k -φ n 1,2k+1 ) + 1 16 (φ n 1,2k+2 + φ n 1,2k+3
). We refer for instance to [START_REF] Harten | Multiresolution algorithms for the numerical solutions of hyperbolic conservation laws[END_REF] for the approximation theory results establishing the link between d n 1,k and the local smoothness of the solution.

In order to monitor the adaptive FV scheme, the absolute value of the details are tested at each time step against a level dependent threshold ε ε = ε2 -L+1 , for = 1, . . . , L -1.

Γ n ε = {( , k), ||d ,k || ≤ ε},
and enlarged into a set Γn+1 ε according a heuristic rule [START_REF] Harten | Multiresolution algorithms for the numerical solutions of hyperbolic conservation laws[END_REF], accounting for singularity displacement (and formation in the nonlinear case) between time t n and t n+1 . Here in the two level case, if |d n 1,k | ≥ ε 1 , then not only D 0,k but also the two neighbors D 0,k-1 and D 0,k+1 should be subdivided to compute the solution at the next time step. The whole predictive procedure of adaptive grid renement is summarized in the general case with L MR levels in Algorithm 1. The solution (φ n ,k ) ( ,k)∈ §n+1 is reconstructed on the adaptive grid at any given time t n with Algorithm 2. It is then updated to time t n+1 with a FV scheme. Theoretically and practically although it implies to renounce the MR computational benets it is also possible to compute the reconstructed solution φ n ε on the nest grid using Algorithm 2 and

setting all d n l,k = 0 for (l, k) ∈ Γn+1 0 \ Γn+1 ε .
In the case of a scalar hyperbolic equation numerically discretized with a Lipschitz ux, the L 1 norm error a n between φ n ε and the FV solution on the nest level φ n unif has been shown in [30] to be bounded

a n := ||φ n unif -φ n ε || ≤ C(t n )ε. (4.15)
The threshold ε should be chosen so that a n is of the same order as the estimate of the discretization error e n

e n := ||φ n unif -φ n exact || ≤ C(t n )∆x p , (4.16)
where p is the numerical order of the reference FV scheme on the uniform grid. Of course in realistic set-ups the exact solution is unavailable and one also wishes to avoid computing the FV reference solution on the ne grid. Estimates (4.15) and (4.16) can be used to calibrate the threshold value on a smaller scale domain in time and space, or for a simplied model. In our case, we will see in the numerical tests that choosing ε between 10 -2 and 10 -3 supplies a good compromise between accuracy and eciency.

Explication of the numerical noise appearance

In this paragraph we describe the mechanism for numerical noise appearance due to the coupling of Multiresolution with doubling uxes.

Suppose that |d n K-i | < ε, for i ≥ 0 and |d n K+1 | ≥ ε. Due to the preceding rule, the mesh D 0,K = D 1,2K ∪ D 1,2K+1 is subdivided, to account for possible displacement of the singularity (see Algorithm 1). The mesh D 0,K-1 is kept at the coarser level, but the value of the solution on its right subdivision D 1,2K-1 , which enters the FV scheme to advance the solution on D 1,2K , is available using the prediction operator

φ n+1 0,K-1 = φ n 0,K-1 - λ 2 φn 1,2K-1 -φ n 0,K-2 , φ n+1 1,2K = (1 -λ)φ n 1,2K + 2λ φn 1,2K-1 , φ n+1 1,2K+1 = (1 -λ)φ n 1,2K+1 + λφ n 1,2K , φ n+1 1,2K+2 = (1 -λ)φ n 1,2K+2 + λφ n 1,2K+1 , φ n+1 1,2K+3 = (1 -λ)φ n 1,2K+3 + λφ n 1,2K+2 .
We should now check the smoothness of the solution after this update. Using (4.13) and (4.4) we can compute d n+1 K as

d n+1 1,K = φ n+1 1,2K -φn+1 1,2K = φ n+1 1,2K - 1 8 φ n+1 0,K-1 + φ n+1 0,K - 1 8 φ n+1 0,K+1 = - 1 8 φ n+1 0,K-1 + 1 2 (φ n+1 1,2K -φ n+1 1,2K+1 ) + 1 16 (φ n+1 1,2K+2 + φ n+1 1,2K+3 ).
We plug in the expressions of the solution at time n + 1 and obtain

d n+1 K = - λ 16 φ n 0,K-2 + λ 2 (2 φn 1,2K-1 -φ n 1,2K ) + λ 16 (φ n 1,2K+1 + φ n 1,2K+2 ) - 1 8 φ n 0,K-1 + λ 16 φn 1,2K-1 + 1 -λ 2 (φ n 1,2K -φ n 1,2K+1 ) + 1 -λ 16 (φ n 1,2K+2 + φ n 1,2K+3
).

The eect of the doubling ux on the detail is an additional term

d n+1 1,K = dn+1 1,K + λ 2 φn 1,2K-1 , (4.17)
compared to the detail dn+1 K if no doubling condition is imposed. This means that if the solution is smooth at time t n , but the solution is large enough before the doubling interface, then the detail may become signicant due to the doubling.

We now explain what happens next, if the detail d n+1 1,K which was negligible at time t n becomes larger than the MR threshold ε at time n + 1. In the quadratic case, as described in Algorithm 1, the mesh D 0,K-1 just upstream of the doubling interface should be predictively rened using (4.6)

φn+1 1,2K-2 = 1 8 φ n+1 0,K-2 + φ n+1 0,K-1 -1 8 φ n+1 0,K , φn+1 1,2K-1 = -1 8 φ n+1 0,K-2 + φ n+1 0,K-1 + 1 8 φ n+1 0,K , where φ n+1 0,K = 1 2 (φ n+1 1,2K + φ n+1 1,2K+1 ), = 1 -λ 2 (φ n 1,2K + φ n 1,2K+1 ) + λ 2 (2 φn 1,2K-1 + φ n 1,2K ). ( 4

.18) transport problem

Here again the dierence between the values of the solution on the subdivisions, which should be small since the solution is only predictively rened, is enhanced by the value φn 1,2K-1 upstream of the interface which contributes twice instead of once.

We see that in a single time step the doubling condition coupled with the MR, has introduced an oscillation of the order of φn 1,2K-1 .

Suppose we start from an initial condition localized in the rst slab before the rst doubling interface, for instance φ 0 k = δ 0,k . Due to the numerical diusion eects described earlier (see (4.10)), after M time steps (M depending on ε and λ), the solution φ M k is nonzero and smooth for k = 0, . . . , M . The MR scheme will therefore coarsen the grid in this region. As the solution is transported over several slabs, its amplitude is doubled at each interface, and depending on its local amplitude we can fall in the situation depicted by (4.17). Furthermore, another factor of noise appearance is the dependence of the MR threshold level on the discretization level according to the rule (4.14).

We see from (4.17) that the decision to rene the mesh after crossing a doubling interface depends not only on the smoothness but also on the local amplitude.

Thanks to (4.14), the same (small) amplitude discretized on a coarse level will be more likely to trigger the renement, and might even trigger the local renement on two consecutive levels (see Algorithm 1). We propose an example for such a situation in Appendix 4.6. We describe a situation with three levels of discretization, where the solution has sensibly the same small amplitude but is rened partly on the intermediate level, and partly on the coarsest one. We show that renement rules together with rules (4.17) and (4.14) lead to coarsen the solution wherever it was previously on the intermediate level, and to rene it on the nest level on some locations where it was before on the coarsest one.

At this point we also stress out that the dierence between the two wavelets (4.12) and (4.13) is quite important, specially if we solve the 1D test case presented above with the 2D FV code. To illustrate this point we perform a simulation on a domain consisting of twelve subdomains of unit width and height with vertical doubling interfaces. The (horizontal) speed is equal to 1. The initial condition is piecewise constant φ(t = 0, x, y) = 10 for 0 ≤ x < 1 and 0.2 ≤ y < 0.3 0 elsewhere

We rst use a two level MR hierarchy, with ve meshes per subdomain in the

x direction on the coarser level. However in both cases the details are later amplied by (4.17) and, in the quadratic case, also by (4.18), which explains the resurgence of rened meshes upstream of the support of the solution (panels b)). In this simple set-up isolated cells appear on the nest level only in the quadratic case, but in more complicated situations the same phenomenon also occurs in the case of the Haar prediction.

Adaptive doubling strategy

To prevent the formation of this "numerical noise" several strategies can be implemented :

Forcibly rening the solution on the nest discretization grid in the vicinity of interfaces, without considering its local smoothness. This may become quite greedy and altogether compensate the benets of the MR strategy. be done carefully to avoid rapid changes in the threshold and subsequently rapid coarsening of the solution. We will document later that the optimal strategy is to adapt the threshold to the total mass of the solution

ε t, = ε M (t) (4.19)
where the level dependent ε is dened by (4.14), with ε calibrated for a unit mass solution at initial time.

Modifying the transmission condition (4.2) with an adaptive strategy consisting in applying the doubling only if the solution is above a given threshold δ

g p φ(t, px + s ) = 2g p-1 φ((p -1)x - s ) if |φ(t, (p -1)x - s )| ≥ δ, g p-1 φ((p -1)x - s ) otherwise. (4.20)
In practice δ should be chosen close to ε, and we will study the relationship more closely in paragraph 4.3.3.

We also assume that the support of the initial condition is entirely located in the rst slab before the rst doubling interface. We have actually exemplied in Appendix

Formalization and generalization of the doubling strategy

The result (4.17) also applies to Haar reconstruction (4.12). This means that the appearance of the spurious noise is due not only to the discontinuities in the solution induced by the localized doubling, but also to the enhancement of the thresholding error. In particular, even if we modied the prediction scheme in the vicinity of 92 Chapitre 4. Adaptive mesh renement strategy for a non conservative transport problem interfaces in order to account for the doubling of the solution, this would not stop the increase of noise.

To formalize this remark, we generalize the MR scheme introduced in Paragraph 2.2 to an arbitrary number of levels L. We introduce the linear prediction operator P -1 such that formula (4.6) can be expressed generically as

φn = P -1 φ n -1 ,
and the reconstruction formulas in Algorithm 2 as Operator P -1 is a 2 N 0 × 2 -1 N 0 matrix, that is zero everywhere except at three consecutive coecients per row (P -1 ) 2k,k-1 = 1/8, (P -1 ) 2k,k = 1, (P -1 ) 2k,k+1 = -1/8, (P -1 ) 2k+1,k-1 = -1/8, (P -1 ) 2k+1,k = 1, (P -1 ) 2k-1,k+1 = 1/8, with periodic boundary conditions

φ n = P -1 φ n -1 + d .
(P -1 ) 0,2 -1 N 0 -1 = 1/8, (P -1 ) 2 N 0 -2,0 = -1/8, (P -1 ) 1,2 -1 N 0 -1 = -1/8, (P -1 ) 2 N 0 -1,0 = 1/8. (4.22)
Note that other types of boundary conditions can be designed but periodic ones are coherent with the boundary conditions used in the application to the biological model. Formula (4.21) can be applied recursively, leading to

φ n L-1 = L-1 =1 P -1 φ n 0 + L-1 =1 L-1 j= +1 P j-1 j d n . (4.23)
The adaptive solution after thresholding with the MR threshold ε can be expressed as

φ n,ε L-1 = L-1 =1 P -1 φ n 0 + L-1 =1 L-1 j= +1 P j-1 j d n ,ε , (4.24)
where d n ,ε is the vector of thresholded details on level

(d n ,ε ) ,k = d n ,k if ||d n ,k || ≥ ε , 0 otherwise.
From (4.23) and (4.24) it is straightforward to obtain an O(ε) estimate of the error between the unthresholded solution φ n L-1 and the thresholded one φ n,ε L-1 . In [30], a strategy was introduced to enlarge the tree of signicant details (with respect to the threshold ε), ensuring that the adaptive solution on the corresponding mesh satises the same error estimate at the next time step time n + 1. We also recall a useful result from [START_REF] Coquel | Convergence of timespace adaptive algorithms for nonlinear conservation laws[END_REF] showing that the reconstructed solution on any intermediate level is transport problem vitiated by an O(ε) perturbation due to MR thresholding and reconstruction, which we can summarize as

φ n,ε ,k = φ n ,k + O(ε). (4.25)
We apply the adaptive doubling strategy (4.20) along with the elementary FV scheme (4.9) generalized to an L level MR hierarchy with λ = 2

-L+1 λ φ n+1,ε ,k = (1 -λ )φ n,ε ,k + λ φ n,ε ,k-1 if k = pK or |φ n,ε ,k-1 | ≤ δ (1 -λ )φ n,ε ,k + λ 2φ n,ε ,k-1 if k = pK and |φ n,ε ,k-1 | > δ,
with p = 1, . . . , N c . We see that we have a conditional maximum principle in the

sense that if φ n,ε ,k ≤ δ and φ n,ε ,k-1 ≤ δ then φ n+1,ε ,k ≤ δ.

Numerical validation of the adaptive thresholds strategies

In this paragraph we perform numerical tests to validate the adaptive strategies (4. [START_REF] Chen | Hyperbolic conservation laws with discontinuous uxes and hydrodynamic limit for particle systems[END_REF]) and (4.20). We then study the relationship between the two thresholds δ and ε.

Eect of the adaptive doubling strategy on the solution

To justify the doubling strategy designed to tackle the spurious noise introduced by the MR we show the inuence of applying (4.20) instead of (4.2), independently of the MR, and even of any numerical scheme, by computing the exact solution of the 1D problem (4.1). As shown in [START_REF]A numerical method for transport equations with discontinuous ux functions : Application to mathematical modeling of cell dynamics[END_REF], the exact solution for a single interface located

at x = x s with condition (4.2) is φ 1 (t, x) =        φ 0 (x -g 0 t) for x < x s , φ1 (t s ) for (x s -g 1 t) ≤ (x -g 1 t) < x s , φ 0 (x -g 1 t) for (x -g 1 t) ≥ x s ,
where φ1 is the trace of the solution on the right of the interface, dened by

φ1 (t) = 1 g 1 k 1 (g 0 φ 0 (x s -g 0 t)), (4.27)
and t s is the delay after which the eect of interface x s is felt at position x and time t dened by

t s = t - (x -x s ) g 1 .
(4.28) transport problem We now show how the exact solution for a sequence of N c interfaces can be dened recursively. Let us denote by φ p-1 (t, x) the solution for a sequence of p-1 interfaces, with p > 1 and add a p th interface at px s with a coecient k p for the transmission condition and a speed g p on the right hand side (see Figure 4.4). For x < px s the eect of the p th interface is not felt and the solution is φ p-1 (t, x). For x ≥ px s , depending on time t, the eect of the interface is felt or not. If (xpx s ) < g p t, events crossing the p th interface are multiplied by the k p coecient and transported at speed g p . The eects of these events are felt at position x after a delay (x-px s )/g p . If (x-px s ) > g p t, the eect of the p th interface is not felt yet and the initial condition dened on x > px s is transported at speed g p . We eventually have

φ p (x, t) =        φ p-1 (x, t) for x < px s , φp (t p s ) for (px s -g p t) ≤ (x -g p t) < px s , φ 0 (x -g p t) for x -g p t ≥ px s , (4.29)
where φp is the trace of the solution on the right of the p th interface, dened by

φp (t) =      1 g p k p g p-1 φ p-1 (px s , t) if p > 1 1 g p k p g p-1 φ 0 (x s -g p-1 t) otherwise (4.30)
and t p s is dened by

t p s = t - (x -px s ) g p .
We mimic set-up (4.1) by solving equation (4.29) numerically using recursiveness, with g p = 1 for all p and an initial condition φ 0 dened by φ 0 (x) = exp -5(x -0.5) 2 .

(4.31) In each case we perform the simulation with or without the adaptive doubling strategy (4.20). The six possible cases are referred in the graphs with the following legends The L 1 -norm of the relative error at a given time t n = n∆t is dened as

δ δ x [0, 20] δ M δ,n = ∆x j φ δ (n∆t, j∆x) M n = ∆x j φ(n∆t, j∆x). Err(δ) = n M δ,n -M n n |M n | , φ δ φ ∆t × ∆x δ δ ε δ t = δM (t), M (t) t M (t) = Nxs 0 φ(t, x)dx. δ E(δ t ) E(δ) O(δ) δ δ = 0.1 O(δ) t =
||E|| L 1 = i,j φ ε,n L-1;i,j -φ n L-1;i,j i,j φ n L-1;i,j , (4.35)
where φ n L-1;i,j is the FV solution at time t n = n∆t on the nest MR level L -1 of discretization, with index i in the x direction and j in the y direction, and φ ε,n L-1;i,j is the adaptive scheme solution, reconstructed at time t n on the nest level L -1.

This error is computed at ve dierent times and displayed in Figure 4.9 against the MR threshold level ε on the left panels, the CPU gain on the center panels, and the memory gain on the right panels. The CPU gain is the ratio of the CPU time for the simulation on the uniform nest grid to the CPU time for the FVMR simulation.

The memory gain is the ratio of the number of meshes in the uniform nest grid to the maximum number of meshes in the adaptive grid during the FVMR simulation.

The left panels show that, whichever the thresholding strategy, the expected asymptotic behavior in O(ε) is satised (see equation (4.15)). Nevertheless, as time increases, the normalization factors in the adaptive strategies (4. [START_REF] Chen | Hyperbolic conservation laws with discontinuous uxes and hydrodynamic limit for particle systems[END_REF]) and (4.34) increase, resulting in more and more distinct curves. In practice, this type of curve is computed for a reduced version of the full scale simulation in order to select a threshold value ε ensuring a correct balance between the MR (4.15) and discretization (4.16) errors. In our case we will use the optimal values of the thresholds δ and ε calibrated from this simple 1D set-up in future 2D and vector simulations of the biological model. In other situations, one could calibrate the MR threshold in 2D tests before addressing the 3D cases where the reference solution is beyond reach. We have represented the error in L 1 norm in space at various times during the simulation, which gives more detailed information than the global L 1 error over the whole timespan, and allows us in particular to see the variation in the error as the time adaptive threshold increases. For a constant threshold (red and green lines) the error remains almost unchanged with time. This is not a surprise since we have already observed in [START_REF] Cohen | Fully adaptive multiresolution nite volume schemes for conservation laws[END_REF] that the theoretical estimate (4.15) predicting a linear Chapitre 4. Adaptive mesh renement strategy for a non conservative transport problem growth in time was very pessimistic. Also quite expectedly, the dierence between the error curves increases with time in the case of time adaptive thresholding with increasing value of ε. Note that the MR error at a given time has built up progressively and does not entirely depend on the current value of ε(t) : at time t = 10 there is a factor of order 10 between the red and dark blue curves, which correspond respectively to the constant and mass dependent threshold (Eq. (4. [START_REF] Chen | Hyperbolic conservation laws with discontinuous uxes and hydrodynamic limit for particle systems[END_REF])and (4.33)), although at that time the mass, and therefore ε(t), has already been multiplied by 10 3 . At the end of the simulation (at t = 20) the error ratio between both curves is still about 10, although ε(t) has been multiplied by 10 6 .

The center and right panels allow one to assess the performance of the thresholding strategies with respect to computational costs, in terms of CPU time as well as memory requirement. For early times all strategies seem equivalent, but after ten cycles, the simulations corresponding to constant MR thresholding start having worse performances than those using either of adaptive strategies (4.19) and (4.34), which remain basically undistinguishable. The simulation with constant MR thresholding and no doubling adaptation, displayed with ×, has the worst performances, because at large times, spurious noise has started to develop, requiring ne mesh discretization and therefore, more memory and more CPU time.

We now validate the adaptive doubling strategy (4.20) by extensive numerical tests.

We compare its eciency when it is coupled with one or another method of MR thresholding. We also estimate the optimal value for the threshold parameter δ.

Link between the MR and the doubling thresholds

In this paragraph we study the relation between the MR threshold parameter ε and the threshold δ in the adaptive doubling interface (4.20). Still for 1D test case (4.1) with the piecewise constant initial condition, we run extensive tests for dierent values of both thresholds. We also study the inuence of the number of levels in the MR hierarchy, the precision of the FV algorithm, and the eect of normalizing the MR threshold by the overall mass of the solution, as it increases with time (see Eq. is the linear trend of the error with ε which is preserved by the adaptive doubling strategy. However normalizing by ε both the error on the vertical axis and δ on the horizontal axis highlights the correlation between ε and δ since the minimum of the error is observed for a similar ratio range for each set of curves. Chapitre 4. Adaptive mesh renement strategy for a non conservative transport problem and the third order FV scheme. The results are displayed in the top panels of Figure 4.11 ; the value of δ minimizing the error is shifted from about ε in the rst order case to about 2ε.

We then couple the adaptive doubling strategy (4.20) with the adaptation of the MR threshold to the increasing size of the solution. The same error study as previously is performed, for four MR levels and the third order FV scheme. The results, displayed in the bottom panels of Figure 4.11, still exhibit that there is an optimal value for δ. It seems to be roughly around 1.5ε.

After these extensive numerical tests, we are condent that coupling both adaptive thresholds (4. [START_REF] Chen | Hyperbolic conservation laws with discontinuous uxes and hydrodynamic limit for particle systems[END_REF]) and (4.20) enables the FVMR strategy to robustly handle doubling ux conditions of type (4.2). We will adopt in the sequel the value δ = ε which is close to the optimal value in all cases.

Application to a biological model 4.4.1 Model and numerical method

In this section we apply the strategy presented above to the numerical simulation of a biological model dedicated to the cell dynamics in ovarian follicular development introduced in Echenim et al. [START_REF] Clément | Sorine Multiscale modeling of follicular ovulation as a reachability problem[END_REF] and references therein. The model is multiscale to account for the selection process of ovulatory follicles which involves the cellular, follicular and ovarian levels. At the microscopic level, the granulosa cell population is structured according to the cell age (position within the cell cycle) and the cell maturity (level of sensitivity towards hormonal control). In each ovarian follicle, the granulosa cell population is described by a density function whose changes are ruled by conservation laws. The multiscale structure arises from the formulation of a hierarchical control operating on the aging and maturation velocities as well on the source terms of the conservation law ; this control depends on the rst moment in the maturity variable of the density. Diagrams summarizing the main features of the model are depicted in Figure 4.12. The functional space in cell age × cell maturity is subdivided into subdomains corresponding to dierent cell states : proliferation (Ω

p 1 ∪ Ω p
2 ) or dierentiation (Ω 3 ), and also with respect to the cell sensitivity towards the hormonal control exerted by the pituitary gland (sensitive phase Ω p 1 ∪ Ω 3 , insensitive phase Ω p 2 ). The mitosis is the endpoint of the cell cycle which is completed when the two daughter cells are separated from each other. In our set-up the mitosis happens at values x s = p for p = 1, . . . , N c -1, on the internal boundaries between Ω p in Ω p 1 and consequently all cells eventually enter Ω 3 . In the dierentiation phase Ω 3 , cells do not proliferate any more and cannot overcome an asymptotical maturity level, corresponding to a null vertical velocity (see [START_REF] Michel | Multiscale modeling of follicular ovulation as a mass and maturity dynamical system[END_REF]).

||E||L 1 /ε δ/ε ε=10 -9 ε=10 -8 ε=10 -7 ε=10 -6 ε=10 -5 ε=10 -4 ε=10 -3 ε=10 -2 ε=10 -1
Three MR levels, rst order FV scheme

||E||L 1 /ε δ/ε ε=10 -9 ε=10 -8 ε=10 -7 ε=10 -6 ε=10 -5 ε=10 -4 ε=10 -3 ε=10 -2 ε=10 -1
Four MR levels, rst order FV scheme Adaptive MR thresholding ε t (4.19) ensure that it remains in Ω 3 . We can thus apply periodic boundary conditions to the density which are easy to implement in our adaptive mesh context on boundaries y = 0 and y = 1. We refer the interested reader to [START_REF]A numerical method for transport equations with discontinuous ux functions : Application to mathematical modeling of cell dynamics[END_REF] for a detailed exposition of the motivation underlying the choice of a localized transmission condition of type (4.2), and to [START_REF] Tomura | Contrasting quiescent G0 phase with mitotic cell cycling in the mouse immune system[END_REF][START_REF] Sakaue-Sawano | Visualizing spatiotemporal dynamics of multicellular cell-cycle progression[END_REF] for biological evidence of localized mitotic phenomenon). We also refer the reader to Appendix 4.5 for the description of the model in the single follicle case and to [START_REF]A numerical method for transport equations with discontinuous ux functions : Application to mathematical modeling of cell dynamics[END_REF] for a more general presentation, including a description of the third order numerical scheme designed to handle the discontinuous ux conditions.

Here we solely address the coupling of the MR with the FV scheme in the context of a localized or distributed mitosis. Actually we even neglect at this stage the macroscopic scale where the dierent follicles are coupled through the interaction between the ovaries and the hypothalamus/pituitary complex, formerly exposed in [START_REF] Aymard | Numerical simulation of the selection process of the ovarian follicles[END_REF], and only consider the case of a single follicle. We use the third order FV scheme designed in [START_REF]A numerical method for transport equations with discontinuous ux functions : Application to mathematical modeling of cell dynamics[END_REF] and we couple it with the MR strategy on four levels. The domain consists of eight cell cycles with ten meshes in each cycle at the coarsest level in the x direction, and ten meshes in the y direction. The CFL number ensuring the stability of the scheme is set to 0.4. Note that the velocity coecients depend on the solution (see equations (5.2-4.44) in Appendix 4.5). The time step value is therefore time-dependent and computed at each time iteration. We adapt the stability rule designed in [START_REF] Aymard | Numerical simulation of the selection process of the ovarian follicles[END_REF] to the adaptive case so that

∆t n = min CF L min D ,(i,j) ∈ Sn ∆x max(|g n ,(i,j) |, |h n ,(i,j) |) , 1 max D ,(i,j) ∈ Sn |Λ n ,(i,j) | (4.36)
where g n ,(i,j) (respectively h n ,(i,j) ) is the discrete value of the speed g given by ) lie lower than their counterparts without the adaptive doubling strategy (with green ×, pink and black •).

In the center and right panels, the performances of the dierent algorithms in terms of CPU time and memory requirements are displayed for a given accuracy measured by the relative error (4.35). The dierences between the dierent strategies are not as spectacular as is the simple test case. Due to the dierentiation phenomenon and the associated cell cycle exit, the overall proliferation rate is around 37 at the nal time t = 20, much smaller -by a factor of 200-than in the 1D test case, so that the spurious noise is less important. Nonetheless, for late times, the curves corresponding to the simulation without MR nor doubling threshold adaptation (with green ×)

clearly exhibit worse performances than the others. It is also noticeable that as time increases, the range of memory gain shifts to the left. Indeed, the size of the adaptive grid increases, because the density spreads out when it crosses the boundary between the proliferation and dierentiation phase. The range of CPU gain also shifts accordingly to the left, since the computing time is directly and almost linearly related to the size of the grid.

The relation between the MR threshold ε and the adaptive doubling threshold δ is studied in Figure 4. [START_REF] Bürger | Risebro A relaxation scheme for continuous sedimentation in ideal clarier-thickener units[END_REF], where the relative error with the uniform grid solution is displayed against δ for dierent values of ε. Here again an optimal value is clearly visible. The curves in the right panel, where the product εE(δ, ε) is displayed against δ/ε are almost all superimposed, except for the two extreme values of ε. This corroborates the asymptotic behavior of the MR error in O(ε), which is always veried except if ε is too large (= 0.1) or too small (in that case the numerical accuracy of the computer penalizes the asymptotic behavior).

||E||L 1 /ε δ/ε ε=10 -9 ε=10 -8 ε=10 -7 ε=10 -6 ε=10 -5 ε=10 -4 ε=10 -3 ε=10 -2 ε=10 -1

Comparison of the numerical performances for both models of mitosis

As previously explained in the Introduction and the beginning of Paragraph 4.4, the mitosis is an almost punctual event in time. Nevertheless cell proliferation is commonly modeled by a distributed gain term [START_REF]Perthame Transport equations in biology[END_REF]. In this paragraph we study the inuence of treating the mitosis in either a localized or distributed manner on the convergence of the MR algorithm with ε. In contrast with the simple model illustrated in Figure 4.7, we are now in a realistic biological context including the 2D dependence of the cell density on age and maturity variables. However, to make the comparison meaningful, the aging velocity is left constant in time and piecewise constant in age and maturity

g(x, y, u) = γ 2 = 1.2 for (x, y) ∈ Θ 1 , 1 for (x, y) ∈ Θ 2 ∪ Ω 3 , (4.37)
instead of using its general closed-loop model formulation (5.2). We can compute the equivalent distributed linear source term B, which results in doubling the cell mass after one cell cycle has elapsed

B = 2 ln(2)γ 2 1 + γ 2 .
In order to reach a realistic mass gain before all cells have left the proliferation stage, we also reduce the time constant of the maturation velocity τ h = 0.2. Figure 4.17 displays the dependence of the relative error between the adaptive and uniform solutions with respect to the threshold ε, the CPU and the memory requirement. The left panels in Figure 4.17 show that the theoretical behavior in O(ε) except for very small values of ε where the error estimates reaches the numerical accuracy. The reference solution on the uniform nest grid with 51200 meshes requires 535 seconds of CPU time. The center and right panels show that an average gain of 10 in CPU can be achieved, for a relative error of 10 -3 , while reducing the memory by a factor of six. Comparison of the top and bottom panels corresponding respectively to the localized and distributed models shows that the adaptive scheme exhibits comparable characteristics in both cases.

Conclusion

We have adapted a MR procedure to FV in order to simulate numerically non conservative transport equations with specic discontinuous ux conditions on interfaces between subdomains. In the absence of any specic care, we almost systematically observe the appearance of spurious numerical noise in long time simulations.

In this paper we have explained on a simple but generic model the reasons underlying this artifact and proposed a strategy to avoid it. We show that the doubling condition should be applied according to an adaptive rule only when the solution is above some threshold δ, to be taken of the order of the MR threshold ε. This stops the spurious noise appearance, while preserving all the robust characteristics of the MR method, namely its O(ε) asymptotic behavior, and interesting gains in CPU time and memory requirements. Accessorily, we have also qualitatively compared the modeling of biological mitosis by a discontinuous ux condition, to the distributed birth term often encountered in the literature, and shown that the later is also well handled by our FVMR scheme. This method will next be applied to real case simulations involving large dimension vector systems.

Appendix : Description of the model for follicular development

This paragraph presents the details of the biological model illustrated by the numerical simulations commented in Paragraph 4.4. It is meant to make the paper self-contained and introduce the notations. Details on the biological bases of the model can be found in the founding paper [START_REF] Echenim | Multi-scale modeling of the follicle selection process in the ovary[END_REF] and the review [START_REF] Clément | Monniaux Multiscale modeling of ovarian follicular selection[END_REF]. Since our goal here is to design the coupling of MR with adaptive doubling strategy, the vector nature of the model is not relevant, so that we only present its scalar version, referring the reader to [START_REF] Aymard | Numerical simulation of the selection process of the ovarian follicles[END_REF][START_REF]A numerical method for transport equations with discontinuous ux functions : Application to mathematical modeling of cell dynamics[END_REF] for a more general presentation of the numerical scheme. Let us denote φ(t, x, y) the density of granulosa cells at time t, age x and cell maturity y.

Ω = {(x, y), 0 ≤ x ≤ N c × D c , 0 ≤ y ≤ 1}
where N c is the number of cell cycles and D c is the duration of one cycle. The domain Ω is divided in 2N c + 1 zones : Ω p 1 , Ω p 2 , for p = 1, . . . , N c and Ω 3 , corresponding to dierent cell states illustrated in Figure 4.12 and hence dierent denitions of the speeds and source terms. Phase Ω p 2 in the model aggregates the three latest phases (S, G2, M) of the p th cell cycle

                 Ω p 1 = {(x, y) ∈ Ω, pD c ≤ x ≤ (p + 1/2)D c , 0 ≤ y ≤ y s }, p = 0, . . . , N c -1, Θ 1 = ∪ Nc p=1 Ω p 1 Ω p 2 = {(x, y) ∈ Ω, (p + 1/2)D c ≤ x ≤ (p + 1)D c , 0 ≤ y ≤ y s }, p = 0, . . . , N c -1, Θ 2 = ∪ Nc p=1 Ω p 2 Ω 3 = {(x, y) ∈ Ω, y s ≤ y}.
Each cell cycle consists of the Ω p 1 ∪ Ω p 2 subdomain and Θ i for i = 1, 2 denotes the disconnected union of the N c corresponding phases Ω i p , for p = 1, . . . , N c . The aging function g appearing in (4.38) is dened by

g(x, y, u) = γ 1 u + γ 2 for (x, y) ∈ Θ 1 1 for (x, y) ∈ Θ 2 ∪ Ω 3 (4.39)
where γ 1 , γ 2 are real positive constants.

The maturation function h is dened by The source term, that represents cell loss through apoptosis, is dened by

h(x, y, u) = τ h (-y 2 + (c 1 y + c 2 )(1 -exp( -u ū ))) for (x, y) ∈ Θ 1 ∪ Ω 3 0 for (x, y) ∈ Θ 2
Λ(x, y, U ) =    Λ exp(-( (y -y s ) 2 γ ))(1 -U )χ {|y-γs|≤Λw} (y) for (x, y) ∈ Θ 1 ∪ Ω 3 0 for (x, y) ∈ Θ 2 (4.41)
where Λ, y s and γ are real positive constants, and χ E (y) is the characteristic function of subset E. The precise denition of the required transmission conditions along the successive cell cycles of the domain has been formulated in [START_REF] Shang | Cauchy problem for multiscale conservation laws : Application to structured cell populations[END_REF]. For each cycle p = 1, . . . , N c , the ux on the x-axis is continuous on the interface between Ω p 1 and Ω p 2

u(t) = min b 1 + exp(b 2 M (t)) b 3 , 1 U (t)
φ(t, x + , y) = (γ 1 u + γ 2 )φ(t, x -, y), x = (p -1/2)D c , 0 ≤ y ≤ y s . (4.45)
The ux is doubling on the interface between Ω p 2 and Ω p+1

1 , which accounts for the birth of two daughter cells from one mother cell at the end of each cell cycle We show on a simple example that spurious noise can appear due to doubling transitions and requires renement on the nest model, while the real discontinuities get smoothed out by the numerical scheme diusion.

(γ 1 u + γ 2 )φ(t, x + , y) = 2φ(t, x -, y), x = pD c , 0 ≤ y ≤ y s .
Suppose for instance that at time t n the solution is locally almost piecewise constant, with a small discontinuity in the mesh D 0,K+1 requiring both this mesh and its immediate neighbors to be subdivided, as in Figure 4.18. To x ideas, denote by α = 4ε/3

the size of the discontinuity

φ n 0,K-i = a, for i ≥ 1 φ n 1,2K = φ n 1,2K+1 = φ n 1,2K+2 = a φ n 1,2K+3 = φ n 1,2K+4 = φ n 1,2K+5 = a + α φ n 0,K+i = a + α, for i ≥ 3
The details at time n can be computed

d n 1,K = φ n 1,2K - 1 8 φ n 0,K-1 + φ n 0,K - 1 8 φ n 0,K+1 = a - 1 8 a + a - 1 8 a + a + α 2 = α 16
d n 1,K+1 = φ n 1,2K+2 - 1 8 φ n 0,K + φ n 0,K+1 - 1 8 φ n 0,K+2 = a - 1 8 a + a + a + α 2 - 1 8 (a + α) = 3 8 α 114
Chapitre 4. Adaptive mesh renement strategy for a non conservative transport problem

α D 0,K-2 D 0,K+1 D 0,K-1 D 0,K D 0,K+1 0,K+2 a Φ n+1 x2 α D 0,K-2 D 0,K+1 D 0,K-1 D 0,K D a Φ x2 n+1 0,K+2
d n 1,K+2 = φ n 1,2K+4 - 1 8 φ n 0,K+1 + φ n 0,K+2 - 1 8 φ n 0,K+3 = a + α - 1 8 a + a + α 2 + a + α - 1 8 (a + α) = -α 16
We have tested these details against the level dependent threshold (4.14) and we have found

|d n 1,K | = |d n 1,K+2 | < ε 1 , ε 1 ≤ |d n 1,K+1 | < ε 2 ,
justifying the discretization on the coarsest and intermediate levels. Let us now update the solution at the next time step, using λ/2 = ∆t/∆x 1 on the intermediate level and λ/4 = ∆t/∆x 0 on the coarsest one (note that to update the solution on mesh (1, 2K) we will need to reconstruct the solution on its neighbor φn 1,2K-1 = a).

We obtain the following values, also sketched out in the top panel of Figure 4.19

                       φ n+1 0,K-i = φ n+1 0,K-1 = φ n 0,K-1 -λ 4 (φ n 0,K-1 -φ n 0,K-2 ) = a φ n+1 1,2K = φ n 1,2K -λ 2 (φ n 1,2K -2φ n 1,2K-1 ) = a + λa φ n+1 1,2K+1 = φ n 1,2K+1 -λ 2 (φ n 1,2K+1 -φ n 1,2K ) = a φ n+1 1,2K+2 = φ n 1,2K+2 -λ 2 (φ n 1,2K+2 -φ n 1,2K+1 ) = a φ n+1 1,2K+3 = φ n 1,2K+3 -λ 2 (φ n 1,2K+3 -φ n 1,2K+2 ) = a + α(1 -λ 2 ) φ n+1 1,2K+5 = φ n+1 1,2K+4 = φ n 1,2K+4 -λ 2 (φ n 1,2K+4 -φ n 1,2K+3 ) = a + α φ n+1 0,K+i = φ n+1 0,K+3 = φ n 0,K+3 -λ 4 (φ n 0,K+3 -φ n 0,K+2 ) = a + α (4.48)
From these updated values we compute the averaged values on the coarsest level transport problem

φ n+1 0,K = a + λa + a 2 = a + a λ 2 φ n+1 0,K+1 = a + a + α(1 -λ 2 ) 2 = a + α 2 λ 2 φ n+1 0,K+2 = a + α
and the details at time n + 1

d n+1 1,K = φ n+1 1,2K - 1 8 φ n+1 0,K-1 + φ n+1 0,K - 1 8 φ n+1 0,K+1 = a + λa - 1 8 a + a + a λ 2 - 1 8 (a + α 2 λ 2 ) = λa 2 + αλ 32
d n+1 1,K+1 = φ n+1 1,2K+2 - 1 8 φ n+1 0,K + φ n+1 0,K+1 - 1 8 φ n+1 0,K+2 = a - 1 8 (a + a λ 2 ) + a + α 2 λ 2 - 1 8 (a + α) = - aλ 16 - αλ 4 + α 8 d n+1 1,K+2 = φ n+1 1,2K+4 - 1 8 φ n+1 0,K+1 + φ n+1 0,K+2 - 1 8 φ n+1 0,K+3 = a + α - 1 8 (a + α 2 λ 2 ) + a + α - 1 8 (a + α) = -αλ 32 + α 8
If we use λ = 1/2 and the value α = 4ε/3 we obtain

d n+1 1,K = -a 32 d n+1 1,K+1 = a 4 + ε 48
d n+1 1,K+2 = 7ε 48
we see that as soon as

4 47 48 ε ≤ a < 16ε
for instance say a = 4ε we have

     ε 2 ≤ |d n+1 1,K |, |d n+1 1,K+1 | < ε 1 , |d n+1 1,K+2 | < ε 1 ,
for level = 1 L -1 do for k = 0 N 0 2 -1 do if ( , k) ∈ Γ n+1 ε then {D ,2k
δ = 0 × δ = ε + Adaptive MR threshold (4.19) δ = 0, ε t = εM (t) δ = ε t = εM (t) * Adaptive MR threshold (4.34) δ = 0, ε t = εSup|Φ|(t) • δ = ε t = εSup|Φ|(t)
φ 0 (a, γ) =                            µ 1 c 0 1 -exp(-c 0 ) exp(-c 0 a) pour 0 < a < 1 et 0.10 < γ < 0.15, µ 2 c 0 1 -exp(-c 0 ) exp(-c 0 a) pour 0 < a < 1 et 0.15 < γ < 0.20,
(0) = 2A(T c ) T c (g 1 u(t = 0) + g 2 )φ 0 (a = 0 + , γ) = 2φ 0 (a = 1, γ) c 0 = ln( 2 (g 1 u(t = 0) + g 2 )
)

g f g f g f (a, γ, u f ) = g 1 u f + g 2 G1, 1 SM ∪ D. a γ g f f γ [0, γ s ] g f u 1 ≤ u 2 (a, γ) g f (a, γ, u 1 ) ≤ g f (a, γ, u 2 ).
Chapitre 5. Calibration Figure 5.2 Exemple de fonction de vitesse de vieillissement (progression selon l'axe des abscisses). En abscisse, l'âge cellulaire. En ordonnée, la vitesse de vieillissement. La valeur de maturité (axe des ordonnées) est xée à γ = 0 car la fonction de vieillissement est constante en γ pour γ ∈ [0, γ s ]. On représente plusieurs courbes, chacune étant associée à une valeur de variable de contrôle u f . La fonction de vieillissement est une fonction croissante du contrôle.

Fonction h f : vitesse de maturation. La fonction de vitesse en ordonnée h f est une fonction de trois variables réelles : les deux variables d'espace a et γ, et une variable de contrôle u f , dénie par

h f (a, γ, u f ) = τ f (-γ 2 + (c 1 γ + c 2 )(1 -e ( -u f ū ))) ) en G1 ∪ D, 0 en SM.
Sur la Figure 5.3, on représente une famille de fonctions h f , pour un f donné, à a xé en G1 (coupe verticale) en fonction de γ, indexée par des valeurs de contrôles distincts.

Zéros de la fonction h f . La fonction h f est un polynôme de degré 2 en γ, dont les racines sont fonctions de u f . On note que ces racines sont indépendantes du paramètre τ f . Cette fonction a alors les mêmes racines que le polynôme

P (γ) = -γ 2 + (c 1 γ + c 2 )θ(u f ), avec θ(u f ) = (1 -e ( -u f ū ) ), à contrôle u f xé. Nous pouvons calculer le discriminant ∆ = c 2 1 θ(u f ) 2 + 4c 2 θ(u f ), Figure 5
.3 Exemple de fonction de vitesse de maturation (progression selon l'axe des ordonnées). En abscisse, la maturité cellulaire, en ordonnée la vitesse de maturation, pour l'abscisse a = 0 xée. On a représenté l'axe h = 0, pour bien identier les zéros de h f dans le demi-plan positif. On a tracé la fonction h f (a, γ, u f ) pour différentes valeurs du contrôle mésoscopique u f , les paramètres c 1 , c 2 , ū étant xés. h f est une fonction constante de u f (mais bornée), si bien que l'écart entre les graphes de h f pour diérentes valeurs de u f diminue quand u f augmente.

125 Chapitre 5. Calibration

avec θ(u f ) = (1 -e ( -u f ū )))
), puis expliciter les deux racines réelles

         r -= -c 2 1 θ(u f ) 2 + 4c 2 θ(u f ) + c 1 θ(u f ) 2 , r + = + c 2 1 θ(u f ) 2 + 4c 2 θ(u f ) + c 1 θ(u f ) 2 ,
sachant que seule la racine positive a un intérêt dans notre contexte.

Monotonie de r + par rapport au contrôle. La position des racines de h f est monotone par rapport au contrôle u f .

Preuve : Supposons que

u 1 ≤ u 2 .
Alors, par décroissance de la fonction u → exp(-u) et par composition, on obtient

θ(u 1 ) = 1 -exp(- u 1 ū ) ≤ 1 -exp(- u 2 ū ) = θ(u 2 ), et par conséquent r + (u 1 ) = + c 2 1 θ(u 1 ) 2 + 4c 2 θ(u 1 ) + c 1 θ(u 1 ) 2 ≤ + c 2 1 θ(u 2 ) 2 + 4c 2 θ(u 2 ) + c 1 θ(u 2 ) 2 = r + (u 2 ). Conclusion, u 1 ≤ u 2 implique r + (u 1 ) ≤ r + (u 2 ).
Contrainte a priori sur la racine positive r + . Étant donné deux réels U max et ūmin (représentant des variables de contrôle), à contrôle u f xé, on a l'encadrement

γ s < r + < 1, si on a les inégalités            γ 2 s c 1 γ s + c 2 < θ(ū min ), θ(U max ) < 1 c 1 + c 2 , ūmin ≤ u f ≤ U max .
Preuve. Supposons que

r + < 1,
Alors on peut écrire, en revenant à la dénition de r

+ , √ ∆ + c 1 θ(u f ) < 2,
puis passer au carré

∆ < 4 + c 2 1 θ(u f ) 2 -4c 1 θ(u f ), et simplier l'expression pour obtenir θ(u f )(c 1 + c 2 ) < 1, et ainsi obtenir la première inégalité θ(u f ) < 1 c 1 + c 2 .
Supposons à présent que γ s < r + .

On peut alors écrire, toujours en revenant à la dénition

2γ s -c 1 θ(u f ) < (∆),
puis en passant au carré et en simpliant, comme précédemment,

γ 2 s < θ(u f )(c 1 θ + c 2 ),
On conclut alors que

γ 2 s c 1 γ s + c 2 < θ(u f ).
Notations. A présent, par souci de simplicité, nous noterons

γ min 0 = r + (ū min ), γ max 0 = r + (U max ).
Comportement asymptotique. Rappelons un résultat concernant le comportement asymptotique des solutions, démontré dans [START_REF] Michel | Multiscale modeling of follicular ovulation as a mass and maturity dynamical system[END_REF]. L'annulation de la vitesse de progression en ordonnée implique qu'en temps long, et à contrôle constant, toute densité tend à se concentrer autour d'une maturité asymptotique, dénie par le zéro supérieur de la fonction h f . On illustre ce phénomène sur la Figure 5.4. On comprend alors l'intérêt d'utiliser cette propriété pour contrôler le résultat d'une simulation.

Variables de contrôle

Fonction U : contrôle macroscopique. On dénit le contrôle macroscopique par

U (t) = S(M 1 (t)) + U 0 (t),
avec S la fonction sigmoïde dénie par :

S(M 1 ) = U min + U max -U min 1 + exp(c(M 1 -M ))
.

(5.1)

Sur la Figure 5.5 on a représenté une telle sigmoïde. On note qu'il y a deux valeurs asymptotiques reliées par une zone de transition.

t = 0, 2, 4, 8 1 γ = 0.4 h f u f Figure 5
.5 Exemple de fonction déterminant la variable de contrôle macroscopique. En abscisse, la maturité globale (ou maturité ovarienne), en ordonnée, la valeur du contrôle macroscopique (ou concentration plasmatique de FSH). C'est une fonction sigmoïde monotone décroissante qui possède un seuil (dit seuil de maturité globale), correspondant au maximum en valeur absolue de la dérivée seconde en temps.

Point d'inexion et pente de la fonction U . Considérons la fonction S :

S(M 1 ) = U min + U max -U min 1 + exp(c(M 1 -M ))
.

Posons S(M ) = 1 1 + exp(c(M 1 -M ))

. Calculons la dérivée de S (en M 1 )

S (M 1 ) = -c exp(c(M 1 -M ))) (1 + exp(c(M 1 -M ))) 2 = -c S(M 1 ) exp(c(M 1 -M )) 1 + exp(c(M 1 -M ))
.

Or, on peut écrire

exp(c(M 1 -M )) 1 + exp(c(M 1 -M )) = exp(c(M 1 -M )) + 1 1 + exp(c(M 1 -M )) - 1 1 + exp(c(M 1 -M ))
.

Par conséquent,

S (M 1 ) = -c S(M 1 )(1 -S(M 1 )). Conclusion S (M 1 ) = (U max -U min ) S(M 1 ) = -c(U max -U min ) S(M 1 )(1 -S(M 1 )).
Calculons à présent la dérivée seconde

S (M 1 ) = -c( S (M 1 )(1 -S(M 1 )) -S(M 1 ) S (M 1 )).
En réinjectant l'expression de la dérivée, on obtient

S (M 1 ) = -c(-c S(M 1 )(1 -S(M 1 )) 2 + c S(M 1 ) 2 (1 -S(M 1 ))), autrement dit S (M 1 ) = c 2 ( S(M 1 )(1 -S) 2 -S(M 1 ) 2 (1 -S(M 1 ))).
puis, en factorisant, et en utilisant l'expression algébrique suivante

a(1 -a) 2 -a 2 (1 -a) = a(1 -a)(1 -a -a) = a(1 -a)(1 -2a), on obtient S (M 1 ) = c 2 S(M 1 )(1 -S(M 1 )(1 -2 S(M 1 )).
On note que

S( M ) = 1 2 , et en particulier S ( M ) = 0. Autrement dit, l'abscisse du point d'inexion de S se situe au point M = M , et l'ordonnée de S( M ) est U max -U min 2 (résultat bien connu d'ailleurs). Conclusion, S (M 1 ) = (U max -U min ) S (M 1 ). On note alors que    S (m) = 0, S (m) = -(U max -U min ) c 2 .
donc le paramètre c règle la valeur de la pente de la sigmoïde.

Fonction b f : contrôle mésoscopique. La variable de contrôle mésoscopique en un follicule f donné est donnée par En abscisse, la maturité folliculaire, en ordonnée, la valeur de b f , fonction de biodisponibilité locale. On note qu'elle comporte deux parties, une partie exponentielle au départ, puis à l'atteinte de la valeur 1, cette fonction devient constante. Autrement dit, la valeur du contrôle mésoscopique ne peut excéder celle de la valeur du contrôle macroscopique.

u f (t) = b f (m f 1 (t))U (t), avec b f (m) = min b 1 + exp(b 2 m) b 3 , 1 ,
Description de la fonction Λ. Le taux de perte Λ(a, γ, U ) en facteur de la densité dans le terme source est une composition de diérentes fonctions : une gaus-

sienne centrée en γ s , une indicatrice 1 [γ - s ,γ + s ] avec γ - s et γ + s des constantes réelles positives telles que 0 < γ - s < γ s < γ + s < 1, et une fonction dépendant linéairement du contrôle macroscopique : Λ(a, γ, U ) =    K exp - (γ -γ s ) 2 γ (1 -U )1 [γ - s ,γ + s ] en G1 ∪ D, 0 en SM.
La gaussienne permet de rendre la source maximale au passage de l'interface γ s , l'indicatrice assure la nullité en dehors d'un certain intervalle, et le terme dépendant de la variable de contrôle macroscopique permet d'annuler la source à contrôle maximal, et de maximiser la source à contrôle minimal. La Figure 5.7 donne l'exemple d'une telle fonction.

Description de l'ovulation : Deux paramètres servent à décrire l'issue du processus de sélection, qui aboutit à l'ovulation.

Le seuil de maturité ovarienne M o dénit de manière implicite l'instant de déclenchement de la décharge ovulatoire

t f = min{t, M 1 (t) ≥ M o } Figure 5
.7 Exemple de fonction de perte cellulaire (apoptose). En abscisse, la maturité cellulaire, en ordonnée, la valeur du taux de perte Λ(a, γ, U ) pour diérentes valeurs du contrôle global U . Cette fonction est constante en âge a (axe des abscisses), sauf en zone SM où elle est nulle. On note qu'elle est maximale en γ = γ s (ici γ s = 0.3), au passage de l'interface entre zone de prolifération et zone diérenciée, et qu'elle est nulle en dehors d'un support compact strictement inclus dans le domaine en zones G1 et D, et identiquement nulle en SM. L'intérêt du modèle brebis est multiple :

M f m f (t o ) ≥ M f N m N c γ s CF L 0 0.5 L 1 γ max = 1 γ s 0 < γ s < 1 N c 1 ≤ N c ≤ 20 N m 1 ≤ N m ≤ 20 0 < CF L < 0.5 1.e -10 0 1 ≤ L ≤ 10 N m = 5 N c = 3 y = γ s = 0.3 L = 4 = 0.01
1. C'est une espèce dont la physiologie ovarienne est beaucoup plus proche de l'espèce humaine que les espèces classiques de laboratoire (rat, souris).

2. On dispose dans cette espèce du jeu de données le plus complet, et rassemblant des informations sur la biologie cellulaire, mais aussi sur l'endocrinologie (séries temporelles de niveaux hormonaux). Dans l'article [START_REF] Pisselet | Monniaux Fraction of proliferating cells in granulosa during terminal follicular development in high and low prolic sheep breeds[END_REF], les auteurs présentent des courbes de fraction de croissance de populations de cellules de granulosa pour des follicules ovariens de brebis, de races Romanov ou Ile de France. La fraction de croissance diminue avec le diamètre folliculaire en suivant un prol en cloche, à partir d'une valeur maximale proche de 1mm, jusqu'à devenir proche de zéro aux diamètres ovulatoires (6 ou 7 mm suivant les cas).

Perte cumulée. Des observations morphologiques par une approche histologique combinées à une technique de marquage in situ permettent de détecter la fragmentation de l'ADN. Cette technique dite TUNEL (voir [START_REF] Gavrieli | Ben-Sasson Identication of programmed cell death In Situ via specic labeling of nuclear DNA fragmentation[END_REF] pour plus de précision) est utilisée dans [START_REF] Jolly | McNatty Morphological evidence of apoptosis and the prevalence of apoptotic versus mitotic cells in the membrana granulosa of ovarian follicles during spontaneous and induced atresia in ewes[END_REF] par exemple sur des cellules de granulosa extraites de follicules de brebis. La présence de taches noires sur les photos au microscope est le témoin de corps pycnotiques, qui sont des condensations du matériel nucléaire (pycnose) que l'on observe suite à l'apoptose. Cependant pour un follicule ovulatoire leur apparition reste minoritaire et on estime empiriquement à 10% le maximum de la perte cellulaire cumulée.

Non prédestination. Une notion fondamentale dans le processus de sélection est celle de non prédestination. A priori, parmi les follicules recrutés pour la phase nale de développement, aucun n'est prédestiné, au sens où, en environnement hormonal favorable (que l'on peut mimer avec une valeur de contrôle macroscopique xée à Durée du processus de sélection La phase folliculaire se termine par l'ovulation, déclenchée en réponse aux niveaux élevés d'÷stradiol (provenant des follicules dominants), par la décharge ovulatoire de la neurohormone GnRH qui induit celle de l'hormone hypophysaire LH. L'ovulation survient environ 24h après le pic de LH, et environ 10 jours à partir de l'instant initial choisi dans le modèle (correspondant au début de la phase terminale du développement folliculaire). Cette durée correspond à un intervalle de 6 à 8 cycles cellulaires parcourus.

U = U max ),

Trajectoires folliculaires

Comme nous l'avons vu précédemment, les follicules ovariens suivent un développement qui peut aboutir à deux situations : ovulation ou atrésie. Nous allons à présent nous appuyer sur les mesures précédentes pour dénir les concepts de trajectoire ovulatoire et de trajectoire atrétique. Cela nous permet de donner des contraintes a priori sur les paramètres de dynamique (comme par exemple le respect de la non prédestination), et de classer les follicules en n de simulation.

Trajectoire ovulatoire. On dira qu'une trajectoire folliculaire est ovulatoire lorsque l'on observe :

Un accroissement de l'eectif cellulaire compris entre 20 et 30 fois l'eectif initial ; Une fraction de croissance qui part de 1 pour chuter près de 0 en environ 10 jours ; Une perte cumulée inférieure à 10%.

Trajectoire atrétique. On dira qu'une trajectoire folliculaire est atrétique lorsque l'on observe :

Un décrochage de la courbe de masse cellulaire induit par la chute des niveaux de FSH ;

Une courbe d'eectif cellulaire décroissante en n de sélection.

Aides à la calibration

Contraintes a priori

A partir des spécications précédemment établies, on peut imposer des contraintes a priori sur les paramètres. Le but n'est pas d'obtenir directement les paramètres admissibles, car on ne sait pas comment les obtenir tous a priori, mais de réduire l'espace des paramètres, par exemple en donnant un bon ordre de grandeur, et en éliminant tous les paramètres conduisant à des situations que l'on peut caractériser comme non souhaitables a priori.

En particulier les paramètres de la fonction de maturation décrite au paragraphe 5.1.2 doivent être choisis de telle sorte que la racine positive de cette fonction, qui correspond à la maturité cellulaire atteint en temps long, satisfasse les deux inégalités suivantes : L'ensemble des contraintes a priori est résumé dans la Table 5.2.

γ min 0 ∈ [γ s , γ + s ], γ max 0 ∈ [γ + s , 1],

Visualisation à une échelle microscopique

Contraintes de visualisation. A l'échelle microscopique, nous pouvons visualiser les densités de cellules de granulosa au cours du temps dans les plans âge-maturité (un plan par follicule). Pour cela, nous pouvons utiliser des codes couleur classiques.

Une première idée est d'utiliser un code couleur qui dépend du temps, et est propre à chaque follicule. A chaque temps t n , une échelle de couleur est déterminée, variant de 0 jusqu'au maximum de l'amplitude au temps t n . Cela permet de bien voir les nuances à chaque pas de temps. Cependant, cela ne met pas en valeurs les diérences entre les follicules.

Pour pallier ce problème, on peut dénir une échelle de couleur commune à tous les follicules. Mais cela pose un nouveau problème. Les densités étant très variables, l'échelle de couleur est bien adaptée pour le follicule avec la densité la plus élevée, mais il est dicile de distinguer les autres. Typiquement, les follicules ovulatoires sont bien visualisés, mais pas les atrétiques.

Paramètre Contraintes Les variations de vitesse, ainsi que les accroissements de masse aux interfaces de type mitose, modient l'amplitude de la densité. On propose d'introduire une saturation non-linéaire des couleurs basée sur la variation par paliers de l'amplitude de la solution, due aux mitoses successives. 140 12 Exemple de panel de diagnostic de simulation numérique. De gauche à droite, de haut en bas : Contrôle macroscopique, maturités mésoscopiques, pertes de masses mésoscopiques instantanées, contrôles mésoscopiques, fractions de croissances mésoscopiques, perte mésoscopiques cumulées de masse, ζ f mésoscopiques, γ f min mésoscopiques, masses mésoscopiques. Dans cet exemple la cohorte comprend deux follicules, qui dièrent uniquement par leurs paramètres de vitesse de vieillissement.

µ 1 0 ≤ µ 1 ≤ 10 µ 2 0 ≤ µ 2 ≤ 10 µ 3 0 ≤ µ 3 ≤ 10 c 1 0.01 ≤ c 1 ≤ 0.1 c 2 0.1 ≤ c 2 ≤ 1 ū 0.001 ≤ ū ≤ 0.01 U max 0.1 ≤ U max ≤ 0.2 U min U min = 1 2 U max c 0 < c < 10 M 1 5 M o N f ≤ M ≤ M o N f M f 1 < M f < 20 M o M o > M f b 1 0.01 ≤ b 1 ≤ 0.1 b 2 1 ≤ b 2 ≤ 10 b 3 1000 ≤ b 3 ≤ 2000 K K > 0 γ 0 < γ < 1 γ - s 0 < γ - s < γ s γ + s γ s < γ + s < 1
1 z = sup a,γ,f |φ f 0 (a, γ)|, 0 ≤ z ≤ z z ≤ z ≤ 2z 2z ≤ z ≤ 4z 4z ≤ z ≤ 8z 8z ≤ z ≤ 16z 16z ≤ z ≤ sup t sup f sup a,γ |φ f (a, γ, t)|
U (t) = U max m f 0 = (t) = [0,Nc]×[0,γs] φ f (a, γ, t)dadγ, m f 1 (t) = [0,Nc]×[0,γs] γφ f (a, γ, t)dadγ,
des densités folliculaires. Rappelons ici que pour un follicule ovulatoire, l'accroissement de masse doit être compris entre 20 et 30 fois l'eectif initial, et qu'un follicule atrétique est caractérisé par un décrochement de la courbe de masse.

Maturité ovarienne et contrôle macroscopique. En haut à gauche de la Figure 5.12, on représente les courbes de maturité ovarienne et de valeur du contrôle macroscopique. Il faut que la courbe de niveau plasmatique de FSH au cours du temps U (t) chute à partir de sa valeur asymptotique U max pour qu'il y ait sélection parmi les follicules.

Contrôle mésoscopique. Au centre à gauche de la Figure 5.12, on trace les courbes de disponibilité locale en ressource FSH (une courbe par follicule). Typiquement, une courbe de ce style croit et atteint U (t) (qui se trouve être la plupart du temps U min ), dans le cas des follicules ovulatoires, et décroît pour les follicules atrétiques.

Fraction de croissance. Rappelons que l'on dénit la fraction de croissance d'un follicule par

GF f (t) = m f (t) m f 0 (t)
, où m f (t) = [0,Nc]×[0,γs] φ f (a, γ, t)dadγ est l'eectif cellulaire dans la partie du domaine correspondant au cycle cellulaire.

Courbes ζ f . En bas à gauche de la Figure 5.12, on représente les courbes de maturité cellulaire moyenne dénies par

ζ f (t) = m f 1 (t) m f 0 (t)
.

Cette grandeur indique ainsi la maturité asymptotique vers laquelle tend la densité en temps long (voir [START_REF] Michel | Multiscale modeling of follicular ovulation as a mass and maturity dynamical system[END_REF])

Courbe γ f min . En bas au centre, on trace la courbe de la valeur minimale de la maturité, que l'on note γ min :

γ f min = min{γ ∈ [0, 1]|∃a ∈ [0, N c ], φ f (a, γ) > 0}.
On l'évalue numériquement en comparant la densité avec une petite valeur de seuil, xée arbitrairement à 10 Pour calculer une estimation de l'index mitotique dans le cadre de notre modèle, nous devons prendre en compte la durée de la mitose qui débute en n de phase SM

I f m (t n ) = Ω i,j ∩Ω M it =∅ |Ω i,j ∩ Ω M it |φ n, i,j Ω i,j ⊂Ω |Ω i,j |φ n, i,j = Ω i,j ∩Ω M it =∅ |Ω i,j ∩ Ω M it |φ n, i,j m f 0 (t)
Il est à remarquer que la limite à gauche de la zone de mitose peut tomber à l'intérieur d'une maille de la grille volume nis (adaptative qui plus est). C'est pourquoi nous prenons soin de pondérer la valeur de la densité sur une maille par la surface contenue eectivement dans la zone de mitose. Perte instantanée et perte cumulée. On dénit la perte de masse instantanée r f pour chaque follicule f = 1, . . . , N f :

r f (t) = [0,Nc]×[0,1]
Λ(a, γ, U )φ f (a, γ, t)dadγ.

On évalue r f (t) par quadrature numérique

r f (t n ) ≈ Nx i=1 Ny j=1 Λ(a i , γ j , U n )φ f (a i , γ j , t n )∆a∆γ.
On dénit la perte cumulée au cours du temps pour chaque follicule

R f (t) = t 0 r f (s)ds, qu'on évalue numériquement par R f (t) ≈ N n=1 r f (t n )∆t n .
On représente ces deux indicateurs sur la Figure 5

Méthode de calibration pour un follicule

Dans cette section, nous présentons une méthode de calibration pour un follicule en deux étapes :

1. Détermination de la maturité asymptotique, à contrôle maximal constant ;

2. Détermination des paramètres aectant la dynamique, à contrôle maximal constant ;

A l'issue de ces étapes, on obtient une situation de référence, que l'on peut ensuite perturber (aléatoirement ou de façon déterministe) pour en obtenir d'autres.

Détermination de la maturité asymptotique

On a détaillé dans le paragraphe 5.1.2 le rôle des zéros de la vitesse de maturation dans le comportement asymptotique en temps long de la densité. Il est important de noter que le paramètre τ f n'a pas d'inuence sur les valeurs des zéros de h f . On peut ainsi faire varier l'amplitude de h f tout en gardant la même valeur asymptotique pour la solution, à contrôle constant. Le paragraphe suivant montre comment choisir les autres paramètres de la vitesse de maturation, qui eux, inuent sur la position des zéros.

c 1 c 2 ū u f = U min u f = ūmin h f γ γ s γ - s γ + s h f γ min 0 γ max 0 ūmin U min h γ max 0 γ + s 1 γ min 0 γ s γ + s c 1 c 2 ū γ 0 (c 1 , c 2 , ū) h f u f = ūmin u f = U min c 1 c 2 ū γ 0 E 1 = {(c 1 , c 2 , ū)|γ min 0 ∈ [γ s , γ + s ]}, E 2 = {(c 1 , c 2 , ū)|γ min 0 ∈ [γ + s , 1]}. E = E 1 ∩ E 2 c 1 , c 2 , ū E E γ min 0 γ max 0 h f c 1 c 2 ū (c 1 , c 2 , ū) h f γ 0 h [γ 1 , γ 2 ] γ 1 = 0.35 γ 2 = 0.45 γ 0 ∈ [γ 1 , γ 2 ] h f (c 1 , c 2 , ū) h f γ 0 ∈ [γ s , γ + s ] u f = ūmin h γ 0 > γ + s u f = U min γ s = 0.3 γ + s = 0. 35
U (t) = U max g f h f g f (a, γ, u f ) = g 1 u f + g 2 G1, 1 SM ∪ D h f (a, γ, u f ) = τ f -γ 2 + (c 1 γ + c 2 )(1 -e ( -u f ū ))
G1 ∪ D, 0 SM Chapitre 5. Calibration dans le cas simplié où les paramètres c 1 , c 2 , ū restent inchangés. Ainsi, pour obtenir la même masse nale, lorsque l'on divise les paramètres g 1 et g 2 par une même quantité p, il sut de diviser τ f par cette même quantité.

Reparamétrisation. On a vu que les paramètres du sous problème étaient liés de façon linéaire à contrôle constant. On peut alors ré-exprimer les paramètres des fonctions g f et h f . Nous posons

           g 1 = p, g 2 = pα, τ f = C p × p, α = g 2 g 1 .
avec p, C p et α trois constantes réelles, que nous désignerons par le terme de "paramètres de travail". Nous allons à présent voir l'utilité de cette reparamétrisation.

Paramètre de capital prolifératif à contrôle constant C p . Sur la Figure La fraction de croissance chute de 1 à près de 0.

Le taux de mort cumulée est inférieur à 10%. On note que seul l'un des follicules est ovulatoire alors que les autres sont atrétiques.

h lim t→∞ b f (t) = 1 lim t→∞ u f (t) = U (t) U (t) = U min lim t→∞ b f (t) = (b 1 + 1 b 3 ) lim t→∞ u f (t) = (b 1 + 1 b 3 )U min ūmin
Ce sont des résultats préliminaires illustrant l'eet de l'hétérogénéité sur un seul paramètre. Il est intéressant de comparer ces résultats avec ceux de la Figure 5.24.

En eet, cette dernière est le résultat de la même simulation, mais sans pression sélective. On note que dans ce cas, tous les follicules sont ovulatoires.

La Les zéros de h f sont les mêmes pour tous les follicules, mais les amplitudes de la fonction sont diérentes (en eet, seul le paramètre τ h varie). Sans pression sélective, cela n'a pas d'eet sur la trajectoire folliculaire, mais en introduisant de la sélection, on observe que seul le follicule avec une grande vitesse de maturation devient ovulatoire, les autres devenant atrétiques. controlled population dynamics in a system of transport equations in 2D, with a linear source term. This model embeds a close loop control acting on the velocities and source term and dened from one of the rst order moment of the solutions, so that the equations are weakly coupled, see Figure 6.1. Non conservative ux conditions at the internal boundaries between sub regions of the domain of interest are another original feature..

From the very beginning, the numerical simulation of this model was a problem (see the perspectives of the PhD thesis [START_REF]Echenim Modélisation et contrôle multi-échelles du processus de sélection des follicules ovulatoires[END_REF]).

A performant code was needed in order to run intensive simulations for calibration and uncertainty quantication on one hand, and to enable one to do real time computation within a "biologist friendly environment" on the other hand. From this point, it was necessary to nd methods that would speed up the computations. To do so, we designed a dedicated scheme, involving parallel computing and adaptive meshing. We also enhanced the precision with a third order scheme. It takes now around 2 minutes on a cluster to obtain the solution of a simulation corresponding to a biologically realistic conguration (see Figure 6.14). This is a tremendous improvement with respect to the initial simulations ( [START_REF]Echenim Modélisation et contrôle multi-échelles du processus de sélection des follicules ovulatoires[END_REF]) which would have required several days for the same set-up.

In order to evaluate the performances of this parallel code, the weak scalability test is the most adapted to our set-up. However, its results are relative since, for reasons that we will develop later, the CPU requirement are very sensitive to the parameters of the simulation. We then proposed a generalization, combining the classical method and Monte Carlo methods, inspired from uncertainty quantication. It consists in perturbing a reference situation, and evaluating the impact on the performances.

The outline of this article is the following. In the rst section, we present the class of models considered in this work, namely the multiscale controlled population dynamics. We note that such models may be written under the form of a block diagram (see Figure 6.1). In the second section, we present the multiscale numerical method, together with the general SIMD algorithm in the last part of the section. Finally, we discuss the tools that we introduced for performance analysis, and apply them to the model of the selection process of ovarian follicles.

Multiscale model

Conservation law. The evolution of the sub-population densities is given by the following conservation law with associated initial condition

       ∂ t φ f (x, y, t) + ∂ x g f (x, y, u f )φ f (x, y, t) + ∂ y h f (x, y, u f )φ f (x, y, t) = -Λ(x, y, U )φ f (x, y, t), φ f (x, y, 0) = φ 0 f (x, y),
where g f and h f are velocity functions, Λ is the source term, U is the global control and u f the control at the mesoscopic level.

The velocity functions depend on the values of the non-local control. In the biological model, both controls depend on time, through the rst order moment of the solution

φ f f ∈ {1, . . . , N f } U u f U b f m f 1 U M 1 f = 1 f = N f Chapitre 6.
Design and performance analysis of a multiscale scheme for controlled population dynamics in the y-direction. They can also be dened as explicit functions of time in an open loop control version.

The source term models a birth term when it is positive, or a loss term when it is negative. It can also be both, for example the sum of a birth term and a loss term.

Moments. The sub-populations moments are the link between the micro scale and the mesoscopic scale. We compute the rst order moments of each sub-population with the following formula

m f 1 (t) = [0,Nc]×[0,1]
yφ f (x, y, t)dxdy.

(6.1)

The link between the mesoscopic scale and the macroscopic scale is dened with the global moment :

M 1 (t) = N f f =1 m f 1 (t).
       G1 = {(x, y) ∈ Ω, k ≤ x ≤ (k + 1/2), k = 0, . . . , N c -1, 0 ≤ y ≤ γ s }, SM = {(x, y) ∈ Ω, (k + 1/2) ≤ x ≤ (k + 1), k = 0, . . . , N c -1, 0 ≤ y ≤ γ s }, D = {(x, y) ∈ Ω, γ s ≤ y}. controlled

population dynamics

Here, the speed functions and the source term are dened by

                           g f (x, y, u) = p(u + α) for (x, y) ∈ G1, 1 for (x, y) ∈ SM ∪ D, h f (x, y, u) =    C p × p × (-y 2 + (c 1 y + c 2 )(1 -exp( -u ū ))) for (x, y) ∈ G1 ∪ D, 0 for (x, y) ∈ SM , Λ(x, y, U ) =    -K × exp(-( (y -γ s ) 2 γ )) × (U max -U ) U max for (x, y) ∈ G1 ∪ D, 0 for (x, y) ∈ SM , (6.3)
where p, α, C p , c 1 , c 2 , ū, U max , K and γ are positive real constants.

The functions S and b f dening the macroscopic and mesoscopic controls respectively in the functional diagram (see Figure 6.1) are given by

             S(M 1 ) = U min + U max -U min 1 + exp(c(M 1 -M )) , b f (m f 1 ) = min b 1 + exp(b 2 m f 1 ) b 3 , 1 , (6.4)
where U min , c, M , b 1 , b 2 and b 3 are positive real constants.

The initial conditions are given by (6.5) with γ 1 < γ 2 < γ 3 < γ 4 < γ s , µ 1 , µ 2 , µ 3 and c k positive real constants. The population index for velocities g f , h f , control u f and initial condition φ 0 f indicates the possible heterogeneity among the follicles which will be introduced by the pa-

φ 0 f (x, y) =              µ 1 c k 1-exp -c k exp(-c k x) if γ 1 < y < γ 2 and 0 < x < 1, µ 2 c k 1-exp -c k exp(-c k x) if γ 2 < y < γ 3 and 0 < x < 1, µ 3 c k 1-exp -c k exp(-c k x) if γ 3 < y < γ 4 and 0 < x < 1, 0 otherwise,
rameters p, α, C p , c 1 , c 2 , ū, b 1 , b 2 , b 3 , (γ i ) i=1,2,3 , (µ i ) i=1,2,3
, c k , which can vary for the dierent follicles, although we do not index them by f to alleviate notations. We further associate transmission conditions between the sub-domains, as follows :

       φ f (t, x + , y) = p(u f + α)φ f (t, x -, y), x = k -1/2, 0 ≤ y ≤ γ s , p(u f + α)φ f (t, x + , y) = 2φ f (t, x -, y), x = k, 0 ≤ y ≤ γ s , φ f (t, x, γ + s ) = 0, k -1/2 ≤ x ≤ k.
The second condition is a doubling ux, modeling cell mitosis.

As an example, on Figure 6.2, we present a result of a numerical simulation of this model. In panel a), we represent with a color code the cell density of one of the follicle on the microscopic level at initial time (top) and at a later time t = 4 (bottom), controlled population dynamics then have 2 × 10 6 grid cells, to be updated at each time step. It is necessary to use speedup techniques in order to simulate the problem in a reasonable time.

Note here that the size of the computational domain in x, which denote the cells age within the successive cell cycles, can be any integer value (we have conventionally set the size of one cell cycle to 1). If we want to keep track of the cells' history, at the microscopic scale, we can use the same number of cell cycles as that needed to follow a whole simulation. One may be only interested in the mesoscopic and macroscopic outputs (size of the population, maturities, growth fraction, etc) as a function of time. In that case we may use only one cell cycle that repeats with periodic boundary conditions. In practice, we often use an intermediate solution,

with sucient successive cell cycles to avoid any ambiguity on the age of the cells (which would occur if densities belonging to dierent cycles could be superimposed due to the periodicity).

Microscopic level : adaptive nite volume scheme

At a microscopic level, the evolution of the densities is given by a system of transport equations. A classical method used for this type of systems is a nite volume scheme.

In this section we present the principles of adaptive nite volume schemes using multiresolution. For the sake of simplicity, we only present the case of two levels of grid, with a simple rst order scheme. The generalization will come in the next paragraph. The reader can refer to [START_REF]Coquel Theoretical and numerical analysis of hyperbolic systems of conservation laws[END_REF] and [START_REF] Bouchut | Nonlinear stability of nite volume methods for hyperbolic conservation laws, and well-balanced schemes for sources[END_REF] for bases on nite volume schemes and to [START_REF] Harten | Multiresolution algorithms for the numerical solutions of hyperbolic conservation laws[END_REF], [START_REF] Muller | Adaptive multiscale schemes for conservation laws[END_REF] and [START_REF] Cohen | Fully adaptive multiresolution nite volume schemes for conservation laws[END_REF] for adaptive mesh renement in the context of nite volume schemes.

Hierarchy of nested grids. We dene a sequence of nested grids (see Figure 6.3), for = 0, ..., L -1,

       x i = i∆x y j = j∆x Ω i,j = [x i , x i+1 ] × [y j , y j+1 ],
with i ∈ {0, ..., N x }, j ∈ {0, ..., N y } and ∆x = ∆x2 (L-1)-. We have the following property :

Ω i,j = Ω +1 2i,2j ∪ Ω +1 2i+1,2j ∪ Ω +1 2i,2j+1 ∪ Ω +1 2i+1,2j+1
We say that Ω i,j is a mother grid cell, and {Ω +1 2i,2j , Ω +1 2i+1,2j , Ω +1 2i,2j+1 , Ω +1 2i+1,2j+1 } is the set of corresponding daughter grid cells.

Finite volume scheme. The FV approximation of the solution is dened on level by

φ n, i,j ≈ 1 (∆x ) 2 Ω i,j φ(x, y, n∆t)dxdy, φ n+1, i,j = φ n, i,j - ∆t n ∆x (G n, i+1,j -G n, i,j + H n, i,j+1 -H n, i,j ) -∆t n Λ n, i,j φ n, i,j              G n, i,j ≈ 1 ∆t n 1 ∆x [t n ,t n +∆t]×[y j ,y j+1 ] g(x i , y, u)φ(x i , y, t)dydt, H n, i,j ≈ 1 ∆t n 1 ∆x [t n ,t n +∆t]×[x i ,x i+1 ] h(x, y j , u)φ(x, y j , t)dxdt, G n, i,j = (g n, i-1,j ) + φ n, i-1,j + (g n, i,j ) -φ n, i,j H n, i,j = (h n, i,j-1 ) + φ n, i,j-1 + (h n, i,j ) -φ n, i,j (f ) + = sup(f, 0) (f ) -= inf(f, 0) ∆t n = λ min i,j, { ∆x (|g n, i,j |) , ∆x |h n, i,j | }, λ 0 < λ ≤ 1 controlled

population dynamics

We can dene a hybrid grid based on a strategy consisting in replacing locally the solution by its second order polynomial approximation, when it is smooth enough.

Indeed, approximation theory provides the relation between the details and the local smoothness ( [START_REF] Cohen | Wavelet methods in Numerical Analysis[END_REF]). In the multiresolution framework the details are used as local regularity sensors to trigger or not renement to the ner grid level. Given a threshold value , for each mother-daughters pair, if all the details are under the threshold value

|d +1 i+k,j+m | ≤ , ∀k, m ∈ {0, 1}, (6.12)
then we can replace the details by 0 (local second order approximation), and delete the corresponding daughter cells.

In the general case of L levels, the threshold value is level dependent = 2 (L-1)- In other words, each time we decrease the level, we divide the threshold value by 2.

This amounts to a more cautious threshold at coarsest levels than at nest ones.

Data: Solution on hybrid grid Result: Solution coded on hybrid tree for = (L -1) to 1 do for cells in level do

Compute the mean values with the Projection operator (6.9)

(φ +1 i,j , φ +1 i+1,j , φ +1 i,j+1 , φ +1 i+1,j+1 ) → (φ i,j , φ +1 i+1,j , φ +1 i,j+1 , φ +1 i+1,j+1 )
end for cells in level l do

Compute details with the Prediction operator (6.10) and detail formula (6.11)

(φ i,j , φ +1 i+1,j , φ +1 i,j+1 , φ +1 i+1,j+1 ) → (φ i,j , d +1 i+1,j , d +1 i,j+1 , d +1 i+1,j+1 )
end end Algorithm 3: Solution encoding : from the physical basis (FVS) representation into multiresolution basis (wavelet). controlled population dynamics Let us now give the details of the computations of the center grid cell. We may write in this example

     φ n+1,0 i,j = φ n,0 i,j - ∆t n (∆x 0 ) 2 (∆x 0 G n,0 i,j - ∆x 0 2 G n,1 i+1,2j - ∆x 0 2 G n,1 i+1,2j+1,0 + ∆x 0 H n,0 i,j -∆x 0 H n,0 i,j )
We note that it is useful then to use the following formula

φ n+1, i,j = φ n, i,j + λ D n, i,j (6.14)
with λ = ∆t n (∆x ) 2 and D n, i,j the numerical divergence and source term of cell Ω n, i,j .

This term may be computed as follows. At the beginning of each time step, we have D n, i,j = 0. Using a loop on the hybrid grid cells, we compute for each of them the numerical uxes (see 6.7) on the left vertical edge G n, i,j and on the bottom horizontal edge H n, i,j , then we modify theses uxes as follows

Gn,

i,j = ∆x G n, i,j , Hn, i,j = ∆x H n, i,j , (

and we update the values of the load balances of the neighbors

           D n, i-1,j -= Gn, i,j , D n, i,j + = Gn, i,j , D n, i,j-1 -= Hn, i,j , D n, i,j + =
Hn, i,j .

(6.16)

At the end we add the source term D n, i,j + = ∆t n Λ n, i,j φ n, i,j .

Data: Solution on hybrid grid at time t n Result: Flux balance D n, i,j at time t n Initialize load balances D n, i,j = 0 ; for = (L - 6.4 Finite volume scheme on a hybrid grid : the ux computation between two grid cells of dierent size (for example the center right ne cells, marked with an horizontal arrow, and the center coarse cell) have to be done in two steps. First, we have to add ghost cells in order to compute the uxes on the ne level, then we delete these cells doing a projection (6.9), and compute the uxes on the coarse level.

Update the hybrid grid. The hybrid grid at time t n is generated by a graded tree, noted T n , dened by recursion ∀n ≥ 1, T n+1 = P (T n )

with P the prediction operator. This operation, consisting in evaluating the grid cells that have to be either added or kept, is done in three steps :

Adding of cells using the heuristic Harten rules (Algorithms 5 and 6) ;

Adding of ghost cells : in order to be able to compute the uxes on the hybrid mesh, we add all the necessary ghost cells (see Figure 6.4) ;

Deletion of unused cells ;

Thus, updating the hybrid grid consists in : Initialize the hybrid grid. The initial hybrid grid T 0 may be dened by using two dierent methods.

The rst method consists in computing the grid from the ner level. One denes the solution on all the ne grids, and then apply the algorithmic sequence : coding, thresholding, decoding. In other words, we compute all the details, and delete all the grid cells where the details are below the threshold. The problem with this method is that it uses a lot of memory at the beginning.

Another approach may be used, without dening the entire ne level. First, we compute the solution on the coarser level. Then we do a two level recursion. At level we compute the predictions and the details at level + 1 (we can do it because we know the exact solution at time t = 0). We split all the grid cells where the details are signicant. Yet, as we can see on Figure 6.5, it is not enough. The grid mesh has to be graded. In order to ensure this property, we do another loop on the level, and split all the grid cells that are in the prediction stencil dened by (6.13) of the new grid cells created just before. This can be a recursive operation if the prediction stencils of these new cells are not already in the grid.

We have illustrated the dierences between both methods on Figure 6.6. The parameters used are : = 0.01, 10 grid cells on the coarsest level, 5 levels of multiresolution. Initialization starting from the coarsest grid is of course much more economical in terms of memory. However, this method should be used with caution when the initial condition is not smooth enough. Indeed the coarsest discretization must be able to capture all the features of the initial condition in order to trigger its renement to the adequate level of discretization. In our case, we use piecewise controlled population dynamics Figure 6.6 Initialization method from the coarsest level. We clearly see that this method use less memory than in the case where we initialize from the nest grid. Precision enhancement. In order to enhance the precision of the scheme, we can improve the numerical scheme. On one hand, the scheme has to be precise in space. It is ensured thanks to a good choice of the numerical ux (6.7).

For more details about the original scheme developed in the framework of the model of the selection process of ovarian follicles, with specic transmission conditions (in particular concerning mitosis), reader can refer to [START_REF]A numerical method for transport equations with discontinuous ux functions : Application to mathematical modeling of cell dynamics[END_REF] and [START_REF] Aymard | Adaptive mesh renement strategy for a non conservative transport problem[END_REF]. In the rst reference, one may nd details about the numerical scheme on uniform grids, and in the second, one may nd details on how to adapt the previous scheme in a framework of adaptive meshing using multiresolution analysis.

On the other hand, we have to get better precision in time, which can be obtained with a Runge-Kutta scheme. Yet, we have to adapt this method if we want to use it on an adaptive grid. Runge-Kutta schemes consist in computing intermediate steps to compute each time step. Since the Finite Volume scheme designed in [START_REF]A numerical method for transport equations with discontinuous ux functions : Application to mathematical modeling of cell dynamics[END_REF] achieves at best third order precision in space, we need to use a third order scheme in time : RK3 (see [START_REF]Demailly Analyse Numérique et Equations Diérentielles[END_REF]). For the rst step, we compute φ * , i,j = φ n, i,j + λ D

(φ i,j , d +1 i+1,j , d +1 i,j+1 , d +1 i+1,j+1 ) → (φ i,j , φ +1 i+1,j , φ +1 i,j+1 , φ +1 i+1,j+1 )
end for cells in level do

If(ghost cell)

Reconstruct mother cell value with projection formula (6.9)

(φ i,j , φ +1 i+1,j , φ +1 i,j+1 , φ +1 i+1,j+1 ) → (φ +1 i,j , φ +1 i+1,j , φ +1 i,j+1 , φ +1 i+1,j+1 ) end end
Algorithm 13: ghostReconstruction 6.2.2 Mesoscopic and macroscopic scale : Numerical moments and control.

Once we have dened a discretization of time and space, and after having computed the densities at the microscopic level, we can compute the numerical rst order moments at the mesoscopic scale

m n 1,f = (i,j, )∈T φ n,f i,j (∆x ) 2 y i,j, , (6.20)
Figure 6.7 Strategy of parallel computing using distributed memory. The strategy mimics biology : at each sub-population we associate a process. This strategy is easy to implement (SIMD algorithm), and involves few and ecient communications (reduction operations). Option 1 : domain decomposition method. We cut the domain into several parts and send them to dierent processes. Option 2 : split the sub-populations and send them to dierent processes. There are less communications using option 2, and the communications that have to be done are ecient (reduction operations) and mesh size independent. controlled population dynamics to get the same performances as in the reference situation. In order to get the real performances, the user wants to know the average performance of the code. For example, the average user time may be of interest.

In this regard, our idea was to use a Monte Carlo method. We introduce perturbations on the parameters, and evaluate the resulting average performances. We can then dene a mean weak scalability method, generalizing the weak scalability method, by plotting the mean user time and its variance with respect to the number of processes.

Principle of the Monte Carlo method. Based on the law of large numbers, the Monte Carlo method is a numerical method that evaluates the mean value of a random distribution of real numbers. The principle is to simulate many realizations (say N realizations) of a random variable X, and then compute the empirical average. The law of large numbers ensures that lim

N →∞ 1 N N i=1 X i = < X >, (6.25)
where X i is the i-th realization of X, and where by < • > we denote the average function. N will be set later for each numerical test. It is possible to visualize the results by using histograms, to represent the tabulated frequency of the data over discrete intervals (bins), with an area proportional to the frequency of the observations in the interval.

In practice, we simulated uniform laws using the Mersenn twister algorithm [START_REF] Matsumoto | Mersenne Twister : A 623dimensionally equidistributed uniform pseudo-random number generator[END_REF].

The seeds used for Mersenn twister method were time(NULL) × (p + 1)

with p the number of process (it starts from 0), in order to get dierent sequences on each process.

Mean load balance test

Occupation rate. As reminded previously, the key point in parallel computing is the data management. One of the classical concepts used is the load balance, which in our case is directly linked to the size of the FVS grid and to the time step. Each process has a certain amount of work to do. We say that the load balance is good when the work is well splitted between the processes, and none of them is working more than the others, constraining them to wait.

The occupation rate, denoted R(t), is dened by

R(t) = m(t) M , (6.26)
where m(t) is the number of hybrid grid cells at time t and M is the total number of grid cells on the nest uniform grid. We note that 4 This test on one processor can be used to predict some features of the parallel case behavior. We expect that the load balance will remain good. However, it is not sufcient to ensure good weak scalability results. Indeed, even if the size of the meshes associated with the dierent populations are well balanced in size, the variability in the velocities can induce a variability in the time step. Since the population has to evolve in a synchronized manner, the smallest time step is used. If we progressively add sub-populations to our system, with randomly chosen velocity parameters, the time step can only decrease and therefore penalizes the overall CPU time.

These numerical simulations give us some statistic informations about the numerical performances of the scheme that we have designed, and furthermore have set a methodology, allowing us to run a systematic study of parameters' sensibility. In a close future, we aim at extend this method of uncertainty propagations to other outputs, such as biological outputs, describing the qualitative behavior of the model (ovulation rate, exit time, nal mass, etc...).

Mean weak scalability test

Method. As explained in paragraph 6.3.1 in the class of models that we study the size of the problem is related not only to the number of processes but also to the heterogeneity in the parameters. We generalize the notion of weak scalability to dene a "mean weak scalability". This test consists in performing many realizations of numerical simulations of the multiscale model for dierent sizes of the problem.

We have tested four situations : the case of a compact domain (useful for visualizing the results on the mesoscopic and macroscopic scales, with fast computations), unrolled domain (useful for visualizing the results on the microscopic scale), precision enhancement on the whole domain (rene coarse level), precision enhancement locally (add one level of multiresolution). For this test we used the biological parameters listed in 6.3). In the right panel they vary randomly, but, as expected, without inuencing the user time statistics. In the bottom left panel, we use the uniform mesh to see the inuence of the random parameters on the time step only. In the bottom right panel the time step is kept xed, but the random parameters inuence the space discretization through adaptive mesh renement.

As for the single processor reference test displayed in Figure 6.11 we note that even in the reference situation, there is some variability in the user time. This is related to the technology used, and it is known that the user time on a deterministic problem is subject to variability.

The range of variation of the user time due to the variability of the adaptive time stepping is roughly three times wider than the one due to variability of the adaptive mesh.

Also, we clearly see, on the panels of Figure 6.16, that the dots are gathered, and the groups of dots are separated by two gaps. First, a gap between 1 and 3 processes, which corresponds to the addition of communications, that are not necessary for one process only. Second, a gap between 7 and 9 processes, due to a change of technology.

Indeed, processors are gathered in nodes of eight on the cluster that we have used, and the communications are faster within a node.

Conclusion

In this chapter, we have designed a general multiscale scheme for controlled population dynamics, and we have proposed a method of performance analysis in the framework of parallel computing. As an example, we applied successfully the scheme and method to our biological problem, the selection process of ovarian follicles. In appendix A and B the reader can nd two "getting started" chapters, on the computational code CodeFollicleMR and the visualization code dualMR, respectively. All the implementation details are detailed in Appendix C.

195 Chapitre 6. Design and performance analysis of a multiscale scheme for controlled population dynamics q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0 10 20 Compact domain (one cell cycle) q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0 10 20 Unrolled domain (eight cell cycles) q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0 10 20 Precision enhancement (more meshes on the coarsest level) q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0 10 20 Precision enhancement (one more level of MR) Figure 6.13 Mean weak scalability test. In abscissa, the number of processes, in ordinate, the user time. The green line corresponds to the mean user time for one process. This is the theoretical limit. The blue line indicates the mean sequential behavior, product of the mean user time for one process by the number of processes. The black dots are random independent realizations of the numerical simulations of our model, with perturbed parameters. We note that the size of the problem grows with the number of processes, but the CPU time remains close to the optimal theoretical value.

196 Chapitre 6. Design and performance analysis of a multiscale scheme for controlled population dynamics q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0 10 20 Compact domain (one cell cycle) q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0 10 20 Unrolled domain (eight cell cycles) q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0 10 20 Precision enhancement (more meshes on the coarsest level) q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0 10 20 The blue dots correspond to the numerical simulations, and the red curve corresponds to the linear regression. The variability tends to decrease with respect to the number of processes. It is interesting to note that there is a gap between the values for a number of process smaller than 8 and the others, due to the technology that we used (the rst eight nodes are gathered in a small and fast cluster). controlled population dynamics q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 5 10 15 Reference (same parameters, uniform mesh, constant time step) q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 5 10 15 Random parameters (uniform mesh and constant time step) q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 5 10 15 Random parameters and adaptive time step (uniform mesh) q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 5 10 15 0 500 1000 1500 number of processes user time (s) q numerical simulations sequential theoretical limit

Random parameters and adaptive mesh (constant time step) 198 Chapitre 6. Design and performance analysis of a multiscale scheme for controlled population dynamics q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 5 10 15 Reference (same parameters, uniform mesh, constant time step) q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 5 10 15 Random parameters (uniform mesh and constant time step) q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 5 10 15 Random parameters and adaptive time step (uniform mesh) q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 5 10 15 Ici encore, les noms des chiers de sorties sont crées de façon dynamique à l'exécution en fonction des numéros de noeuds de calcul. Par exemple le chier microp.i contient la solution du processeur p à l'itération i (par exemple micro1.32 contient la sortie microscopique relative au follicule 1 au pas de temps 32).

U (t) m f 0 (t) m f 1 (t) GF f (t) R f (t) ∆t n f r f (t) M 1 (t) R f (t)

Sommaire 1 . 1 4 1. 2 . 1 21 1. 1

 11421211 Historique et Motivations . . . . . . . . . . . . . . . . . . . . 1 1.2 Modèle multiéchelles . . . . . . . . . . . . . . . . . . . . . . . Inconnues du problème . . . . . . . . . . . . . . . . . . . . . 5 1.2.2 Système de lois de conservations . . . . . . . . . . . . . . . . 6 1.2.3 Contrôle non local . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 Apports de la thèse . . . . . . . . . . . . . . . . . . . . . . . . 12 1.3.1 Numerical simulation of the selection process of the ovarian follicles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.3.2 A numerical method for transport equations with discontinuous ux functions : application to mathematical modeling of cell dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.3.3 Adaptive mesh renement strategy for a non conservative transport problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.3.4 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.3.5 Schéma numérique multiéchelles et analyse de performances par propagation d'incertitudes . . . . . . . . . . . . . . . . . Historique et Motivations L'objectif principal de cette thèse est de simuler numériquement, de façon able et rapide, un modèle mathématique représentant un processus physiologique complexe, le développement terminal des follicules ovariens et la sélection des follicules Chapitre 1. Introduction ovulatoires.

Figure 1 . 1

 11 Figure 1.1 Présentation schématique des diérentes échelles impliquées dans le développement folliculaire (Figure extraite de[START_REF] Aymard | Numerical simulation of the selection process of the ovarian follicles[END_REF]). A gauche : boucle de rétrocontrôle entre les ovaires et l'hypophyse modulant les niveaux de FSH. Au milieu : coupe schématique d'un follicule ovarien. La granulosa, dont certaines cellules entourent l'ovocyte, est délimitée par une cavité liquidienne, l'antrum et par une autre couronne cellulaire, la thèque. A droite : diérents états cellulaires rencontrés par les cellules de granulosa.

Figure 1 . 3 1

 131 Figure 1.3 Plan age-maturité. En abscisse, l'âge (position dans le cycle cellulaire), en ordonnée la maturité cellulaire. A chaque sous-domaine correspond une phase du cycle cellulaire des cellules de granulosa. Les zones hachurées correspondent aux zones où les cellules peuvent disparaître par apoptose, pendant les phases G1 ou D.

6 Chapitre 1 .

 61 Introductiondensité nulle φ f = 0 aux bords inférieur γ = 0 et supérieur γ = 1 du domaine ; des termes de contrôle :

Figure 1 . 4

 14 Figure 1.4 Interfaces et conditions de transmission (Figure extraite de[START_REF] Aymard | Adaptive mesh renement strategy for a non conservative transport problem[END_REF]). Le champ de vitesse est régulier sur chaque sous-domaine Ω 1 , Ω 2 , Ω 3 , mais est potentiellement discontinu aux interfaces séparant ces derniers (en x s et en y s ). An de fermer mathématiquement le problème, il est nécessaire d'imposer des conditions de transmission.

Figure 1 . 5

 15 Figure 1.5 Fonction de vieillissement g f sur une coupe horizontale (à gauche), en zone de prolifération, du domaine de calcul 2D (à droite), pour diérentes valeurs de contrôle mésoscopique u f . Un contrôle élevé entraîne une vitesse élevée, favorisant la production de cellules. En eet, dans ce cas, les cellules ont alors un cycle plus court.

Figure 1 . 7

 17 Figure 1.7 Fonction de perte par apoptose Λ sur une coupe (à gauche) du domaine de calcul (à droite) pour diérentes valeurs du contrôle global U . A contrôle maximal, ce terme est nul. Il s'active lorsque le contrôle baisse, ce qui entraîne une perte de cellules par apoptose.

Figure 1 . 8

 18 Figure 1.8 Contrôle hormonal : niveau plasmatique de FSH en fonction de la maturité ovarienne. Tant qu'une maturité seuil n'est pas atteinte, le contrôle est maximal. A l'atteinte du seuil, le contrôle chute, vers une seconde valeur asymptotique.

  résultats ont fait l'objet d'une publication dans SIAM Journal of Scientic Computing ([7]). Une fois obtenu un schéma numérique parallèle, d'ordre élevé, même aux interfaces, nous avons cherché à accélérer encore les calculs, en utilisant des techniques d'adaptation de maillage. En eet, on remarque que les solutions du système sont des fonctions de densités très localisées dans le domaine de calcul. Ainsi, l'utilisation d'un maillage uniforme n'est pas optimale. En eet, le maillage est très rané dans des zones où la densité est, et reste, la plupart du temps, nulle. Cela fait beaucoup de calculs inutiles, et beaucoup de mémoire utilisée pour rien. Nous avons utilisé des techniques de multirésolution (voir [10], [30]), pour faire des simulations sur des maillages adaptatifs en temps, localement ranés dans les zones où la solution varie. Nous avons développé un code de calcul dédié, en C++/MPI appelé CodeFolli-cleMR. Sur la Figure 1.14, on donne l'exemple d'un maillage adaptatif obtenu avec Chapitre 1

  le code. Une notice utilisateur est disponible en Annexe A de la thèse. Une notice programmeur, détaillant la structure du code source, les formats de chier d'entrée et de sortie, est disponible en Annexe C de la thèse. Après le problème du maillage et de la méthode de calcul s'est posé le problème de la visualisation des résultats. Le problème étant assez spécique, nous avons proposé un code de visualisation dédié. Il a fallu pour cela dénir un cahier des charges, qui s'est précisé avec le temps, au l des échanges avec les biologistes. Nous avons besoin de visualiser des résultats sur deux niveaux d'échelle diérents : micro et macro. Au niveau micro, on veut visualiser les densités de chacun des follicules (champs de scalaire dans un domaine borné pour chaque follicule). Une première idée fut de pouvoir visualiser dans une fenêtre les résultats pour un follicule, avec une option pour passer d'un follicule à l'autre. Cette méthode est pratique, mais n'est pas adaptée à la comparaison inter follicules. Nous avons alors proposé une visualisation globale et simultanée de tous les follicules, en achant les densités les unes au dessous des autres. Pour le niveau macro, nous voulons visualiser des courbes temporelles (masses, maturités, contrôle macroscopique et contrôles mésoscopiques, ...). Nous avons proposé de visualiser ces quantités simultanément à l'aide du logiciel gnuplot (voir par exemple Figure 1.19).Le code couleur au niveau micro a soulevé plusieurs questions, et a beaucoup changé au cours du temps. Au départ, il était redéni à chaque pas de temps, pour chaque follicule, à partir du maximum de la densité au temps courant. An de comparer plus facilement les follicules entre eux, nous avons déni un code couleur valable pour tous les follicules, à tous les pas de temps, basé sur la densité maximale, tous follicules confondus, au cours de la simulation. Le problème est que cette méthode favorise la visualisation des densités pour des temps proches du temps nal. Nous avons alors proposé un code couleur adapté à notre problème de dynamique de Chapitre 1

Figure 1 .

 1 Figure[START_REF] Adimurthi | Optimal entropy solutions for conservation laws with discontinuous ux-functions[END_REF].[START_REF] Bürger | Conservation laws with discontinuous ux : a short introduction[END_REF] Exemple de visualisation obtenue avec le code de visualisation C++ -OpenGL dualMR, dédié à la visualisation des résultats du modèle EDP de processus de sélection des follicules ovariens. On ache les densités de deux follicules correspondant à des valeurs diérentes des paramètres, dans le plan âge (abscisse x), maturité (ordonnée y) au même instant t. Le code couleur est adapté à la visualisation de notre problème : il est non linéaire, avec paliers, correspondant à diérents doublements de population.

Figure 1 .

 1 Figure 1.16 Apparition de bruit numérique en temps long : des mailles parasites apparaissent dans le maillage dans des zones où il n'y a pas de densité signicative (panel a) ). Avec notre méthode de réduction de bruit, ces mailles parasites disparaissent (panel b) ).

Figure 2 . 1

 21 Figure 2.1 Follicle development as a multiscale process. Left : endocrine feedback loop between the ovaries and the hypothalamo-pituitary-axis. Midle : schematic 2D view of an ovarian follicle. Right : dierent cell states encountered by the granulosa cells during follicular development.

2. 1 . 1 . 3

 113 Microscopic scale : granulosa cell kinetics Each granulosa cell can be encountered in either of three dierent cell states : proliferation, dierentiation or apoptosis (programmed cell death) (gure 2.3). At the beginning of terminal development, most granulosa cells are progressing along the cell division cycle, that can be split into a G1 phase, where cells are sensitive to FSH control, and a SM phase, where cells are preparing for mitosis and are insensitive to FSH control. At the end of mitosis, one single mother cell gives birth to two daughter cells. During the FSH sensitive phase, cells become more and more mature, up to a point where they reach a threshold maturity and exit the cell division cycle.

Figure 2 . 3

 23 Figure 2.3 Division of the spatial domain according to the cell phases for unit cycle duration D a = 1. Left : schematic view of a granulosa cell cycle. Right : (a, γ) plane. The bottom part represents the successive cell cycles, each composed of the G1 and SM phases. The top part corresponds to the dierentiation phase.

Figure 2 .0. 7 .

 27 Figure 2.6.e) shows that, at the nal time, only three follicles have reached the ovulatory stage, where the follicle maturity dened by (4.42) is higher than M f =

2 .

 2 Numerical simulation of the selection process of the ovarian follicles a) Initial condition. Piecewise constant in γ and constant in a in the rst cell cycle. b) Local increase in the density due to the decrease in the ageing function at the interface between the G1 and SM phases. c) Doubling of the cell density at the end of the second cycle and partial transfer into the dierentiation phase. d) Final density consisting of several clouds of fully dierentiated cells.

Figure 2 . 4 Figure 2 .

 242 Figure 2.4 Snapshots of the cell density at dierent times, starting from a piecewise constant density in the rst cell cycle. The color code is time dependent. The full movie is available at https ://www.ljll.math.upmc.fr/aymard/movie/ovulatory.mpg

Chapitre 3 . 1 Figure 3 . 3

 3133 Figure 3.3 1D test case 1 : ux continuity condition and velocity jump at interface

Figure 3 . 4

 34 Figure 3.4 1D test case 1 : ux continuity condition and velocity jump at interface x s = 0.5. In panel a) the total mass remains constant with time. In panel b) the L 1norm error goes to 0 with ∆x. The convergence rates (around 2.4) are gathered in Table 3.4.1.

(3. 23 ) 3

 233 centered on (c x , c y ) = (0.3, 0.15) and the same variance σ 2 = 0.002 as before. The corresponding results are displayed on Figure time t = 0.25.

Figure 3 . 5 2 Figure 3 . 6

 35236 Figure 3.5 2D visualization of test case 1 : ux continuity condition and velocity jump at interface x s = 0.5, u L = 0.5, u R = 1. Snapshot of the density (initial time, passing interface, nal time). (CF L = 0.4, ∆x = 0.001).

  Figure 3.8 displays the 2D generalization of the second test case. Chapitre 3. A numerical method for transport equations with discontinuous ux functions : application to mathematical modeling of cell dynamics

Figure 3 . 7

 37 Figure 3.7 1D test case 2. Doubling ux condition and velocity jump at interface x = x s . The mass in panel a) has doubled when all the density has reached the second subdomain, as expected. In panel b) the L 1 -norm error goes to zero with ∆x. The convergence rates (around 2.4) are displayed in Table 3.4.1.

Figure 3 . 8

 38 Figure 3.8 2D visualization of test case 2. Doubling ux condition and velocity jump at interface x = x s , u L = 1, u R = 0.5. Snapshots of the density (initial time, passing interface, nal time). The mass is doubling. (CF L = 0.4, ∆x = 0.001).

  Figure3.9 1D test case 3 : ux continuity condition and velocity jump at interface x s = 0.5, with source on the left. Snapshot of the density at initial time and when it crosses the interface between zones Ω 1 and Ω 2 (t = t N ).

Chapitre 3 .Figure 3 . 3 Figure 3 .

 3333 Figure 3.10 1D test case 3 : ux continuity condition and velocity jump at interface x s = 0.5, linear source on the left. In panel a) the mass decreases with time before the density has passed the interface (CF L = 0.4, ∆x = 0.001). In panel b) the L 1 -norm error goes to zero with ∆x. The convergence rates (around 2.4) are displayed in Table 3.4.1.

Figure 3 .

 3 [START_REF] Bürger | Risebro A relaxation scheme for continuous sedimentation in ideal clarier-thickener units[END_REF] displays the density at time t = 0.25 for three dierent h B values. For h B = 0.9, the solution displayed on panel a) looks very much like that obtained with the reference case. For h B = 0.001, the solution displayed in panel c) looks very much like that obtained with the waterproof boundary in Figure3.15.Since the exact solution is not available except for the extreme values h B = 0 or 1, we compute the error at time t N using in (4.35), instead of the exact solution, a reference solution φ f ine computed on a very ne grid with N f ine x grid meshes, and Chapitre 3. A numerical method for transport equations with discontinuous ux functions : application to mathematical modeling of cell dynamics a) Initial time. The density bump lies in zone Ω 1 . b) Intermediate time. Density crossing interface at t = 0.15. c) Final time t = 0.25.

Figure 3 .

 3 Figure 3.13 First 2D test case, shear phenomenon. Snapshot of the density (initial time, passing interface, nal time). (CF L = 0.4, ∆x = 0.001).

  b) 2D test case 2 : waterproof.

Figure 3 .

 3 Figure 3.14 2D test cases. L 1 -norm relative error with respect to ∆x. The convergence rate is 2.24 in the shear case (panel a) and drops to 1.03 in the waterproof case (panel b).

Chapitre 3 .

 3 A numerical method for transport equations with discontinuous ux functions : application to mathematical modeling of cell dynamics a) First order approximation. b) Third order approximation. c) Exact solution.

Figure 3 .

 3 Figure 3.16 From constant diagonal speed to waterproof interface. Solution at time t = 0.25 for three dierent values of the vertical speed in the Ω 2 zone. a) h B = 0.9, b) h B = 0.5, c) h B = 0.001

  et al.[START_REF] Echenim | Multi-scale modeling of the follicle selection process in the ovary[END_REF] and references therein. This multiscale model describes the latest stages of the follicular development before ovulation. The cell population of each follicle is studied as a density function depending on time, and structured according to two functional space variables, age and maturity. The density functions of the interacting follicles are solutions of weakly coupled PDE-s, where the age and maturity velocities and the loss term depend on the integro-moments of the solution for all the follicles. The moment-based integral formulation accounts in a compact way for the feedback loop involving the ovarian hormones, secreted from follicular cells, and the pituitary hormone FSH (follicle-stimulating hormone) that targets in turn the follicular cells.The model follows the development of the follicles starting from an initial stage where all cells are proliferating. As the granulosa cells progress through subsequent cell cycles their maturities increase up to a threshold beyond which they exit the proliferation cell cycle and enter the dierentiation stage. The control exerted by FSH on the cells depends on the cell state, that is on whether the cell be within or outside the division cycle. Moreover, cells become insensitive to any external control during a part of the cycle, leading to a drift dynamics behaving as pure transport.Cell proliferation is underlain by the process of mitosis, through which a mother cell gives birth to two daughter cells. Mitosis is the endpoint of the cell cycle that consists of the 4 phases G1, S, G2 and M, and ensures proper DNA replication and repartition between the new cells. The most common way of representing mitosis in age-structured models of cell populations is to add a gain term in the right hand side, that controls the (average) doubling time in the cell population (see the renewal

  domain. The cell cycles Ω p 1 ∪ Ω p 2 , p = 1, . . . , N c of unit width are periodically reproduced. Ω p 1 corresponds to phase G1, while Ω p 2 aggregates phases S, G2 and M of the pth cell cycle. b) Initial condition, identical for both follicles. The simulation covers N c = 8 cell cycles. Each cell cycle is divided into Ω p 1 ∪ Ω p 2 , for p = 1, . . . , 8. The cell population is initially in Ω 11 , in the rst cell cycle. A color code is used for the cell density ; a null density is encoded in blue, while the highest value is encoded in red. φ max = 75 denotes the maximum observed density over the whole domain at t = 0.

Figure 3 .

 3 Figure 3.18 Initial condition and computational domain for the full model

  where N c is the number of cell cycles and D c is the duration of one cycle. The domainΩ is divided in N = 2N c + 1 zones : Ω p 1 , Ω p2, for p = 1, . . . , N c and Ω 3 , corresponding to dierent cell states illustrated in panel a) of Figure 4.12 and hence dierent denition of the speeds and source terms. Phase Ω p 2 in the model aggregates the three latest phases (S, G2, M) of the p th cell cycle

Figure 3 . 1 0

 31 Figure 3.19 Follicular development model. Symbols × and + indicate the numerical errors respectively obtained with the order 1 and order 3 schemes. Dotted lines indicate the corresponding numerical convergence rates (0.78 and 1.87) obtained by least-square t.

) where b 1 ,

 1 b 2 and b 3 are real positive constants that may depend on the follicle f . The numerical value of the biological constants appearing in denitions 5.2, 4.40 and 4.44 are gathered in Table 3.4.1. To simplify the notation we do not explicitly indicate their possible dependence as a function of the follicle even if it is implied by the f index notation.

1 1 1 0 = 9 . 3 . 4 . 3 .

 119343 happen, as well as the waterproof phenomenon between zones Ω1 2 -Ω 3 . The relative error is computed with respect to a converged solution obtained using a very ne grid of 1920 grid meshes per half cycle (see equation 3.25). It is displayed as a function of the space step ∆x on Figure 3.19. The asymptotic order of convergence computed by least square tting is almost 2. The biological model combines all the numerical diculties studied separately in the six previous test cases. As expected, the order of convergence is intermediate between the best and worst order achieved for these toy problems. Finally we illustrate the competition between two follicles. The initial condition, with all cells in phase Ω , is represented in the bottom panel of Figure 4.12. To illustrate the dierent phenomena at interfaces, snapshots of the density at signicant times of the follicular development are displayed on Figures 3.20 and 3.21. For this simulation, a space discretization ∆x = 0.0125 (80 grid meshes per half cycle) is used, leading to a varying time discretization of ∆t ≈ 0.0012 which meets condition (3.17) with CF L = 0.4. The snapshots illustrate two important aspects of the model. First, the transitions between adjacent phases are well captured using the numerical method described in the paper. Second, the dierences between two competing follicles whose biological properties dier only by the value of one parameter, the maturation velocity, are shown to increase with time. At initial time the two follicles have the same cell population m 1 0 = m 2 0 = 1, and in the nal snapshot at t = 6.30 (panel b) of Figure 3.21), which corresponds to a proliferation process covering seven cell cycles, their cell populations are very dierent in term of cell mass and cell repartition within the domain. Follicle 1 (top), which is 70% slower in the maturity axis (y) than follicle 2 (bottom), is still evolving towards the asymptotic maturity level and its density is still split into distinct clouds corresponding to cells having entered the dierentiation phase during dierent cell cycles, with a total mass m At the same time, follicle 2 has almost reached its asymptotic state with a nal total mass of only m 2 0 = The dierence in mass can be explained by the fact that being slower in the y direction, follicle 1 spends more time in the proliferation phase than follicle 2.

2 and Ω 3 5 2 for follicle 1 , φ max = 510 Figure 3 .Figure 3 .Figure 3 .

 2351510333 Figure 3.20 Full model : snapshots of the density at dierent times of the follicular development, ∆x = 0.0125. The color code is time related. Blue and red colors indicate respectively null and maximum observed density φ max over the whole domain at the snapshot times. Follicle 1 (top) and follicle 2 (bottom) dier by τ h = 0.7 and 1.0 respectively

( 4 .

 4 14) transport problemIn the numerical simulations, we refer to the MR threshold to indicate the threshold value on the nest MR level ε = ε L-1 . The set of indices of the signicant details at time t n is designed as

( 4 . 6 )

 46 in the quadratic case. transport problem Let us compute the solution at time n + 1 in the neighborhood of the doubling interface (see Figure 4.1)

Figure 4 .

 4 2 shows the solution computed at time 20 on the uniform ner grid, which counts ten meshes per subdomain in the x direction, along with the solution computed with the FVMR algorithm and Haar or quadratic reconstruction. Note that due to the detail amplication (4.17) the MR solutions with both Haar and quadratic reconstructions exhibit more rened meshes upstream than downstream of the support of the solution.

Figure 4 . 2

 42 Figure 4.2 Snapshots of the solution φ(t, x, y) at t = 20 on the whole computational domain and color code from 0 to 10 7 . Panels from top to bottom : a) Zoom on the support of the solution, MR with Haar reconstruction, b) MR with quadratic reconstruction, c) MR with quadratic reconstruction and doubling threshold strategy (4.20) d) ner grid no MR, e) full ner grid no MR. Notice the presence of the spurious noise (rened meshes upstream of and contiguous to the support of the solution in panel a) and b), as well as isolated upstream rened meshes in panel b)), and the absence of noise (in panel c)).

Figure 4 . 3

 43 Figure 4.3 Zoom on the support of the solution at t = 20 and computational grid with color code from 0 to to 10. From top to bottom a) MR with Haar reconstruction, b) MR with quadratic reconstruction, c) MR with quadratic reconstruction and doubling threshold (4.20). The thresholding error is more visible in the Haar case (panel a)) than in the quadratic reconstruction (panels b) and c)).

4. 6

 6 that a small and smooth solution lying on both sides of an interface, could right away generate spurious oscillations. Restricting the initial support ensures that no additional oscillations will be created by coupling the MR with the doubling condition. The bottom panels of Figures 4.2 and 4.3 display the eect of the adaptive doubling strategy in the case of second order reconstruction, setting the parameter δ equal to ε = 10 -2 . The numerical noise has completely disappeared.

(4. 26 )

 26 Relations (4.25) and (4.26) together show that parameter δ should be correctly calibrated in relation with the MR threshold ε, to prevent the MR perturbation in (4.25) from being amplied and becoming the unwanted spurious noise.

Figure 4 . 4

 44 Figure 4.4 Set-up for the p th interface. The characteristic curves in the right part of the domain transport either the initial condition if they cross the x-axis or the trace of the solution on the px s interface, multiplied by the k p coecient, if they cross the vertical x = px s axis

( 4 .

 4 [START_REF] Cohen | Postel Adaptive multiresolution for nite volume solutions of gas dynamics[END_REF], the third order FV scheme and a 4 level MR hierarchy with the quadratic reconstruction. We test by turns, the time adaptation of the MR threshold (4.[START_REF] Chen | Hyperbolic conservation laws with discontinuous uxes and hydrodynamic limit for particle systems[END_REF], and another adaptation of the MR threshold based on the maximum value of the solution ε t, = ε max x,y |φ(t, x, y)|.

( 4 .

 4 [START_REF] Chen | Hyperbolic conservation laws with discontinuous uxes and hydrodynamic limit for particle systems[END_REF]).

Figure 4 .

 4 Figure 4.10 displays the L 1 norm of error at nal time t = 20, between the adaptive solution and the solution computed on the uniform nest grid. The adaptive solution is computed with dierent values of the MR threshold ε, xed once for all times, and for values of the threshold δ varying between ε/100 and 10ε. For two, three and four levels in the MR hierarchy we display the results in two dierent fashions : The standard error visualization is shown in the left panels, where the interesting feature

Figure 4 .

 4 Figure 4.10 displays results computed with the rst order upwind FV scheme (4.7-4.8). The value of δ which minimizes the error lies between 0.2ε and 2ε and increases with both ε and the number of levels. We also made the same test for four levels 100

2 and Ω p+1 1 , and between Ω Nc 2 and Ω 1 1

 11 when the density reaches the right side of the domain due to the periodic boundary condition at this point. On the intermediate internal boundaries between Ω p 1 and Ω p 2 , at values x s = p + 1/2 for p = 1, . . . , N c , the ux condition is continuous. The vertical component of the velocity is positive

Figure 4 .

 4 Figure 4.10 Relation between the MR threshold parameter ε and the threshold δ in the adaptive doubling interface (4.20). L 1 norm of the error with the solution computed on the uniform nest grid at nal time. Simple 1D test case (4.1) with rst order FV scheme. Left panels : error as a function of δ. Right panels : error/ε as a function of δ/ε, for dierent numbers of MR levels (top panels : L = 2, middle panels : L = 3, bottom panels : L = 4).

Figure 4 .

 4 Figure 4.11 Relation between the MR threshold parameter ε and the threshold δ in the adaptive doubling interface (4.20). L 1 norm of the error with the solution computed on the uniform nest grid at nal time. Simple 1D test case (4.1), four MR levels, third order FV scheme. Left panels : error as a function of δ. Right panels : error/ε as a function of δ/ε. Top panels : constant MR threshold ε, bottom panels : adaptive MR threshold ε t .

Figure 4 .

 4 Figure 4.12 Computational domain for the biological model of follicular development. In panel a) the cell cycles Ω p 1 ∪ Ω p 2 , p = 1, . . . , N c of unit width are periodically reproduced. Ω p 1 corresponds to phase G1, while Ω p 2 aggregates phases S, G2 and M of the pth cell cycle. Ω 3 is the dierentiation phase, where the cells do not proliferate anymore. Panel b) is a zoom on the rst cell cycle. The source term is active in the zone between the horizontal dash lines, where cells are subject to apoptosis. The arrows indicate the possible directions of the transport in the cell phases.

Figure 4 .Figure 4 .

 44 Figure 4.13 Snapshots of the density of granulosa cells computed with the FVMR strategy, adaptive doubling (4.20) and MR threshold (4.19), four MR levels, third order scheme, ε = 10 -2 , and biological constants from Table 4.2. For each time, the adaptive grid is shown below the density, displayed with a time-dependent color code. The boundaries between biological cell cycles are marked with white lines.

Figure 4 .

 4 Figure 4.15 L1 norm of relative error (4.35) at dierent times, with respect to the MR threshold ε (left panel), the CPU gain (center panel) and the memory gain (right panel). Eect of adaptive MR threshold ε t and doubling threshold δ in the case of the biological model. The line labeled O(ε) in the left panels indicates the theoretical asymptotic behavior of the error as a function of ε which is a straight line of slope one in logarithmic axes. The complete legend code is described in Table 4.1.

Figure 4 .

 4 Figure 4.16 Relation between the MR threshold parameter ε and the threshold δ in the adaptive doubling interface (4.20). L 1 norm of the error with the solution computed on the uniform nest grid at nal time. Biological model, four MR levels, third order FV scheme. Error as a function of δ (left), error as a function of δ/ε, error/ε as a function of δ/ε, for dierent values of ε. Constant MR threshold ε.

Figure 4 .

 4 Figure 4.17 L1 norm of relative error (4.35) with respect to the threshold parameter ε (left panels), CPU gain (center panels) and memory gain (right panels). Top panels : Localized doubling with uncontrolled aging velocity (4.37), bottom panels : equivalent distributed doubling. The complete legend code is described in Table 4.1.

(4. 40 )Chapitre 4 .

 404 Adaptive mesh renement strategy for a non conservative transport problemwhere τ h , c 1 , c 2 and ū are real positive constants.

(4. 44 )

 44 where b 1 , b 2 and b 3 are real positive constants.

(4. 46 )

 46 A homogeneous Dirichlet condition holds to the north of the interface between Ω p 2 and Ω 3 φ(t, x, y + s ) = 0, (p -1/2)D c ≤ x ≤ pD c .

Figure 4 .

 4 Figure 4.18 Set-up for Appendix 4.6. Three level discretization with doubling interface located between meshes D 0,K-1 and D 0,K on the coarsest level. At time t n the solution is piecewise constant with a discontinuity of size α located between meshes D 1,2K+2 and D 1,2K+3 , resulting in a gradually adaptive grid. 4.6 Appendix : Example of spurious numerical noise appearance 4.7 Numerical noise appearance

Figure 4 .

 4 Figure 4.19 Discrete solution in example 4.6. Top panel : updated solution at time t n+1 with formula (4.48). Bottom panel : solution computed with detail thresholding (4.49) according to Algorithms 1 and 2.

(4. 49 )

 49 meaning that meshes D 1,2K and D 1,2K+1 have to be subdivided on the nest level while subdivisions D 0,K+1 and D 0,K+2 have to be aggregated again. The resulting reconstructed solution, obtained after applying Algorithms 1 and 2, is sketched out in bottom panel of Figure4.19. This is of course a theoretical situation but it shows nevertheless how some spurious structure requiring ne meshes may appear in an upstream region and be well detached from the support of the genuine solution. transport problem Require: The solution is known by its mean values on the coarsest grid, and the details for (k, ) ∈ Γ n+1 ε ;

µ 3 c 0 1 - 3 = 5 .

 135 exp(-c 0 ) exp(-c 0 a) pour 0 < a < 1 et 0.20 < γ < 0.25, 0 sinon. On donne un exemple d'une telle fonction sur la Figure 5.1 avec µ 1 = 8, µ 2 = 7 et µ Ce type de condition initiale correspond à une population entièrement proliférante, avec une distribution asynchrone des cellules dans le cycle cellulaire et une répartition constante par morceaux en maturité. Comme détaillé par exemple dans [78] dans le cas d'un modèle compartimental ODE, il y a dans ce cas exactement deux fois plus de cellules qui viennent de naître que de cellules qui vont se diviser γ A

fonction de biodisponibilité locale. La Figure 5 . 6

 56 donne un exemple d'une telle fonction. Qualitativement, le comportement est exponentiel (par construction) jusqu'à atteinte d'une valeur limite, et la fonction devient alors constante. Propriété de la fonction b f . Si b f < 1 alors db f dm (m) = b 2 b 3 exp(b 2 m), donc les paramètres b 2 , b 3 règlent la dérivée de la fonction b f . (voir Figure 5.6, où l'on voit l'impact de diérents choix de b 2 ).

Figure 5 . 6

 56 Figure 5.6 Exemple de fonction déterminant la valeur des contrôles mésoscopique.En abscisse, la maturité folliculaire, en ordonnée, la valeur de b f , fonction de biodisponibilité locale. On note qu'elle comporte deux parties, une partie exponentielle au départ, puis à l'atteinte de la valeur 1, cette fonction devient constante. Autrement dit, la valeur du contrôle mésoscopique ne peut excéder celle de la valeur du contrôle macroscopique.

Chapitre 5. Calibration 5 . 2

 52 Spécications biologiquesL'objectif de cette section est de donner des critères de validation d'une simulation basés sur les connaissances biologiques. Les principaux problèmes que l'on rencontre sont d'une part la rareté des données cinétiques et quantitatives dont nous disposons à l'heure actuelle, d'autre part le fait que ces données sont disponibles principalement aux niveaux macroscopique (échelle ovarienne) et mésoscopiques (échelle folliculaire), que l'on utilise pour calibrer des paramètres au niveau microscopique. Ce dernier point pose aussi le problème de l'unicité des valeurs de paramètres conduisant à des sorties macroscopiques identiques.Nous allons dans un premier temps présenter les diérentes mesures dont nous disposons et la méthode expérimentale associée (sans entrer dans les détails). Puis nous dénirons les concepts de trajectoire ovulatoire et de trajectoire atrétique.Nous nous intéressons ici au cas de la brebis, qui est le modèle étudié par les collègues biologistes de l'équipe BINGO (INRA Tours) mais les méthodes seraient transposables à d'autres espèces si l'on pouvait disposer du même ensemble d'observations.

3 .

 3 Il existe des mutations naturelles aectant le nombre d'ovulations. 5.2.1 Mesures disponibles Diamètre folliculaire. La technique d'échographie ovarienne (ultrasonographique) permet d'obtenir des images 2D (ou 3D, suivant le type d'équipement), à partir desquelles on peut mesurer le diamètre de l'antrum (cavité à l'intérieur du follicule, la seule structure visible à l'échographie). On en déduit le volume du follicule en supposant que ce dernier est sphérique. Cette technique ne détecte que les follicules au-dessus d'une certaine taille (≈ 1mm) et n'est able en terme de suivi individuel qu'au-dessus d'une taille encore plus élevée (≈ 2mm). Dans l'article [66], par exemple, les auteurs présentent des courbes de volume de follicule (diamètre en mm) au cours du temps (en jours) chez la brebis. Eectif cellulaire. Les mesures d'eectif cellulaire ne sont pas directement disponibles en fonction du temps : elles sont données en fonction du diamètre. La mise en correspondance de données de cinétique cellulaire (invasives), telles que : eectifs cellulaires, index mitotique, temps de doublement, et de mesures directes (échographie) ou indirectes (histologie) du diamètre en fonction du temps permet l'élaboration d'une relation entre eectif cellulaire et temps (voir [26]). Nous n'abor-derons pas cette question en temps que telle ici, et notre objectif est plutôt de déterminer des valeurs de paramètres compatibles avec les contraintes biologiques. Fraction de croissance. Mathématiquement, la fraction de croissance est une fonction du temps, dénie comme le quotient de la masse cellulaire en prolifération (c'est-à-dire des cellules en phases G1 et SM) par la masse cellulaire totale. Cette notion est mésoscopique, au sens où chaque follicule possède sa propre fraction de croissance.

  les follicules ont plus de chance d'ovuler et donc à la fois d'atteindre un eectif cellulaire compatible avec l'ovulation (notion d'expression du capital prolifératif ) et d'échapper à l'atrésie. L'exemple typique du non respect de ce principe serait celui d'un follicule dont la maturité asymptotique est dans la zone du domaine où l'apoptose est active pour toute valeur du contrôle. Un tel follicule serait atrétique par nature, donc la sélection serait biaisée. L'atrésie est avant tout un phénomène induit par des conditions d'environnement hormonal défavorables. Par ailleurs, en conditions d'environnement favorables, la plupart des follicules expriment leur potentiel de prolifération et peuvent atteindre un eectif cellulaire compatible avec l'ovulation.Courbes de FSH. L'évolution du taux plasmatique de FSH au cours du cycleovarien est un examen assez fréquent, eectué au moyen de prélèvement sanguins dans le cadre du suivi de cycles ovariens. Sur les mesures expérimentales (voir par exemple dans la thèse[START_REF] Mansanet | Contrôle génétique et physiologique de la prolicité en race ovine lacaune : caractérisation de la mutation causale et rôle fonctionnel du gène FecL[END_REF]) on note que pendant la phase dite folliculaire, on a une chute du taux de FSH d'environ la moitié de la valeur maximale. Cette variation de la quantité de FSH sécrétée par l'hypophyse est régulée par la sécrétion d'hormones (inhibine et ÷stradiol) ovariennes, en quantité croissante jusqu'à l'ovulation, et qui peut être reliée directement à la maturité ovarienne. Ce rétro-contrôle est modélisé par la relation entre la quantité de FSH et la maturité ovarienne U = S(M ).

5 M

 5 où rappelons-le, les indices min et max correspondent respectivement au contrôle le plus défavorable et le plus favorable. Ces inclusions assurent qu'en contrôle favorable, la maturité asymptotique permet une stabilisation de la masse cellulaire (sortie du cycle cellulaire en temps ni), tandis qu'en contrôle défavorable cette maturité est dans la zone où la perte cellulaire est active, entraînant une chute de masse cellulaire. Les données expérimentales sur le taux de FSH permettent d'imposer une relation sur les bornes de la sigmoïde dénissant le contrôle global U min = 1 2 U max L'observation simultanée des taux de FSH plasmatique et des taux d'÷stradiol et d'inhibine permet d'introduire des contraintes a priori reliant la dynamique de la FSH au seuil d'ovulation 1 o N f ≤ M ≤ M o M f D'autres bornes a priori sont imposées sur les autres paramètres du modèle, en se basant sur les simulations numériques antérieures, et gagneraient à être anées.

Table 5 . 2

 52 Résumé des contraintes a priori sur les paramètres. Nous proposons alors un code couleur adapté à la visualisation des densités dans le cadre de la dynamique de la population de follicules. Cette échelle de couleur est indépendante du follicule considéré, et même indépendante du temps. Code couleur adapté à la dynamique de population. Nous utilisons pour cela le cercle chromatique (voir Figure 5.9). Il contient toutes les couleurs que l'on peut obtenir à partir de couleurs de base en visualisation informatique : les couleurs du système RGB (pour Red, Green, Blue). Nous restons dans l'esprit d'une échelle de température. Nous utilisons ainsi, dans l'ordre : blanc, bleu, cyan, vert, jaune, rouge, magenta, pour obtenir l'échelle de couleur représentée sur le panel droite de la Figure 5.9. L'idée est d'utiliser une relation non linéaire pour passer d'une couleur à l'autre, contrairement à la plupart des codes couleur classiques, qui utilisent une relation linéaire. Dans notre problème, la densité est amenée à traverser diérentes interfaces.

Figure 5 .

 5 Figure 5.10 Visualisation de deux follicules, l'un ovulatoire (en bas, avec un accroissement cellulaire d'environ 20) et l'autre atrétique (en haut), avec un code couleur non linéaire adapté à la dynamique de la population de follicules. Dans ce cas N c = 4, et la densité du follicule ovulatoire (en bas) a déjà parcouru entièrement une fois le domaine (rappelons que l'on a des conditions aux limites périodiques).

Figure 5 .

 5 Figure5.12 Exemple de panel de diagnostic de simulation numérique. De gauche à droite, de haut en bas : Contrôle macroscopique, maturités mésoscopiques, pertes de masses mésoscopiques instantanées, contrôles mésoscopiques, fractions de croissances mésoscopiques, perte mésoscopiques cumulées de masse, ζ f mésoscopiques,

  et nit par la division cellulaire sur l'interface entre la zone SM du cycle en cours et la zone G1 du cycle suivant. Pour cela nous introduisons en plus du domaine de calcul Ω = {(a, γ), 0 ≤ a ≤ N c , 0 ≤ γ ≤ 1}, où N c est le nombre de cycles cellulaires, la zone de mitose Ω M it = {(a, γ), p -T m ≤ a < p, for p = 1, . . . N c , 0 ≤ γ < γ s } Ceci nous permet de dénir précisément l'index mitotique numérique

Sur la Figure 5 . 13 ,Figure 5 .

 5135 Figure 5.13 Courbes d'index mitotique correspondantes aux deux conditions initiales envisagées dans le modèle (voir paragraphe 5.1.1), dans le cas entièrement proliférant (h f = 0) et sans discontinuité de vitesse entre la zone G1 et la zone SM (g 1 = 0, g 2 = 1).

5. 18 ,

 18 on trace les sorties mésoscopiques et macroscopiques pour diérentes valeurs du paramètre de travail C p dans la plage de paramètres : 0.56, 0.57, 0.58, 0.59, 0.6, 0.61, 0.62, 0.63, les autres paramètres étant les paramètres par défaut, pour une cohorte de 8 follicules. Le paramètre C p semble contrôler la masse nale de la solution à contrôle constant. La plage de valeurs dans laquelle on fait varier le paramètre de travail C p a été choisie de manière à obtenir un accroissement de masse compris entre 20 et 30. Paramètre de dynamique à contrôle constant p. Sur la Figure 5.19, on trace les solutions pour diérentes valeurs du paramètre de travail p 0.75, 1, 1.25, 1.5, 1.75, 2, 2.25, 2.5, 2.75, 3, à paramètre de travail C p = 0.6 xé. On remarque que les courbes de masse et de maturité n'ont pas suivi la même trajectoire au cours du temps mais aboutissent à la même valeur nale. Le paramètre de travail p permet ainsi de régler la cinétique d'accroissement cellulaire.5.4.3 Exemple de situation de référence à 1 folliculeNous concluons ce paragraphe en combinant les deux méthodes précédentes (détermination de la maturité asymptotique, détermination des paramètres aectant la dynamique) pour dénir un exemple de situation de référence. La Figure5.20 présente le panel de diagnostic associé à cette situation. On note que le follicule satisfait tous les critères d'une trajectoire ovulatoire.La valeur du contrôle global chute de moitié entre le début et la n de la simulation.

L

  'accroissement d'eectif cellulaire est compris entre 20 et 30 fois l'eectif initial.

Figure 5 .

 5 Figure5.22 Abaque de fonction h f à deux follicules. En abscisse, la maturité cellulaire, en ordonnée, la vitesse de maturation pour diérentes valeurs du contrôle. La fonction b f est identique dans les deux cas. Les valeurs extrêmes du contrôle sont ūmin = 0.0060517 et U min = 0.075. Les courbes rouges correspondent à c 1 = 0.07, c 2 = 0.9 et ū = 0.03. Les courbes vertes correspondent à c 1 = 0.62, c 2 = 0.168 et ū = 0.06. Pour ce jeu de paramètres les valeurs de contrôle u f telles que ūmin ≤ u f ≤ u a = 0.0188384 correspondent à une racine r + (u f ) < γ s donc à une situation d'équilibre dans la zone de prolifération. On a matérialisé en noir la courbe correspondant au contrôle u a .

Figure 5 .= γ s γ s = 0. 3 Figure 5 .

 535 Figure5.26 Abaque de chevauchement à 10 follicules. Graphes de la fonction de maturation h f pour f = 1, . . . , 10. Les zéros de h f sont les mêmes pour tous les follicules, mais les amplitudes de la fonction sont diérentes (en eet, seul le paramètre τ h varie). Sans pression sélective, cela n'a pas d'eet sur la trajectoire folliculaire, mais en introduisant de la sélection, on observe que seul le follicule avec une grande vitesse de maturation devient ovulatoire, les autres devenant atrétiques.

(6. 2 )

 2 Control. The control functions are non-local functions in the sense that they are dened with respect to the moments of the densities. The global control function uses information from the sum of the rst order moments only. Each sub-population is controlled at the mesoscopic level by a closed loop control, where the global control also plays a part.6.1.1 Example : selection process of ovarian folliclesAn example of such models is the selection process of ovarian follicles introduced in[START_REF] Echenim | Sorine Multiscale modeling of follicular ovulation as a reachability problem[END_REF] and[START_REF] Echenim | Multi-scale modeling of the follicle selection process in the ovary[END_REF], which we briey describe in this paragraph. The unknowns in this case are the granulosa cell densities of each follicle, denoted φ f (x, y, t), with f ∈ {1, ..., N f }. N f has a value around 20. The computational domain is Ω = [0, N c ] × [0, 1], with N c the number of cell cycles. The space variables x and y are functional variables denoting respectively the age of the cells within the cell cycle and their maturity. An horizontal boundary at y = γ s splits the domain into a proliferating part (bottom), and a dierentiated part (top). The corresponding subdomains are dened as follows :

  Data: Inner cell, level + 1 Result: Additional cells For cells in stencil prediction of mother cell If(level < ) Divide to create border cells (Algorithm 12) Algorithm 11: Complete stencil of prediction : if needed, subdivision of the mother cells of the neighbors of the new divided cells, in order to complete their prediction stencils. Data: Border cell, level Result: Additional cells Create cells of stencil of prediction if needed (possible recursivity) Initialize 4 daughters cells values Algorithm 12: Divide to create border cell

2 Figure 6 . 8

 268 Figure 6.8 Parallel computing strategy. There are 2 options for data management.Option 1 : domain decomposition method. We cut the domain into several parts and send them to dierent processes. Option 2 : split the sub-populations and send them to dierent processes. There are less communications using option 2, and the communications that have to be done are ecient (reduction operations) and mesh size independent.

Figure 6 .

 6 Figure[START_REF] Aymard | Numerical simulation of the selection process of the ovarian follicles[END_REF].14 Mean weak scalability test. Zoom of the panels of Figure6.13. The blue dots correspond to the numerical simulations, and the red curve corresponds to the linear regression. The variability tends to decrease with respect to the number of processes. It is interesting to note that there is a gap between the values for a number of process smaller than 8 and the others, due to the technology that we used (the rst eight nodes are gathered in a small and fast cluster).

Figure 6 .

 6 Figure[START_REF] Aymard | Numerical simulation of the selection process of the ovarian follicles[END_REF].[START_REF] Bürger | Conservation laws with discontinuous ux : a short introduction[END_REF] Mean weak scalability test. In these cases, we identify the origin of the variability observed in the general case of an adaptive mesh, an adaptive time step, and random parameters (Figures 6.13 and 6.14). Top left panel : reference situation with uniform mesh, constant time step, constant parameters (classical weak scalability test). Top right panel : uniform mesh, constant time step, random parameters. Bottom left panel : uniform mesh, adaptive time step, random parameters. Bottom right panel : adaptive mesh, constant time step, random parameters.

Figure 6 . 7 Conclusion

 67 Figure[START_REF] Aymard | Numerical simulation of the selection process of the ovarian follicles[END_REF].[START_REF] Bürger | Risebro A relaxation scheme for continuous sedimentation in ideal clarier-thickener units[END_REF] Mean weak scalability test. Zooms on the panels of Figure6.15. The bottom left panel corresponding to adaptive time step, uniform mesh and random coecients exhibits the highest variability. We clearly see the gaps between one and three processors, and between seven and nine processors. The rst gap corresponds to the communications that have to be done, and the second to a change of technology in the network that is used for communications (processors are gathered in nodes of eight).

Figure A. 1 pFigure A. 3

 13 Figure A.1 Achage du terminal à l'issue d'une simulation avec CodeFollicleMR.

Figure A. 4 1

 41 Figure A.4 Exemple d'interface graphique, réalisée en collaboration avec Serge Steer (Inria Paris-Rocquencourt), dédiée à la simulation numérique du processus de sélection des follicules ovariens.

209

  Figure B.1 Exemple d'achage de résultats macroscopiques, avec le logiciel dualMR, à l'issue d'un calcul eectué avec CodeFollicleMR.

Figure C. 3 Figure C. 4

 34 Figure C.3 Convention utilisée pour le stockage des cellules du stencil volume ni.

Figure C. 7

 7 Figure C.7Table de haschage. Cette structure de données permet un accès rapide aux cellules, via leur clés. Elle est utilisée pour la création et la suppression de mailles (algorithme de prédiction, algorithme d'initialisation).

  élément. MR.cpp Ce chier contient les fonctions relatives à la multirésolution. Initialisation par le niveau n et le niveau grossier, codage, décodage, seuillage, reconstruction du niveau le plus n (pour les calculs d'erreur).

FVS

  .cpp Ce chier contient les fonctions relatives à la méthode des volumes nis. Tout d'abord les fonctions applicables à la méthode classique en maillage uniforme : calcul de ux horizontaux et verticaux (prend en compte les conditions de transmission), calcul de ux sur tout le maillage, méthode de Runge Kutta pour mettre à jour les mailles, calcul du pas de temps. Ensuite, les fonctions utiles à l'adaptation de maillage : prédiction du maillage, reconstruction des fantômes à l'ordre élevé. follicleMR.cpp Ce chier contient le programme principal codant l'algorithme général. input.cpp Ce chier contient la fonction de lecture des paramètres compatibles avec le parser. output.cpp Ce chier contient les fonctions de sorties du code : création de chier de sorties mésoscopiques et macroscopiques, création de chier de sorties microscopiques (format OpenGL ou Paraview), sorties dans le terminal. biologie.cpp Ce chier contient les fonctions liées à la biologie : diérentes conditions initiales, vitesses de vieillissement et de maturation. 218 Annexe C. CodeFollicleMR : Notice programmeur errorMR.cpp Ce chier contient une fonction permettant d'évaluer l'erreur entre deux solutions. Elle est principalement utilisée via les scripts du Benchmark pour faire des courbes de convergence. dualMR.cpp Ce chier contient le code source du logiciel de visualisation. C.3 Entrées/sorties Paramètres en entrée Le passage de paramètre dans le code de calcul se fait via un parser (ce parser a été développé par Marc Schoenauer, chef de l'équipe-projet TAO à l'Inria Saclay). Tous les paramètres ont une valeur par défaut dénie dans input.cpp. On peut passer un chier de paramètres à l'exécution. Note importante, il n'y a qu'un seul chier de paramètres en entrée, contenant les paramètres de tous les processeurs, et non pas un chier de paramètres par processeur. Chacun des noeuds de calcul lit une copie de ce chier de paramètres et va récupérer les paramètres qui le concerne. Cela se fait en récupérant le numéro du noeud de calcul, et en créant dynamiquement les noms de paramètres sous le format FpParametre F (pour follicle) précède tous les paramètres en parallèle, p est le numéro du noeud de calcul, et Paramètre est le nom du paramètre (par exemple F0g2 désigne le paramètre g2 du noeud de calcul 0). Un ensemble de jeux de paramètres est disponible dans le dossier Input. Output macroscopiques et mésoscopiques A chaque itération, plusieurs valeurs macroscopiques et mésoscopiques sont mises à jour. On sauvegarde leur valeur dans un chier au format macrop.dat Comme pour les chiers de paramètres, les noms de chier sortie sont créés dynamiquement en fonction du numéro p de processeur (par exemple macro2.dat contient les sorties macroscopiques et mésoscopiques relatives au follicule 2). Output microscopiques Deux types de sorties microscopiques peuvent être faites. Dans les deux cas, ces dernières sont faites au format binaire, pour limiter les erreurs de conversion au format décimal. Des sorties à intervalle régulier, espacées de plotStep, permettent de réaliser des animations. Ces sorties peuvent se faire en deux format : OpenGL pour être utilisés avec le code de visualisation dualMR, ou le logiciel open source Paraview. On peut aussi faire une unique sortie à la n de la simulation, pour faire des calculs d'erreur avec errorMR par exemple. Pour cela la solution, dénie sur le maillage hybride, est calculée sur un maillage uniforme, par complétion (on ajoute des détails à zéro et on reconstruit sur le niveau n par interpolation).

  Historique et Motivations . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Modèle multiéchelles . . . . . . . . . . . . . . . . . . . . . . . . . . .

	Table des matières Nous présentons dans ce manuscrit les résultats obtenus au cours d'une thèse de doc-1 Introduction 1.1 1.2.1 Introduction torat en Mathématiques Appliquées, eectuée sous la direction de Frédérique Clé-
	ment (Mycenae, Inria Paris-Rocquencourt) et de Marie Postel (Laboratoire Jacques-
	Louis Lions, UPMC -Paris 06).
	Nous avons développé une stratégie permettant de réaliser des simulations numé-

riques ables et performantes d'un modèle multi-échelle représentant la dynamique cellulaire au cours du processus de sélection des follicules ovariens. Ce modèle peut être considéré comme un systèmes d'équations de transport non conservatives et faiblement couplées, et il présente plusieurs spécicités qui soulèvent des problèmes originaux et diciles du point de vue numérique, qui ont nécessité la mise en ÷uvre de méthodes numériques nouvelles, dont la présentation est l'objet principal de ce manuscrit.

  Mesoscopic scale : nite volume scheme . . . . . . . . . . . . 33 2.2.4 Macro/Meso scale Parallelization . . . . . . . . . . . . . . . . 35 2.2.5 Parallel algorithm and improvement to order 2 in time . . . . 36 2.3 Numerical simulations . . . . . . . . . . . . . . . . . . . . . . 36 2.3.1 Initial condition . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.3.2 Sets of default parameters . . . . . . . . . . . . . . . . . . . . 37 2.3.3 Time evolution of the cell density for one follicle . . . . . . . 38 2.3.4 Competition between ten follicles . . . . . . . . . . . . . . . . 39 2.3.5 Convergence rate test . . . . . . . . . . . . . . . . . . . . . . 40 2.3.6 HPC test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

		Chapitre 2
	2000 Numerical simulation of the 4000 q numerical simulations sequential theoretical limit selection process of the ovarian 3000 user time (s) follicles
	1000
	Ce chapitre a fait l'objet d'une publication dans ESAIM Proceedings [6] lors du
	0
	CEMRACS 2011.
	30	40	50	60
	number of processes

Figure

1.20

Test de scalabilité faible moyenne (résultats visualisés avec le logiciel R). On eectue N = 20 réalisations à partir de paramètres perturbés, et on récupère le taux moyen d'occupation du maillage adaptatif pour un follicule (à gauche), et le temps de calcul moyen en fonction du nombre de processeurs (à droite). La courbe verte (th.lim.) représente la limite théorique d'optimalité, la courbe bleue (seq.) représente le temps de calcul séquentiel théorique, et les points noirs sont des résultats des simulation. Le taux d'occupation mémoire, par rapport à un maillage uniforme sur la grille la plus ne présente dans le maillage adaptatif, est de 21% ± 1%. L'équilibrage des charges reste raisonnable et le test de scalabilité faible moyenne donne de bons résultats. Sommaire 2.1 Biological and biomathematical background . . . . . . . . . 27 2.1.1 Biological background . . . . . . . . . . . . . . . . . . . . . . 27 2.1.2 Biomathematical model . . . . . . . . . . . . . . . . . . . . . 29 2.2 Numerical Method . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.2.1 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.2.2 Macroscopic scale : piecewise constant approximation of the hormonal control . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.2.3

  2.1.2.3 Closure equationsThe equations in the PDE system(4.38) are linked together through the argument u f (t) appearing in the speeds g(a, γ, u f ) and h(a, γ, u f ) and the argument U (t) in the source term λ(a, γ, U ). U (t) and u f (t) represent respectively the plasma FSH level and the locally bioavailable FSH level and depend on some maturation moments of the densities.

follicular cell mass (local, one by follicle)

Table 2 .

 2 The parameters values are gathered in Table1and 2, except for g 2 = 1.2 and τ h = 1.2. Also, in this mono-ovulatory situation we set the follicular and ovarian maturity thresholds equal to M o = M f = 16.5. The spatial discretization for this test uses N m = 30 cells per half granulosa cell cycle, and the domain Ω contains eight cycles. The grid size is therefore 8 × (2 × 30) 2 = 28800.

	1, for the global parameters, that are identical for all follicles (space discretization,
	CFL condition,. . . ), another, in Table 2.2, for the local parameters (initial condition,
	velocity parameters,. . . ), which can depend on the follicle.
	Parameter	Description	Value
	N m	number of grid cells by half granulosa cell cycle	30
	N c	number of cycles	8
	CFL	CFL condition	0.4
	M f	follicular maturity threshold	0.7
	M o	ovarian maturity threshold	5.
		FSH plasma level (eq. (5.1))
	U min	minimum level	0.075
	c	slope parameter	2.0
	M	abscissa of the inection point	4.5
		Apoptosis source term (eq. (4.41))
	K	intensity factor	6.0
	γ	scaling factor	0.02
	t max	maximum time (to avoid excessively long computations)	7
	Table 2.1 Values of the global parameters used in Figures 2.4 to 2.9
	The number of follicles, N f , and the number of grid cells by half granulosa cell
	cycle, N m , may depend on the simulation.

  ovarian follicles Figure 2.7 Convergence rate in the SM phase. Green curve : ∆γ → O(∆γ 2 ). Blue curve : L1 norm error between the exact solution and the solution with space step

	Chapitre 2. Numerical simulation of the selection process of the ovarian follicles Chapitre 2. Numerical simulation of the selection process of the ovarian follicles
																Chapitre 3
	0.9 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 A numerical method for transport U(t) 0.073 0.074 0.075 0.077 0.078 0.079 0.08 0.082 0.083 0.01 0.1 0(∆γ 2 ) E equations with discontinuous ux 0.081 u f (t) g 2 g 2 g 2 g 2 g 2 g 2 g 2 g 2 g 2 g 2 g 2 g 2 g 2 g 2 g 2 g 2 g 2 g 2 g 2 0.50 0.55 0.60 0.65 0.70 0.001 E( functions : application to ∆γ ) mathematical modeling of cell 0.076 0.75 0.80 0.85 0.90 0.95 0.0001 dynamics
			0	0.2	0.4	0.6	0.8	1	1.2	1.4	1.6	0	0.2	0.4	0.6	0.8	1	1.2	1.4	1.6
	50		t a) Plasma FSH level 1e-05 0.0001	0.001	t b) Locally bioavailable FSH level 0.01 0.1 700
	40						3		0.50 g 2		∆γ 600
	1.5 2 2.5 Ce chapitre a fait l'objet d'une publication [7] dans la revue SIAM Journal of Scien-m 0 (f,t) 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 ∆γ. 20 30 Real computing time 200 300 400 500 tic Computing. Sommaire User computing time 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
	10		1 3.2 Toy models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 100
	0	0	0.5 10 3.3 Numerical scheme . . . . . . . . . . . . . . . . . . . . . . . . . 54 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 2 4 6 8 12 14 16 0 0 2 4 6 8 10 12 14 3.3.1 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . 54	16
				t 3.3.2 Treatment of discontinuous coecients . . . . . . . . . . . . . 56 # follicles # follicles
	c) Cell mass of each follicle 0.4 0.5 0.6 0.7 0.8 0.9 1 m(f,t) 0.50 g 2 g 2 g 2 g 2 g 2 g 2 g 2 g 2 g 2 g 2 g 2 g 2 g 2 g 2 g 2 0.55 0.60 0.65 0.75 0.80 0.85 0.90 0.95 M f computation. Right : cumulative computed time on all processors 3 3.5 4 4.5 5 5.5 6 M(t) E( 0.001 ∆γ ) 0.70 0.01 0.1 a) Real computing time c) User computing time 3.4 Numerical validation . . . . . . . . . . . . . . . . . . . . . . . 58 0(∆γ 1.95 ) E 3.4.1 1D test cases . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 Figure 2.9 Parallelization performance test. Simulation of an increasing num-3.4.2 2D test cases . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 ber of identical follicles. One follicle per processor. Green curve : theoretical limit. Blue curve : computing time (in seconds). Left : elapsed time from the start of the 3.5 Application to cell dynamics . . . . . . . . . . . . . . . . . . . 69
			2.5								0.3
			1.5 2		0.0001			M o M(t)		0.1 0.2
			0	0.2	0.4	0.6	0.8	1	1.2	1.4	1.6	0	0.2	0.4	0.6	0.8	1	1.2	1.4	1.6
							t									t
					1e-05
			d) Ovarian maturity 0.0001		0.001	e) Maturity of each follicle 0.01 0.1
											∆γ
	Figure 2.6 Competition between ten follicles diering by their ageing function
	parameter g 2 Figure 2.8 CV rate on the G1 phase. Green curve : ∆γ → O(∆γ 1.95 ). Blue curve : L1 norm error between the reference solution and the solution with a space step ∆γ.

3.5.1 Numerical simulation . . . . . . . . . . . . . . . . . . . . . . 74 3.6 Appendix : Exact solutions . . . . . . . . . . . . . . . . . . . 75 3.6.1 Exact solutions of the 1D problem with piecewise constant speeds 75 3.6.2 Exact solution of the 2D problems with piecewise constant speeds 77

  Chapitre 4. Adaptive mesh renement strategy for a non conservative transport problem4.3.2 Time adaptation of the MR threshold ε tTo study the inuence of the time adaptation of the MR threshold ε t (4.19), we perform dierent simulations of simple test case (4.1) with smooth initial condition

		ε = 10 -3
	t = 7.48		t = 7.48
			7.48	t = 8
	t = 8		t = 8
	t = 8	φ(t, x, y) t = 7.48	t = 7.48
		t = 8

  The equations in the PDE system(4.38) are linked together through the argument u(t) appearing in the speeds g(x, y, u) and h(x, y, u) and the argument U (t) appearing in the source term Λ(x, y, U ). U (t) and u(t) represent respectively the plasma FSH level and the locally bioavailable FSH level and depend on the rst maturity

	moment of the density
	1	NcDc
	M (t) =	yφ(x, y, t)dxdy.
	0	0

(4.42) The plasma FSH level U (t) showing up in the arguments of the source term in (4.38) is dened by

U (t) = U min + U max -U min 1 + exp(c(M (t) -M )) ,

(4.43)

where U min , c and M are real positive constants. The locally bioavailable FSH level u(t) showing up in the arguments of the speeds in (4.38) is dened by

  , D ,2k+1 } ∈ S n+1 and D -1,k / ∈ S n+1

	compute φ n ,2k = φn ,2k + d n ,k
	and φ n ,2k+1 = φn ,2k+1 -d n ,k end if end for end for
	Algorithm 2: Construction of the solution on the adaptive grid S n+1 from
	the set of signicant details Γ n+1 ε
	Unconditional doubling (4.2)	Adaptive doubling (4.20)
	Constant MR threshold

Table 4 .

 4

	1 Legend codes for Figures 4.9 and 4.15

Table 4 .

 4 2 Values of the parameters for the biological model simulations

	Chapitre 5. Calibration
	de vitesse), qui permet dans une certaine mesure, en la couplant aux connaissances
	biologiques, d'énoncer des contraintes sur les paramètres ou des combinaisons de pa-
	ramètres, et l'utilisation de méthodes de tir et de visualisation adaptées. On extrait
	un maximum d'information mathématique de ces dernières.
	Dans une première section, on présente les diérents degrés de liberté faisant inter-
	venir les paramètres du modèle. Ensuite, on présente les spécications biologiques
	du modèle. L'objectif est de caractériser des situations, qui permettent de valider
	ou d'inrmer une simulation numérique. Pour cela, on rappelle les principales me-
	sures biologiques actuellement à notre disposition, puis on dénit des trajectoires
	folliculaires.
	Dans la troisième partie nous combinons les résultats des précédentes sections pour
	dénir des contraintes : contraintes a priori sur les paramètres, contraintes a poste-
	riori sur les sorties du modèle.

Dans une quatrième partie, nous présentons une méthode permettant de calibrer le modèle dans le cas d'un seul follicule, étape préliminaire à la calibration d'une cohorte.

La dernière partie est consacrée à la calibration d'une cohorte, et repose sur les résultats obtenus en troisième partie. Nous montrons comment régler la pression sélective, comment introduire (à bon escient) de l'hétérogénéité, tout en s'assurant de la non prédestination des follicules.

5.1 Degrés de liberté

5.1.1 Conditions initiales

Les conditions initiales xent les valeurs prises par les densités de cellules au sein de chaque follicule au temps 0 de la simulation. Ce sont des fonctions localement constantes par morceaux, de la forme :

  Cette grandeur reste constante dans le cas de populations entièrement proliférantes et désynchronisées, alors qu'elle présente un prol complexe dans le cas de populations subissant d'autres processus cellulaires comme la diérenciation et/ou l'apoptose (ce qui est le cas des cellules de granulosa). La durée T m du phénomène de mitose peut être considérée comme relativement stable et d'environ 30 minutes, à rapporter à une durée de cycle de prolifération qui peut varier entre 12 et 18 heures. Dans notre système de variable normalisée, où la durée du cycle est de

	Chapitre 5. Calibration
	Courbe d'index mitotique. L'index mitotique est une grandeur cinétique dénie
	au temps t comme le rapport entre le nombre de cellules en train de se diviser et
	l'eectif total
	I f m (t) =	mitotic cells total number of cells
	La littérature fait état de mesures expérimentales d'index mitotique (voir par exemple
	[75] et [76]). 1, nous aurons donc
	1 36	≤ T m ≤	1 24	.
	cette densité.
			143

-2

. Autrement dit, si la densité est une partie compacte du plan, cette quantité représente l'ordonnée minimale parmi tous les points composants

  .14. On constate que le taux de perte instantanée (sur le panel de gauche) est nul avant l'arrivée dans la zone où le terme source est actif, puis non nul pendant l'intervalle de temps où la densité traverse la zone d'apoptose (γ -5.14 Perte instantanée et perte cumulée. Résultat d'une simulation numérique réalisée avec les paramètres par défaut (voir Tableau 5.3) avec un contrôle en boucle fermée. Panel de gauche : Exemple de fonction de perte instantanée. En abscisse, le temps, en ordonnée, la perte cellulaire instantanée. Cette fonction est toujours positive ou nulle. Panel de droite : Exemple de fonction de perte cumulée, associée à la courbe présentée dans le panel de gauche.est entièrement dans la zone γ > γ + s . La perte cumulée, représentée sur le panel de droite, est la primitive de la perte instantanée, le niveau atteint à la n de la simulation peut être comparé au seuil approximatif de 10% que l'on s'est xé comme limite maximale pour un follicule ovulatoire.

s ≤ γ ≤ γ + s ), et redevient nulle quand la densité

Figure

Table 5 .

 5 Réglage de la pression sélectiveRégler la pression sélective consiste à régler les paramètres de la fonction déterminant le contrôle global. Pour cela, on fait varier les paramètres c et M . Tout d'abord, le paramètre M doit être réglé de façon à ce que la chute de FSH ait bien lieu lorsque la maturité globale est susante, ni trop tôt, ni trop tard. Ensuite, le paramètre c doit être choisi de façon à ce que la pente du prol de chute de FSH ne soit pas trop abrupte en temps. Enn, on peut régler le paramètre K déterminant l'intensité de l'apoptose, et ainsi respecter la perte maximale de 10% pour les follicules ovulatoires. Sur la Figure 5.21, on reprend les paramètres utilisés pour la Figure 5.19, mais cette fois-ci on introduit le contrôle en boucle fermée. On règle empiriquement les valeurs de c, M et K, dans cet ordre. La chute de FSH a pour eet de créer un phénomène de sélection. Sans sélection (en boucle ouverte), tous les follicules atteignaient la même masse nale (voir Figure 5.19). A présent, les masses nales s'étalent, et on

	3 Paramètres par défaut (utilisés pour les simulations représentées dans
	la gures 20, et les suivantes).
	5.5.1

  1) to 1 do for cells in level do

	controlled population dynamics
	Figure
	If (not ghost)
	Compute uxes (6.15)
	Update load balances (6.16)
	end for cells in level do
	If (ghost)
	Delete three ghost cells with Projection (6.9)
	end
	end for cells in level 0 do
	Compute uxes
	end
	Algorithm 7: Flux computation

  At each intermediate step, instead of deleting ghost cells after the ux computation, we keep them in the hybrid tree. Then after updating the solution, we compute the new values of the ghost cells, and the hybrid grid is ready for a new intermediate step.We free the memory corresponding to the ghost cells only at the nal intermediate step, when the cells are no longer necessary.

	controlled population dynamics
	and for the last step,
	φ n+1, i,j	= φ n, i,j + λ (	1 6	D n, i,j +	1 6	D * , i,j +	2 3	D * * , i,j ).	(6.19)
	with D * , i,j the ux balance (including the source term) computed from the interme-
	diate solution φ * , i,j , and D * * , i,j the ux balance (including the source term) computed
	from the intermediate solution φ * * , i,j .
	Data: Solution coded on hybrid tree Result: Solution on hybrid grid for = L -1 to 1 do for cells in level do
	If(ghost cell)
	Reconstruct three daughter cell values with Prediction operator (6.10)
	and detail formula (6.11)
						n, i,j ,	(6.17)
	for the second step,
	φ * * , i,j = φ * , i,j + λ (	1 4	D n, i,j +	1 4	D * , i,j ),	(6.18)

Table 6 .

 6 1-L < R ≤ 1. Indeed, there is controlled population dynamics 1 Biological parameters tions of the same reference situation, with the parameters equal to the mean values of Table6.4. As expected, the occupation rate is the same for all these simulations and equal to 0.1717. The user time is not constant because it can be perturbed by the computer load. It is distributed with a beta-like distribution around the mean value 27.27. results of the random simulations are exposed in Figure6.10. There are four rows of gures, each of them corresponding to a set of Monte Carlo simulations where we let one parameter vary randomly with a uniform distribution in the range specied in Table6.3, while the three other parameters remain xed and equal to their mean value (in the left column of Table6.4). On each row, we represent the

	Parameter	Description	Value
		Global control
	U min	minimum level	0.075
	U max	minimum level	0.15
	c	slope parameter	2.0
	M	abscissa of the inection point	25
		Local control
	b 1	basal level	0.08
	b 2	exponential rate	2.2482
	b 3	scaling factor	1450.
		Source term
	Λ	intensity factor	0.6
	Λ w	half width	0.05
	γ	scaling factor	0.1
	γ s	cellular maturity threshold	0.3
		Dynamics
	C p	capacity	0.47
	p	speed	2.2
	α	contrast	1
	c 1		0.032
	c 2 ū	Maturation function	0.148 0.05
	The histogram of a parameter in input (left column), the histogram of the occupation
	rate in output (center column) and the histogram on the user time in output (right
	column).
	The rst two lines correspond to variations of the parameters appearing in the velo-

city functions (p and C p ). The distribution of the occupation rate looks uniform in the p case and more skewed towards the lower values in the C p case. In both cases the mean value of the occupation rate (the number in parenthesis in Table

6

.4) is very close to the occupation rate for the reference case. The distribution of the user controlled population dynamics

Table 6 .

 6 4 Results of the mean load balance test. Median values, maximum deviation in percentage and mean value of the output (in parentheses). Left column : perturbation on 4 dierent input parameters. Center column : occupation rate at nal time. Right column : User time the reference situation (see Figure6.11), which indicates that variations on several parameters simultaneously combine into a systematic slowdown of the simulation.

Table 6

 6 Parameter p.User time.Parameter µ 1 .Parameter M . 6.12 Mean load balance test : perturbation of the reference situation and impact on the occupation rate and CPU time. The four parameters of Table6.3 are perturbed at the same time. controlled population dynamics p, µ 1 and M are constant for the 20 simulations (equal to the mean values in Table

				Histogram of x						Histogram of x				Histogram of x
		1.5																0.15
											20
		1.0									15							0.10
	Density									Density	10						Density
		0.5																0.05
											5
		0.0									0							0.00
			0.8	1.0		1.2		1.4			0.15	0.16	0.17	0.18	0.19	0.20		25	30	35	40	45	50
					p								occupation rate				user time
											Occupation rate.		User time.
				Histogram of x						Histogram of x				Histogram of x
		15									150
																		0.4
		10									100							0.3
	Density									Density							Density
																		0.2
		5									50
																		0.1
		0									0							0.0
		0.56	0.57	0.58	0.59	0.60	0.61	0.62	0.63		0.165	0.170		0.175	0.180		26	28	30	32
					cp								occupation rate				user time
			Parameter C p .			Occupation rate.
				Histogram of x						Histogram of x				Histogram of x
		0.7
																		0.25
		0.6									400
		0.5																0.20
	Density	0.3 0.4								Density	300 200						Density	0.10 0.15
		0.2
		0.1									100							0.05
		0.0									0							0.00
			6.0	6.5	7.0		7.5			0.169	0.170	0.171	0.172	0.173		25	30	35	40
					initial condition							occupation rate				user time
											Occupation rate.		User time.
				Histogram of x						Histogram of x					Histogram of x
		0.7								500
		0.6
										400							0.3
		0.5
	Density	0.3 0.4							Density	300 200						Density	0.2
		0.2								100							0.1
		0.1
		0.0								0							0.0
		1.5		2.0	2.5		3.0		3.5	0.168	0.169	0.170	0.171	0.172	0.173		26	28	30	32	34	36
					control fall						occupation rate					user time
										Occupation rate.		User time.

.1 and the numerical parameters from Table

6

.2. For the perturbed parameters p, C p , µ 1 and M , we used Table

6

.3. Moreover, in the Monte Carlo method (6.25), we set N = 20. controlled population dynamics Figure 6.10 Mean load balance test : perturbation of an input parameter (left column) and impact on the occupation rate at nal time (center column) and user time (right column). Dierent parameters are tested : p and C p appearing in the velocity functions, µ 1 appearing in the initial condition and M appearing in the global control. controlled population dynamics Figure 6.11 Mean load balance test : Reference test without perturbations on the input parameters.

Figure

ni à partir de conditions sur l'eectif diérencié) consistait à utiliser des taux de transfert de type bang-bang, et de préciser la chronologie des événements ovulatoires dans ce cas (instants de commutation, de déclenchement de la décharge ovulatoire et d'ovulation)[START_REF] Clément | Optimal control of the cell dynamics in the granulosa of ovulatory follicles[END_REF].A ce stade, la diculté majeure pour représenter la dynamique cellulaire, non pas en fonction de l'âge folliculaire, mais de l'historique d'exposition d'un follicule en terme d'environnement hormonal, venait de ce que l'action de cet environnement (c'est-à-dire de l'hormone FSH) varie suivant le degré de maturité cellulaire. Schématiquement, cette hormone favorise la prolifération des cellules moins matures (majoritaires dans les follicules en début de développement terminal) tandis qu'elle favorise la diérenciation terminale dans les cellules plus matures. Il fallait donc pouvoir distinguer les eets de FSH en terme de prolifération et en terme de diérenciation et les moduler selon l'état cellulaire.Une deuxième étape a donc consisté à étudier plus nement la transduction du signal FSH, en suivant une approche de type modélisation biochimique, tout en tenant compte de modications à long terme dans cette signalisation (phénomène d'autoamplication)[START_REF] Clément | A mathematical model of FSH-induced cAMP production in ovarian follicles[END_REF].La troisième et dernière étape a consisté à réintégrer les dynamiques précédemment élaborées au niveau intracellulaire et cellulaire, dans un formalisme spatialisé (EDP en 2D), ce qui permet de suivre l'évolution des populations cellulaires d'un follicule en fonction des conditions d'environnement hormonal, mais aussi de coupler entre elles les trajectoires folliculaires, pour rendre compte du rétrocontrôle exercé par l'ensemble de la population de follicules en développement sur les niveaux de FSH.Ce modèle EDP a fait l'objet d'un ensemble de résultats théoriques. Dans[START_REF] Echenim | Sorine Multiscale modeling of follicular ovulation as a reachability problem[END_REF], les auteurs ont étudié un problème d'atteignabilité, déni à partir de la formulation du modèle sous forme de courbes caractéristiques, pour délimiter les conditions initiales compatibles respectivement avec un état nal de type ovulatoire ou atrétique. Il s'est avéré que ces conditions initiales sont peu distinguables et que le contrôle est déterminant dans la divergence des trajectoires ovulatoires ou atrétiques. Dans le même article, les auteurs avaient avancé des arguments basés sur les traces des solutions pour aborder la question du caractère bien posé du modèle. Cette

Figure1.13 Cas test 1D (Figures extraites de[START_REF]A numerical method for transport equations with discontinuous ux functions : Application to mathematical modeling of cell dynamics[END_REF]) : doublement de ux au passage de l'interface x s = 0.5. La masse (à gauche) a doublé lorsque toute la densité a passé l'interface, comme attendu. L'erreur en norme L1 (à droite) en fonction de la taille des mailles décroît à zéro avec un taux de 2.4.population. Il est non linéaire, et permet de visualiser tous les follicules, à tout temps, sans favoriser le temps nal.Sur la Figure1.15, on donne un exemple de visualisation obtenu avec le code de visualisation dualMR (une notice utilisateur est disponible en Annexe B de la thèse).On ache avec le même code couleur les densités de cellules de deux follicules dans le plan âge (abscisse x), maturité (ordonnée y) au même instant t. Les paramètres dénissant les fonctions de vitesses g f et h f , f = 1, 2 ont des valeurs diérentes et les densités de cellules des deux follicules sont dans des régions diérentes de l'espace fonctionnel. De plus, l'eectif cellulaire du follicule sur le panel du bas (M 0 = 20.46, en dessous à gauche du panel) est plus de trois fois plus grand que celui du follicule sur le panel du haut (m 0 = 6.33).1.3.3 Adaptive mesh renement strategy for a non conservative transport problemEn l'absence de soin particulier la stratégie de maillage en présence de doublement de ux met en évidence un problème en temps long. En eet, le maillage adaptatif se rane dans des zones où il n'y avait pas de densité signicative (voir Figure1.16, a) ).Ce phénomène est dû au couplage entre l'adaptation de maillage par multirésolution et les conditions de doublement aux interfaces (modélisant le phénomène de mitose). En raison des opérations de ranement et déranement successifs, la méthode adaptative introduit des perturbations par rapport à la solution calculée sur une grille uniforme. La Figure1.17 visualise les diérences entre les deux solutions dans le cas d'une équation de transport, pour une valeur exagérément grande du

entre les diérents processeurs, ajoutent du temps de calcul, et éloignent la courbe de performance de l'optimal.Par ailleurs, les paramètres du modèle inuent eux aussi sur les performances en temps de calcul et en occupation mémoire, dans le cas d'un schéma volumes nis adaptatif explicite en temps, dont les paramètres de discrétisation doivent satisfaire une contrainte de stabilité.On peut obtenir d'excellents résultats avec un test de scalabilité faible, eectué sur un jeu de paramètres de référence. Cependant, si une perturbation sur les paramètres en entrée (erreur de mesure, valeur modiée par un utilisateur, ...) entraîne une diminution du pas de temps ou bien un ranement du maillage, les performances réelles seront loin de celles de la situation de référence. An d'avoir des résultats réalistes, du point de vue de l'utilisateur, nous avons proposé une méthode plus générale, inspirée des méthodes de propagation d'incertitudes. Le principe est le suivant : on dénit un jeu de paramètres de référence, et des lois de probabilités perturbant ces paramètres. Par une méthode de type Monte Carlo, on eectue un grand nombre de réalisations (simulations numériques basées sur ces paramètres), et on évalue les valeurs moyennes de certaines sorties, comme le taux d'occupation moyen du maillage adaptatif et le temps utilisateur moyen. Cela dénit un test de scalabilité faible en moyenne.

http ://www.pas.rochester.edu/~bearclaw/

We use this index notation for the use of the forthcoming Multiresolution development

and further tested and extended in[START_REF] Coquel | Entropysatisfying relaxation method with large time-steps for Euler IBVPs[END_REF] or[START_REF] Hovhannisyan | On the stability of fully adaptive multiscale schemes for conservation laws using approximate ux and source reconstruction strategies[END_REF] is currently implemented in this new conguration, where the presence of source terms and integral terms will require special considerations as exemplied for instance in[START_REF] Chiavassa | Martinez-Gavara Cost-eective multiresolutions schemes for shock computations[END_REF].

Process in cell proliferation through which a mother cell gives birth to two daughter cells

Table3.1 1D test cases description, asymptotic order p of the L1-norm error O(∆x p ) and relative error value for ∆x = 10 -3 . Each test case is run for four dierent speed contrasts, leading to slightly dierent asymptotic behavior.

lenges since they require both a large domain of calculation and sucient accuracy.It is therefore advantageous, in terms of computational costs, to use a time varying adaptive mesh, with small cells in the region of interest and coarser cells where the solution is smooth. Biological models involving cell dynamics fall for instance within this framework and are often non conservative to account for cell division. In that

à droite). A chaque follicule on associe un bloc, délimité en abscisse par le numéro de follicule, et en ordonnée par des indicateurs correspondants à diérentes valeurs de contrôle. Précisément, il s'agit des racines de la fonction h f à gauche, et des valeurs de la fonction g f en zone G1, à droite. On sépare ces blocs en deux parties, la partie rouge correspond à des valeurs de contrôle défavorables (ū min ≤ u f ≤ U min ) et la partie verte à des valeurs de contrôle favorables (U min ≤ u f ≤ U max ). Une lecture verticale permet de connaître des informations locales sur un follicule, et une lecture horizontale permet de quantier l'hétérogénéité.Abaque multi follicule (2ème espèce). Une autre option consiste à généraliser l'abaque à deux follicules en xant la valeur du contrôle. On a alors une famille de courbes, indexées non plus par la valeur de contrôle, mais par le numéro de follicule.Cette représentation est très pratique pour visualiser les diérences d'amplitudedans les fonctions h f . On donne un exemple dans la Figure5.26.5.5.3 Exemple de situation de référence à 10 folliculesEn combinant les résultats précédents, on dénit une méthode en deux étapes pour calibrer une simulation multi follicules. 1. Contrôle en boucle ouverte Détermination de la maturité asymptotique ; Détermination des paramètres de cinétique cellulaire ; 2. Contrôle en boucle fermée Réglage de la pression sélective ; Réglage de l'hétérogénéité. A titre d'exemple nous allons utiliser cet algorithme pour identier une situation de référence à 10 follicules.

Remerciements Acknowledgment

The authors would like to thanks Pascal Joly, Philippe Parnaudeau and Kyrill Pichon Gosta 3 for their help concer- ning parallel computing, the CIRM 4 for the excellent working conditions, the INRIA Large Scale Initiative Action REGATE 5 for the funding of the project.

transport problem (5.2) (resp. h given by (4.40)) and Λ n ,(i,j) is the discrete value of the source term Λ(x, y, U (t)) given by (4.41) on the mesh D ,(i,j) ∈ Sn . A simpler condition would consist in using the uniform case denition with the space step corresponding to the nest MR level ∆t n = min CF L ∆x L-1 max D ,(i,j) ∈ Sn (|g n ,(i,j) |, |h n ,(i,j) |)

, 1 max D ,(i,j) ∈ Sn |Λ n ,(i,j) | but since the speed and source coecients are space dependent, the mesh has to be fully surveyed at each time iteration in any cases and formula (4.36) is more accurate and possibly less stringent. The numerical value of the biological constants appearing in denitions (5.2), (4.40) and (4.44) are gathered in Table 4.2. In the following paragraph we detail a set of numerical experiments which highlight the robustness of our numerical method in this biological set-up. We show in particular, that for this standard set of parameters spurious noise will appear at late times of the simulation unless the adaptive strategies presented earlier are used.

Numerical performances

The reference solution is computed on the nest level of resolution, comprising 51200 meshes and it takes 540 seconds on a mono processor Apple laptop to reach the nal time t = 20. Snapshots of the solution at representative times are displayed in Figure 4. [START_REF] Boutin | Coupling techniques for nonlinear hyperbolic equations. III. well-balanced approximation of thick interfaces[END_REF], along with the adaptive mesh corresponding to ε = 0.01. Noticeably, at initial time (top panel), due to the periodicity of the MR reconstruction operator (see Eq.

(4.22)), the mesh is rened near the right edge of the domain although the density is zero in this area.

As in the simple test case, we perform simulations with or without time adaptation of the MR threshold (4. [START_REF] Chen | Hyperbolic conservation laws with discontinuous uxes and hydrodynamic limit for particle systems[END_REF]) and (4.34), and with or without the adaptive doubling strategy (4.20). Figure 4.14 dedicated to snapshots of the solution at t = 20 shows the inuence of adaptive thresholds on the spurious noise removal. A zoom of the density computed with the dierent methods is displayed in the left panels. Both solutions look very similar although if we let the simulation run longer the solution in the bottom panel eventually deteriorates. In the right panels we display the adaptive grid superimposed to the density with a color code adapted to the amplitude of the spurious noise.

We then compute the L1 norm of the relative error (4.35) as a function of ε at dierent times, corresponding to the snapshots displayed in Figures 4. [START_REF] Boutin | Coupling techniques for nonlinear hyperbolic equations. III. well-balanced approximation of thick interfaces[END_REF] with respect to the solution obtained using the uniform nest grid.

In Figure 4.15 we display the error curves against the MR threshold ε (left panel), the CPU gain (center panel) and the memory gain (right panel). The same legend as in Figure 4.9 is used, and described in

end if end for end for Algorithm 1: Prediction of the set Γ n+1 ε of signicant details at time t n+1 from the details at time t n . General case with L MR levels numbered from the coarsest 0 to the nest L -1

Chapitre 6

Design and performance analysis of a multiscale scheme for controlled population dynamics Sommaire Abstract. In this chapter, we discuss two main subjects. First, we introduce a numerical scheme for a general multiscale model of controlled population dynamics.

Originally designed for the selection process of ovarian follicles, the method is presented in a general frame that may be afterward adapted to several situations. Second, we introduce a new method for analyzing the performance of a parallel code.

Keywords. Multiscale models, population dynamics, non-local control, nite volume schemes, adaptive mesh, multiresolution, high order scheme, parallel computing, performances analysis, weak scalability, load balance, Monte Carlo method, uncertainty propagation.

Introduction. In this chapter, we are concerned with multiscale controlled population dynamics, and present two main results. First, we introduce a new multiscale method, and second we present a new method for performance analysis of a parallel code. Originally, the multiscale numerical method was designed for the model of the selection process of ovarian follicles (see [START_REF] Echenim | Multi-scale modeling of the follicle selection process in the ovary[END_REF] and [START_REF] Echenim | Sorine Multiscale modeling of follicular ovulation as a reachability problem[END_REF] ). The mathematical model consists t = 5 y < γ s Chapitre 6. Design and performance analysis of a multiscale scheme for controlled population dynamics with the age x in abscissa and cell maturity y in ordinate. The sub-domains (cells cycles and dierentiation phase) are bounded with straight lines. In this model the mesoscopic moments (eq 6.1) are indicators of the follicle maturities and the macroscopic moment (eq 6.2) represents the ovarian maturity. In panel b) of Figure 6.2, we display the follicle maturities as a function of time for all ten follicles of the numerical experiment. In panel c) we display the global control function U = S(M 1 ) as a function of the maturity.

For these specic speed functions, an initial density localized in the bottom left of the computational domain (cycle 1) is transported to the top right of the domain (see Figure 6.2, panel a)). During this process, the mesoscopic moments begin by increasing, because the density crosses, several times, a mitosis boundary, where the density is locally doubled (see Figure 6.2, panel b)). Once the density has reached the upper part of the domain (y > γ s ), the cells stop proliferating, the maturities either stabilize or decrease due to cell apoptosis. At the beginning, the global control is high, but around t = 4.5, it becomes sensitive to the increase in the global maturity and decreases steadily until reaching its lowest level (see Figure 6.2, bottom right panel). The minimum and maximum values of the global control U in equation 6.4 are for this simulation U min = 0.075 and U max = 0.15. They are asymptotically reached for values of the global moment M 1 well within the range of the nal and initial values of this quantity M 1 (t) as a function of time. Therefore for this choice of parameters U (t), represented in panel c) of Figure 6.2 has the same shape as S(M ). We will not go here into the details of the underlying biological model, which is the topic addressed in the Chapter "Calibration". However, it is interesting to note that after the control has dropped to half its initial value only one follicle keep a stationary maturity while all other follicular maturities decrease. This is a good instance of a selection process triggered by hormonal control.

Conclusion.

In this section, we have presented a class of models for controlled multiscale population dynamics. This model may be adapted to many other scenarios, either by changing the conservation law, or by changing the controls.

Multiscale numerical method

In this section, we present the numerical method that we have designed in order to simulate the multiscale model presented in the rst section.

Size of the problem. In order to x ideas, let us precise the size of the problem in a typical realistic situation in the biological model. We have N f coupled PDEs in a 2D computational domain that is a compact subset of R 2 , namely [0, 1]×[0, N c ], with N c of order 10. If we take 100 × 100 grid cells to discretize the domain [0, 1] × [0, 1], then we have 100 × 100 × N c mesh cells for each sub-population hence 100 × 100 × N c × N f grid cells for the whole population. Setting N c = 10 and N f = 20, we controlled population dynamics Multiresolution. In this paragraph, we present the basis of the multiresolution analysis. Since we are interested in adaptive nite volume schemes, we use a multiresolution approach based on a mean value representation of the solution, by contrast to point values decomposition scheme (see [START_REF] Bihari | Multiresolution schemes for the numerical solution of 2-D conservation laws[END_REF] for more details).

For the sake of simplicity, we will consider only two levels of multiresolution in order to introduce the main concepts.

The solution in a mother cell can be computed from the daughter cells with the following formula

This formula denes a 2D projection operator.

Knowing the mean values of the solution on the coarser level, approximated mean values on the ner level (daughter cells) may be obtained by second order reconstruction :

This denes a 2D prediction operator.

The error between real values and predicted values are called details (the terminology comes from image processing), and may be computed with the formula

For each mother-daughters pair we dene the change of basis, from mean values to details,

Note that due to property (6.9), it is sucient to know only three details and the mean value of a mother grid cell to compute the value of the last detail. The corresponding algorithm (called "Encoding") in the general multilevel case is dened by Algorithm (3). This operation is invertible, and the corresponding algorithm (called "Decoding") in the general multilevel case is given by Algorithm (4). controlled population dynamics Data: Solution coded on hybrid tree Result: Solution on hybrid grid for = 1 to (L -1) do for cells in level do

Reconstruct three daughter cell values with the Prediction operator (6.10) and the detail formula (6.11)

end for cells in level do Reconstruct last value with the Projection formula (6.9)

end end Algorithm 4: Solution decoding : from the multiresolution (wavelet) representation into the physical basis (FVS).

Tree structure. In order to preserve the linear complexity of Algorithms 3 and 4 it is crucial to ensure that the adaptive grid corresponds to the hierarchy of details organized in a tree structure. To make this concept more precise we introduce the notion of stencil of prediction SP i,j , which is the set of mesh indices entering in the prediction of the solution values on the subdivisions of mesh Ω i,j . For the quadratic prediction that we use

After identifying the meshes that cannot be coarsened using rule 6.12 we must also include at this level all the meshes belonging to their prediction stencils (6.13) even if those satisfy rule 6.12. To do so, it is practical to introduce the notion of "border cells" to denote cells that are added at a given level only because they belong to the stencil of prediction of "inner" cells. It is important to distinguish these border cells from the others because they should not be tested for subdivision at the ner level.

Finite volume scheme on an hybrid grid. The key point in adaptive nite volume schemes using multiresolution is the following : if the solution is smooth enough (in other words, if the details are under a threshold value) we can compute the scheme on the coarser level.

In fact, instead of computing the details, we compute regularity sensors, according to the article of Bihari and Harten [START_REF] Bihari | Multiresolution schemes for the numerical solution of 2-D conservation laws[END_REF]. The linear relation between details and regularity sensors is given by the following formula controlled population dynamics

These formulas lead to dierent threshold tests, that we call the Harten rules (see We will show on a simple example, without any source term and involving only two multiresolution levels, how to update a solution dened on a multiresolution grid.

If two neighboring grid cells do not have the same size (see Figure 6.4), we cannot apply directly the classical nite volume scheme (6.6), which is dened for uniform grid at level . First, we add ghost cells (the coarse cell in the center is divided), in order to compute the uxes entering the cells at the ner level (ne grid cells on the right on Figure 6.4). Then, we delete these ghost cells, and compute the uxes entering the cell on the coarser level, at the center. We can do that using the uxes of the ner level, by computing the sum of the ne uxes. Indeed, using additivity theorem, the ux across a reference vertical edge [0, ∆x] during a reference time step [0, ∆t] satises

gφdydt, which leads at the discrete level to

and a similar formula for uxes across horizontal edges. At the end, we compute the uxes at the coarse level. controlled population dynamics constant or slowly varying initial condition and we safely adopt this strategy.

Figure 6.5 Initialization of the hybrid grid from the coarsest level. In this special case, we see that the prediction stencil of the small grid cell in the center (grey part of the domain) is not full (there are cut lines at the places where grid cells are needed).

In order to get a graded mesh tree when initializing from the coarsest level, it is necessary to do two loops on each level. The rst one identies the grid cells that have to be created because the details are signicant, and the second one is needed to grade the hybrid tree.

the numerical global control,

and the numerical controls at the mesoscopic scale,

(6.23)

Parallel computing

In order to speedup the code, we use parallel computing techniques. Let us rst recall some basics about parallel computing (the reader can nd an introduction to parallel computing in the lecture [START_REF] Roux | Analyse numérique matricielle avancée et calcul parallèle[END_REF]).

Schematically, a computer has three main components : the CPU (Control Process Unit), the disk cache memory and the global memory. The disk cache is a small memory with fast data access, as opposed to the main memory, that is larger, but with slow data access. In parallel computing, computations are done on a computer with several processors, each of them endowed with a disk cache and a global memory. Tasks that can be performed independently and simultaneously will be splitted between the dierent processors. In the framework of distributed memory computation, which is the set-up that we have selected, there is no global memory, but small local memories, associated to each CPU. If the processes need to communicate, this communication is done via a network. There are several well-known problems that may arise in parallel computing, such as those described below.

If the amount of computation to be done is not the same for each process then naturally some of the processes are forced to wait for the others. (unbalanced load).

The load needs to be distributed as equally as possible among the processes.

As opposed to classical computing, communications between processes are needed to share data. Therefore, a certain amount of work is added. When designing a parallel code, we should take care of not spending too much time on communications. Indeed, if there are too many communications, the parallel method may take more time than the sequential one.

Since the disk cache is a fast data access memory, in the best scenario, the whole data is in the disk cache. If this is not the case, the process has to access the small local memory and this action penalizes the user time.

We will now propose a parallel scheme for our multiscale model. Trying to solve our specic problem in this parallel framework will lead us to choose between two options.

Option 1 : domain decomposition method. The rst option involves using a classical domain decomposition method (see Figure 6.8). The idea is to split the 182 Chapitre 6. Design and performance analysis of a multiscale scheme for controlled population dynamics computational domain into several parts, and send each of them on a process. To do so, we have to add several communications, at each time step, as follows :

FVS evolution : Each process has to send the table of the boundary cells whose mean value enters the ux computation on the boundary of the grid mesh of its neighbors. It also has to receive from the other processes the corresponding tables on the other part of the boundary. The bottom line is that each process has to send two tables and receive two tables (except for those on the outside boundaries, which send and receive only one table), of the size of the mesh. This communication is mesh size dependent.

Moments computation : each process has to compute its own part of the rst order moment, for each sub-population, and share the table of local rst order moments. This can be done with a reduction operation. Each process has to make N f reduction operations, that will compute the N f local rst order moments.

When all the involved processes are aware of all local maturities, the global rst order moment is computed as the sum of the local maturities (reduction operation Option 2 : biology inspired method. A second option is to try to mimic biology. As we know, one of the problems arising in parallel computing is to deal correctly with the available data, as well as to keep a maximal granularity, avoid communications and try to localize data in space and time in order to optimize disk cache accesses. The idea behind this option is to send each sub-population on a process (see Figure 6.7). In some way, it can be viewed as an ecient domain decomposition method. Indeed, if we replicate the computational domain (one for each sub-population), and represent them as a stack (see Figure 6.8 for an example with 2 sub-populations), this method can be considered as a domain decomposition method, which avoids a large number of communications. The only communications to be done are the following :

Computation of the global rst order moment as the sum of the local maturities (reduction operation).

Computation of the global time as the minimum of the local time steps (reduction operation).

Moreover, these communications are the fastest as possible, as they use only reduction operations, which are moreover mesh size-independant.

Conclusion. The second option was chosen, for the following reasons :

First, as we have seen, it involves fewer and more ecient communications.

Second, the load balance is better with this option. Indeed, the densities are quite localized in the computational domain, and they start from the same area in their controlled population dynamics respective domains at initial time. If we look at Figure 6.8, with the rst option, process 1 will be very busy at the beginning, because it contains all the active grid cells, while process 2 is not busy. Then, when the densities move to the second part of the domain, the roles are inversed. Using option 2 ensures that the load balance remains good with time.

We now summarize the methods described in the previous sections and dene the general SIMD (Single Instruction Multiple Data) algorithm for our problem.

Initialize hybrid grid from coarse level (Algorithm 9).

while

Performance analysis

In this section, we will present a new method dedicated to the performance analysis of a parallel numerical scheme, based on uncertainty quantication. We apply this method to the example of follicular development introduced in the rst section.

For an introduction to the bases of probability and uncertainty quantication, the interested reader is referred to [START_REF] Lelièvre | Méthodes numériques probabilistes[END_REF] and [START_REF] Tryoen | Adaptive stochastic Galerkin methods for parametric uncertainty propagation in hyperbolic systems[END_REF]. controlled population dynamics a) [0, 1] ordinate scale b) Zoom ordinate scale Figure 6.9 Load balance in a simulation of a selection process among 20 subpopulations. In abscissa, the time, in ordinate, the occupation rate (6.26) of each sub-population (the amount of computations done by the process supporting the subpopulation is proportional to this quantity). In this case, the computationnal domain is long (N c = 8), and we set dierent speed parameters for each sub-population.

In panel a) we display the occupation rate in the scale [0,1], to highlight the overall very good performance of the multiresolution compared with the uniform nest mesh. In panel b) we zoom on the range of interest to emphasize the dierences among follicles.

Method

The importance of using a stochastic method. There exist several classical methods for evaluating the performance of a parallel code. In problems such as ours, where the size of the problem grows with the number of processes, the classical test is a weak scalability test.

The theoretical set-up for the evaluation of the weak scalability would consist in increasing the number of processors, and letting the parameters unchanged. This method was successfully applied in [START_REF] Aymard | Numerical simulation of the selection process of the ovarian follicles[END_REF], in the framework of uniform meshes.

Since we have dierent parameters on each processor, we face two problems.

Due to the stability condition associated with the explicit numerical scheme, the parameters dening the control and velocities determine the local time step, hence the number of time steps necessary to reach a xed nal time. In our problem, the processors have to be synchronized, and to use the same global time step, computed as the minimum of the times steps computed on the mesoscopic scale. If the increase in the number of processes corresponds to a change in the time step, then the notion of weak scalability is modied because the size of the problem does not depend only on the number of processes.

Another source of variability between processors is the time adaptive meshing. Since the solutions are distinct on dierent processors, the meshes are not the same, so that the number of grid cells are also dierent.

In practice, the user who modies or creates his own set of parameters is not ensured Chapitre 6. Design and performance analysis of a multiscale scheme for controlled population dynamics always at least the coarsest grid cells if no grid cell is rened, and in the worst case the whole nest grid is active.

The number of arithmetic operations in a simulation, in our case, is proportional to the number of active grid cells. Then, we can say that the occupation rate is a good indicator of the amount of computation, and we can use it in order to evaluate the load balance of the code.

On Figure 6.9, we have plotted the occupation rate curves in the case of a selection process among 20 sub-populations, which dier by their speed parameter. In this case, N c = 8, the domain of computation is very large compared to the size of the support of the solution, which amounts to a very good overall occupation rate. At most 8.5% of the size of the uniform nest grid is required at any time of the simulation. We note that at the beginning, the occupation rates are the same, because all the follicles start from the same initial condition, and then, due to the dierences between the velocity parameters, they become dierent from one another.

In the right panel, we see the inuence of the heterogeneity among the follicles on the occupation rate which varies from 4.8% to 8.5%. Moreover its time dependance is also very dierent from one follicle to the next. Some curves exhibit a clear maximum around t = 4, others around t = 6, and others have a more regular behaviour throughout the simulation. Indeed we have noticed that the support of the density may become larger during the time span where some cells exit the proliferation cycle, if this event happens over several cycles. This behaviour is of course completely dependant on the individual velocity parameters of each follicle and its heterogeneity of the dynamics reects on the occupation rates.

We next try to answer the question of the sensitivity of the computationnal performances with respect to the parameters of the simulation.

Method. The mean load balance test consists in performing many realizations of the numerical simulation of our model with perturbed parameters for one subpopulation (sequential case). The aim is to evaluate the mean load balance. We add uniformly distributed perturbations to four distinct parameters : p and C p appearing in the dynamics, µ 1 in the initial condition and M in the global control term. The range of variation is the largest one ensuring that the parameters satisfy the biological specications established in the chapter on the calibration of the model. Table 6.1 and Table 6.2 indicate the biological and numerical parameters.

For the mean load balance test, we have used these parameters and we considered the perturbations of Table 6.3. Furthermore, in the Monte Carlo method (6.25), we have set N = 200.

Hardware. This test was done on a laptop Dell precision M4400 with an Intel Core 2 Duo processor of 2.26 G-Hz, 3Mo of disk cache.

Results. In order to interpret the Monte Carlo simulations we have plotted on Table 6.3 Laws of probability used to generate the random parameter perturbations for the simulations displayed in Figure 6.10. The ranges of variation are chosen in order to respect the biological constraints.

time is clearly non uniform and has a beta-like shape in both the p and C p cases. In the p case the mean value (≈ 30) is slightly larger than the mean user time in the reference case (≈ 27).

On Table 6.4 we have gathered the median values and the maximum deviation in percentage for the results displayed in Figure 6.10, computed with the following formula median value = X max + X min 2 and max deviation in percentage = X max -X min X max + X min where X min and X max are the extremal values reached by the random input parameter or by the output measurement. Because of the constraints on the parameters imposed by the biological specications the random variations have dierent ranges for the four selected parameters. It is therefore dicult to appreciate there relative sensitivity. However the comparison of the maximum deviations in percentage between the occupation rate and the input parameters seems to indicate that variations in µ 1 and M have little inuence and that C p is relatively more inuent than p.

As opposed to Figure 6.10, which treats separately the impact of perturbing each of the four parameters, Figure 6.12 depicts the global situation where all four parameters are perturbed at the same time. With the perturbations of Table 6.3 on parameters p, C p , µ 1 and M in input, the occupation rate varies around a median value of 0.17 with a maximal deviation of ±18% of the median value. There has obviously been a compensation phenomena. The user time on the other hand varies between ≈ 30 and ≈ 45. This interval is altogether above the average user time for Chapitre 6. Design and performance analysis of a multiscale scheme for controlled population dynamics

Hardware. Cluster HPC1 of the Jacques-Louis Lions Laboratory is a SGI Altix UV 100, with 20 CPUs Intel Xeon 64 bits Nehalem EX 7550 of 8 cores of 2 G-Hz (160 cores), 640 Go of RAM, using Numalink interconnect system (15 GB/s).

Results. Figure 6.13 shows the results of this test for dierent cases.

On each panel, the blue curve is the theoretical sequential time. We computed the mean user time for one process, denoted < t seq >, and we draw the curve c(N p ) =< t seq > ×N p , with N p the number of processes. The green curve is the theoretical limit. By using this type of parallelization strategy (distributed memory using MPI), the computational cost for several processes cannot be inferior to that corresponding to the behavior we could expect if the same computation was performed on all processors without any communications. It is important to note that this is not the case in a domain decomposition method, where the user time should decrease with the number of processes. In our case, the overall size of the problem grows with the number of processes. The black points correspond to realizations of numerical simulations using our method.

The parallel method that we use is more performant, in all cases, than the sequential method. Moreover, the user time is close to the theoretical limit.

On Figure 6.14, we represent zooms of the panels of Figure 6.13. The black dots are realizations of numerical simulations, and the red curve corresponds to the linear regression. On these panels, we see that there is a variability in the user time. This variability seems to decrease with the number of processes. This would be logical since the time step seems to be the major limiting factor on the CPU time. Indeed, adding processes can only force the local time step to decrease, and the processes are synchronized. Then, adding process can only slow all the processes.

In order to identify the source of this variability we rst perform the mean weak scalability test on the reference situation with a uniform mesh, a constant time step, and the same parameters for all the populations and simulations. We then introduce variability through the four selected parameters C p , p, µ 1 and M , that vary simultaneously among the populations and also from one simulation to the next.

We perform 20 independent simulations per number of processes. Since a change in the parameters inuences the time step through the stability condition (6.8) and also the load balance through the adaptive mesh size, we study these two factors independently. For the uniform meshes, we have set N m = 80, L = 1 and = 0. For the constant time step we have set ∆t = 0.001. For the other parameters we have used, when necessary, Tables 6.1, 6.2 and 6.3. In particular, the simulations with random parameters are performed with uniformly distributed random realisations of the four parameters C p , p, µ 1 and M simultaneously.

The results are displayed in Figures 6. [START_REF] Bürger | Conservation laws with discontinuous ux : a short introduction[END_REF] On peut ensuite visualiser les résultats.

A.2 Visualisation

On peut visualiser les résultats de plusieurs façons. Listes chaînées de niveaux Plusieurs algorithmes nécessitent de parcourir le maillage hybride niveau par niveau : codage, décodage, calcul des ux, mise à jour