Nicole Spillane
email: spillane@ann.jussieu.fr

Victorita Dolean
email: dolean@unice.fr

Patrice Hauret
email: patrice.hauret@fr.michelin.com

Frédéric Nataf
email: nataf@ann.jussieu.fr

Clemens Pechstein

Robert Scheichl

An Algebraic Local Generalized Eigenvalue in the Overlapping Zone Based Coarse Space : A first introduction

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Version française abrégée

Ce travail s'intéresse à la résolution d'un système linéaire (3) issu de la discrétisation par éléments finis (2) d'un problème aux limites elliptique donné sous forme variationnelle (1) où les coefficients peuvent être discontinus. Afin d'obtenir des méthodes de décomposition de domaine extensibles (robustes vis à vis du nombre de sous-domaines), nous considérons des méthodes à deux niveaux [START_REF] Toselli | Domain decomposition methods: algorithms and theory[END_REF]. Ces méthodes sont étroitement liées aux méthodes multigrilles et de déflation. Elles sont définies par deux ingrédients : une grille grossière V H composée de m vecteurs avec m petit devant la taille du problème initial et une formulation algébrique de la correction qui consistera ici en la méthode de Schwarz à deux niveaux (4) que l'on utilise comme préconditionneur pour un solveur de type gradient conjugué. Ce choix nous permet d'appliquer des résultats connus qui ramènent l'étude de la convergence de l'algorithme à celle du conditionnement de l'opérateur préconditionné. La contribution clé de ce travail consiste en une définition systématique de la grille grossière fondée sur les plus basses fréquences de problèmes spectraux généralisés locaux (Définitions 3.3 et 3.4), voir aussi [START_REF] Efendiev | Robust Domain Decomposition Preconditioners for Abstract Symmetric Positive Definite Bilinear Forms[END_REF] et les références citées. Par rapport à [START_REF] Efendiev | Robust Domain Decomposition Preconditioners for Abstract Symmetric Positive Definite Bilinear Forms[END_REF], notre grille grossière présente l'avantage d'être construite à partir de la matrice avant assemblage sans calculs supplémentaires de contributions élémentaires. De plus, l'estimation ne dépend pas d'une hypothèse de stabilité d'un interpolant éléments finis. Enfin, on fournit un critère de sélection optimal du nombre de vecteurs propres. Cette approche étend au cas de systèmes et dans un cadre variationnel la méthode analysée et validée par [START_REF] Dolean | Analysis of a two-level Schwarz method with coarse spaces based on local Dirichlet-to-Neumann maps[END_REF] et [START_REF] Nataf | A coarse space construction based on local Dirichlet to Neumann maps[END_REF] dans le cas scalaire. En particulier, elle permet l'obtention d'une convergence efficace de l'algorithme de Schwarz indépendamment des paramètres du problème, y compris des hétérogénéités. Le résultat théorique (Théorème 3.5) démontre en effet l'existence d'une décomposition stable, au sens de la Définition 3.1, de toute fonction discrète sur la grille grossière et les sous-espaces locaux. La section 4 constitue une première illustration de la méthode introduite sur un cas élémentaire mais fortement hétérogène d'élasticité bidimensionnelle.

Introduction

This work is the extension to systems of PDEs of the coarse space introduced and validated in [START_REF] Dolean | Analysis of a two-level Schwarz method with coarse spaces based on local Dirichlet-to-Neumann maps[END_REF] and [START_REF] Nataf | A coarse space construction based on local Dirichlet to Neumann maps[END_REF] for the Darcy equation (scalar PDE). In that specific case, in order to define the coarse space, generalized eigenvalue problems defined on the interfaces between subdomains were solved. The theoretical proof used weighted Poincaré inequalities [START_REF] Pechstein | Weighted Poincaré inequalities[END_REF] to write inequalities between quantities defined on the interfaces and ones defined in the whole overlapped regions. In order to be more general and avoid the need for weighted Poincaré type inequalities, this time we will define the generalized eigenproblems directly in the overlapped regions. We also bypass any stability assumption for the finite element interpolation. First we define the framework in which we will introduce the new coarse space.

Given a Hilbert space V 0 , a symmetric and coercive bilinear form a : V 0 × V 0 → R and an element f in the dual space V 0 , we consider the abstract variational problem: Find u ∈ V 0 such that

a(u, v) = f, v for all v ∈ V 0 . (1)
This variational problem is associated with an elliptic boundary value problem on a given polygonal (polyhedral) domain Ω ⊂ R d (d = 2 or 3) with suitable boundary conditions (possibly homogeneous Dirichlet on part of the boundary), posed in a suitable space of functions V 0 on Ω. We consider a discretization of the variational problem (1) with finite elements (FE). To define the FE spaces and the approximate solution, we assume that we have a mesh T h of Ω: Ω = τ ∈T h τ .

clemens.pechstein@numa.uni-linz.ac.at (Clemens Pechstein), r.scheichl@bath.ac.uk (Robert Scheichl).

The corresponding space of finite element functions w.r.t T h is then denoted by V h , and the subspace of functions from V h that fulfill the homogeneous Dirichlet boundary conditions by V h,0 = V h ∩ V 0 . In the case where a is a bilinear form derived from a system of PDEs, V h is a space of vector functions. In our analysis, we will also need restrictions of FE functions into subdomains D ⊂ Ω that are resolved by T h . The space of restrictions of the functions in V h to D is denoted by V h (D). Similarly, the space of restrictions of functions from V h,0 which vanish on Ω \ D is denoted by V h,0 (D).

The finite element discretization of ( 1) is: Find

u h ∈ V h,0 such that a(u h , v h ) = (f, v h ) for all v h ∈ V h,0 . (2)
Let {φ i } n i=1 be a basis for V h,0 with n := dim(V h,0 ), then (2) can be compactly written as

Au = f , (3)
where A i,j := a(φ j , φ i ), f i = f, φ i , i, j = 1, . . . , n and u is the vector of coefficients corresponding to the unknown FE function u h in [START_REF] Toselli | Domain decomposition methods: algorithms and theory[END_REF].

In order to automatically construct robust two-level Schwarz type methods for (2) we first partition our domain Ω into a set of non-overlapping subdomains {Ω j } N j=1 using for example a graph partitioner such as METIS or SCOTCH. Each subdomain Ω j is then extended to a domain Ω j by adding one or several layers of fine grid elements, thus creating an overlapping decomposition {Ω j } N j=1 of Ω. Let us suppose that the domains {Ω j } N j=1 are large enough so that for every k

(1 ≤ k ≤ n) there is a subdomain j (1 ≤ j ≤ N ) such that supp(φ k ) ⊂ Ωj .
Having defined overlapping subdomains we now introduce restriction operators R j . For 1 ≤ j ≤ N , R j is a mapping between the dual of V h,0 and the dual of V h,0 (Ω j ) -it restricts residuals. The adjoint R T j of R j extends functions from V h,0 (Ω j ) to V h,0 by zero. The corresponding matrix (also denoted by R T j for simplicity) takes a local vector and makes a global vector by inserting zeros. Finally, let us assume that we have a subspace V H ⊂ V h,0 (the so called coarse space) and an extension operator R T H from V H to V h,0 , then the two level preconditioner that we use is defined as

M -1 AS,2 = R T H A -1 H R H + N j=1 R T j A -1 j R j , A H := R H AR T H and A j := R j AR T j . (4)
The paper is organized as follows, in Section 3 we finish introducing the method by specifying our choice for the coarse space and give the theoretical bound for the condition number of the preconditioned operator; in Section 4 we give numerical results for the case where the two dimensional elasticity equations with heterogeneous coefficients are considered.

An automatic coarse space construction

Let us first define some notation. Since the bilinear form a originates from a second order elliptic partial differential equation in Ω, it has the following property: there exists a family of bilinear forms {a D } D indexed by all subsets D ⊂ Ω such that for any D ⊂ Ω, D ⊂ Ω verifying D ∩ D = ∅, the following holds

a D∪D (u |D∪D , v |D∪D ) = a D (u |D , v |D ) + a D (u |D , v |D ) ∀u, v ∈ V h,0 .

In addition for any D ⊂ Ω and any

v ∈ V h (D), let v 2 a,D = a D (v, v)
. This is usually referred to as the energy seminorm on V h (D), which defines a full norm on V h,0 (D).

Since both the preconditioner and the matrix A of the problem that we want to solve are symmetric and positive definite it is well known that the rate of convergence of the preconditioned conjugate gradient method depends only on the condition number of M -1 AS,2 A. In turn, a bound for this condition number relies on the existence of a stable decomposition of the solution onto the coarse space and the local subspaces (see [START_REF] Toselli | Domain decomposition methods: algorithms and theory[END_REF] or [START_REF] Scheichl | Domain decomposition methods in computational mechanics[END_REF] and the references therein). Next, we give the definition of such a stable decomposition. Definition 3.1 Given a coarse space V H ⊂ V h,0 , the local subspaces {V h,0 (Ω j )} 1≤j≤N and a constant C 0 , a C 0 -stable decomposition of u ∈ V h,0 is a family of functions {z j } 0≤j≤N such that z

0 ∈ V H , z j ∈ V h,0 (Ω j )
for all 1 ≤ j ≤ N and u = N j=0 z j , which verifies

z 0 2 a + N j=1 z j 2 a,Ωj ≤ C 2 0 u 2 a .
What we aim to achieve is to define the coarse space in such a way that there is a constant C 0 for which any u ∈ V h,0 admits a C 0 -stable decomposition and C 0 depends only on the ratio diam(Ωj ) δj between the sizes of the subdomains (diam(Ω j )) and the width of the overlap (δ j ). In particular C 0 will remain independent of the decomposition into subdomains and the heterogeneities defined by the bilinear form a. First, we introduce a partition of unity operator.

Definition 3.2 (i) For each k, 1 ≤ k ≤ n define N k = {j; 1 ≤ j ≤ N and supp(φ k ) ⊂ Ωj }. (ii) For each j, 1 ≤ j ≤ N define M j = {k; 1 ≤ k ≤ n and supp(φ k ) ⊂ Ωj }.
(iii) Then, for each j, 1 ≤ j ≤ N , let Ξ j : V h → V h be defined by:

Ξ j (u) = k∈Mj 1 #(N k ) u k φ k (x),
where φ k are the vector shape functions and u(x) = n k=1 u k φ k (x). Notice that N j=1 Ξ j (u) = u and supp(Ξ j u) ⊂ Ωj . Now, let Ω • j = {x ∈ Ω j ; ∃j = j such that x ∈ Ω j } denote the boundary layer of Ω j that is overlapped by neighbouring domains, and let δ j denote the width of Ω • j at the narrowest place. It can be shown that (Ξ j (u)) |Ωj \Ω • j = u |Ωj \Ω • j . Then the following eigenproblems enable us to ensure the stability of the decomposition by isolating the terms which slow down convergence: Definition 3.3 For any given j, 1 ≤ j ≤ N , let (λ j k , p j k ) 1≤k≤dim(V h (Ω • j )) be defined as the solutions of: Find the eigenpairs (λ j k , p j k ) ∈ (IR + × V h (Ω j )) of the generalized eigenproblem

λ j k a Ω • j (Ξ j (p j k ), Ξ j (v)) = a Ωj (p j k , v) ∀ v ∈ V h (Ω j ), ( 5
)
normalize them in the • a norm and order them in increasing eigenvalue order. These eigenproblems have a clear relationship with but are different from those introduced in [START_REF] Efendiev | Robust Domain Decomposition Preconditioners for Abstract Symmetric Positive Definite Bilinear Forms[END_REF]. Here we no longer need to make any assumption on a stability result for the finite element interpolant, this simplifies matters greatly: all assumptions hold also in a discrete setting so our theory actually covers the implemented method. Moreover, implementing [START_REF] Dolean | Analysis of a two-level Schwarz method with coarse spaces based on local Dirichlet-to-Neumann maps[END_REF] does not require any additional elementary matrix computations. In addition, we provide here a practical criterion to optimally select the number of vectors to be included in the coarse space. Definition 3.4 For a given positive real number K j , define m j (K

j ) = min k | λ j k ≥ 1 K j -1.
After choosing K j , we prove the following result for the new preconditioner. Theorem 3.5 Assume that each point in Ω belongs to at most k 0 subdomains and the coarse space is constructed as follows: for each 1 ≤ j ≤ N , solve the generalized eigenproblem given in Definition 3.3, then select the m j first eigenvectors according to the strategy given in Definition 3.4 for K j = diam(Ωj ) δj , and set V H to

V H = span 1≤j≤N ,1≤k≤mj Ξ j (p j k ) . (6)
Then for any u ∈ V h,0 there exists a C 0 -stable decomposition with

C 2 0 = 2 + 4 k 0 (2k 0 + 1) 1 + max 1≤j≤N diam(Ω j ) δ j . (7)
We insist on the fact that, thanks to the choice of K j , C 0 depends only on k 0 and on the proportion of overlap in each subdomain diam(Ωj ) δj

. From this, a bound on the condition number of the preconditioned operator and thus on the convergence rate of the preconditioned conjugate gradient method can be derived immediately using standard domain decomposition theorems (again, see [START_REF] Toselli | Domain decomposition methods: algorithms and theory[END_REF] or [START_REF] Scheichl | Domain decomposition methods in computational mechanics[END_REF]). These bounds will depend on the same quantities (k 0 and diam(Ωj ) δj

) leading to a method that should be scalable and robust with regard to high heterogeneities in the coefficients. The particular choice K j = diam(Ωj ) δj is made to match the well known estimate in the constant coefficient case.

Numerical Results

The purpose of this section is to illustrate the behaviour of our new preconditioner on the two dimensional linear elasticity equations with heterogeneities. We have used FreeFem++ [START_REF] Hecht Freefem++ | [END_REF] to define the test cases and build all the finite element data and Matlab for the actual solver. The equations are the following:

-div(σ(u)) = f , where u = (u 1 , u 2 ) T , and the stress tensor σ(u), the Lamé coefficients λ and µ and the right hand side are

       σ ij (u) = 2µε ij (u) + λδ ij div(u), ε ij (u) = 1 2
∂u i ∂x j + ∂u j ∂x i , f = (0, g) T = (0, 10) T , µ = E 2(1 + ν) , λ = Eν (1 + ν)(1 -2ν) . (8)
Here E and ν denote respectively Young's modulus and Poisson's ratio and are discontinuous. The test case is as follows (see Figure 1): we take a bar of height 1 and length N made up of 21 × (N × 20 + 1) nodes and divided into N subdomains which we extend by two nodes to form the overlapping partition.

The coefficient distribution consists of four horizontal layers. We impose Dirichlet conditions u = 0 on the left hand side boundary denoted by ∂Ω D , and Neumann conditions σ(u) • n = 0 on the remaining boundaries. The spaces are: V 0 = {v ∈ H 1 (Ω) 2 ; v |∂Ω D = 0}, V h = (P 1 (Ω)) 2 (piecewise linear with respect to τ h ). Throughout this section we compare three methods. The first one is the one level additive Schwarz method (referred to as AS), defined by the preconditioner

M -1 AS,1 = N j=1 R T j A -1 j R j .
The second one is the standard two level method (referred to as RBM), given by (4) with the coarse space which consists of all rigid body modes (so three in 2D) simply weighted by partition of unity functions. The third one is the new two level method (referred to as NEW). As a stopping criterion we use u-ū ∞ ū ∞ < 10 -7 where ū is the solution of (2) obtained via a direct solver on the global problem. Every time we give an iteration count for the NEW method we add in brackets the size of the coarse space compared to the total number of d.o.f.s in the overlaps. Table 1 gives the number of iterations that are needed to reach convergence for different decompositions and a coefficient distribution which mimics layers of steel and rubber (without dealing with incompressibility): (E 1 , ν 1 ) = (2 • 10 11 , 0.3) and (E 2 , ν 2 ) = (2 • 10 7 , 0.45). The new method is a lot more efficient and it is also more robust with regard to the number of subdomains. Table 2 gives the number of iterations that are needed to reach convergence for different jumps in the coefficients which are indexed by different values of (E 2 , ν 2 ). We use a regular decomposition into N = 8 subdomains and (E 1 , ν 1 ) = (2 • 10 11 , 0.3). When (E 1 , ν 1 ) = (E 2 , ν 2 ) (last line) we are looking at a homogeneous steel domain. As expected the method selects the three rigid body modes and only the three rigid body modes in all floating subdomains (plus two in the remaining subdomain).

Conclusion

We have constructed a coarse space for a two-level overlapping Schwarz method that is robust with regard to jumps in the coefficients in the equation. This coarse space is computed locally and automatically. It is suitable for parallel implementation. We have a theoretical proof for the bound on the condition number and the numerical tests for two dimensional elasticity are in agreement with theoretical conclusions. Since the method is constructed at the variational level, numerical experiments could be conducted for any other second order elliptic system of equations.

Figure 1 .

 1 Figure 1. Decomposition into N = 8 subdomains: regular (left), with METIS (middle) and Coefficient distribution (right) Décomposition en N = 8 sous-domaines: régulière (gauche), avec METIS (milieu) et Choix des coefficients (droite)

  Iteration count vs. number of subdomains Nombre d'itérations en fonction du nombre de sous-domaines (E 2 , ν 2 ) AS RBM NEW (V H ) (2 • 10 7 , 0.49) 152 97 36 (60/2352) (2 • 10 8 , 0.45) 97 86 35 (45/2352) (2 • 10 9 , 0.4) 79 68 33 (45/2352) (2 • 10 10 , 0.35) 87 42 30 (30/2352) (2 • 10 11 , 0.3) 50 33 31 (23/2352) Table 2 Iteration count vs. jump in the coefficients Nombre d'itérations en fonction du saut dans les coefficients

		Regular			Metis
	AS RBM NEW (V H )	AS RBM NEW (V H )
	4 sub 51 55	28 (22/1008)	56	59	24 (29/1040)
	8 sub 108 115 35 (46/2352)	121	124 31 (66/2410)
	16 sub 282 207 53 (94/5040)	260	239 36 (131/5146)
	32 sub 747 442 66 (190/10416) > 1000 537 41 (268/10706)
	Table 1


Julie Lions

Ian Gale

Erik Nygaard

Heike Rütters

Franz May

Stanley Beaubien

Mehran Sohrabi

Dimitrios G Hatzignatiou

Ludmilla Basava-Reddi

Potential impacts on groundwater resources of deep CO2 storage: natural analogues for assessing potential chemical effects

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Potential impacts on groundwater resources of deep CO 2 storage: natural analogues for assessing potential chemical effects Julie Lions (1*), Ian Gale (2*), Erik Nygaard (3*), Heike Rütters (4*), Franz May (4*), Stanley Beaubien (5*), Mehran Sohrabi (6*), Dimitrios G. Hatzignatiou (7*), Ludmilla Basava-Reddi (8), and the team of CO2GeoNet Members involved in the present study (1) BRGM, Water Division, Orléans, France (j.lions@brgm.fr), (2) British Geological Survey, UK, (3) Geological Survey of Denmark and Greenland, Denmark, (4) Bundesanstalt für Geowissenschaften und Rohstoffe, Germany, (5) Università Sapienza di Roma, Italy, (6) Heriot-Watt University, UK, (7) International Research Institute of Stavanger, Norway, (8) IEA Greenhouse Gas R&D Programme, UK * CO2GeoNet memberswww.co2geonet.eu; info@co2geonet.com.

Carbon dioxide Capture and Storage (CCS) is considered as one of the promising options for reducing atmospheric emissions of CO 2 related to human activities. One of the main concerns associated with the geological storage of CO 2 is that the CO 2 may leak from the intended storage formation, migrate to the near-surface environment and, eventually, escape from the ground. This is a concern because such leakage may affect aquifers overlying the storage site and containing freshwater that may be used for drinking, industry and agriculture.

The IEA Greenhouse Gas R&D Programme (IEAGHG) recently commissioned the CO2GeoNet Association to undertake a review of published and unpublished literature on this topic with the aim of summarizing 'state of the art' knowledge and identifying knowledge gaps and research priorities in this field. Work carried out by various CO 2 GeoNet members was also used in this study.

This study identifies possible areas of conflict by combining available datasets to map the global and regional superposition of deep saline formations (DSF) suitable for CO 2 storage and overlying fresh groundwater resources. A scenario classification is developed for the various geological settings where conflict could occur. The study proposes two approaches to address the potential impact mechanisms of CO 2 storage projects on the hydrodynamics and chemistry of shallow groundwater. The first classifies and synthesizes changes of water quality observed in natural/industrial analogues and in laboratory experiments. The second reviews hydrodynamic and geochemical models, including coupled multiphase flow and reactive transport. Various models are discussed in terms of their advantages and limitations, with conclusions on possible impacts on groundwater resources. Possible mitigation options to stop or control CO 2 leakage are assessed. The effect of CO 2 pressure in the host DSF and the potential effects on shallow aquifers are also examined. The study provides a review of CO 2 storage-specific regulations in the main countries undertaking CCS evaluation and research. It aims to identify the constraints imposed by existing regulations on the protection of groundwater resources and highlight the inconsistencies and gaps between CCS regulations and Water Protection regulations.

The present paper focuses specifically on potential risks on groundwater quality caused by CO 2 storage in DSF assessed via natural CO 2 analogues from both the literature and detailed European case studies (France, Italy, Germany, Denmark).


Frédéric Wertz

Philippe Blanc

Antonin Fabbri

Fabrizio Gherardi

Brgm Orléans

Emannuel Jobard

Jérôme Sterpenich

Jacques Pironon

Inpl Nancy

Geochemical Model of CO2 impact at the Well-Caprock-Reservoir interfaces

Increase due to neighbor dissolution Increase due to neighbor dissolution

Constant calcite increase limited by HCO 3 -diffusion

Straetlingite dissolves and precipitates into CSH gel and boehmite

Mechanisms Analysis -pH-Buffering and Volumes • H+ and HCO 3 -Diffusion is the limiting factor for the global cement degradation process > Limitations of the model:

• Cement carbonation produces a lot of H 2 O (not considered in the mass/volume balance)

• Gas phase not considered (some experiments show a stronger effect of wet CO 2 than acidified brine)

• No heterogeneities

>

  Modeling Geochemical interactions due to CO 2 injection at the Well-Reservoir-Caprock INTERFACE • Characterization of reactions pathways • Identification of new resulting minerals • Chemical stability of cement-caprock interface • Consequences in Porosity Evolution of the 3 zones > Injection Well cement alteration due to acid brine only -CO 2 Injection downhole => Reservoir Brine acidification > Diffusion only of HCO 3 -and H + (from CO 2aq ), no flow, no gas phase -Cement degradation due to brine contact -Cement leaching due to caprock contact > Code Used : ToughReact • (finite volumes) > 1m*2m-scale axisymmetric model • 2753 cells • ΔR = ΔZ = 2 mm at the interface • Δt = 9 sec for kinetic convergence

Frame•

  Possibility to extract simplified scheme to use it in a Poro-Mechanical study (part of the Project) > Sensibility Analysis done on the cement tortuosity • Variation of the Damkholer Number (here diffusion/kinetics)

Initial state, equilibrated water at t=0 and at the 2 Limit Conditions border-Cells (LC∞) Injection of Supercritical CO Results

  Ca 1.6 SiO 3.6 :2.58H 2 O : triple point cement alteration KatoiteSi1 Ca 3 Al 2 SiO 4 (OH) 8 Ettringite Ca 6 Al 2 (SO 4 ) 3 (OH) 12 :26H 2 O Hydrotalcite Mg 4 Al 2 O

	Volume
	Fraction
		Anhydrite
		CaSO 4	Ca 2 Al 2 SiO 2 (OH) 10 : Straetlingite	Magnesite MgCO 3
					2.5H 2 O
					14.0%
	Dolomite	CaMg(CO 3 ) 2		10.2%	11.0%
	Illite-Mg
		SiO 2			-	24.0%
					5.8%	6.0%
	Siderite	FeCO 3			4.1%	-
	Kaolinite	Al 2 Si 2 O 5 (OH) 4		2.0%	-
	Magnesite	MgCO 3			0.3%	-
	Pyrite	FeS 2			-	1.0%
	Chamosite Portlandite Celestite Ca(OH) 2	Fe 5 Al(AlSi 3 )O 10 (OH) 8 CaCO 3 SrSO 4 Calcite	--	1.0% 0.7%
					Composition
	Class_G Cement	Formula
					(% Vol)
	CSH_1.6 Portlandite CSH1.6 Ca 1.6 SiO 3.6 :2.58H 2 O	Ca(OH) 2			38.1% 26.1%	Dogger Reservoir
	Ettringite	Ca 6 Al 2 (SO 4 ) 3 (OH) 12 :26H 2 O CSH1.2	13.8%	Φ = 20%
	Katoite_Si1	Ca 3 Al 2 SiO 4 (OH) 8 Ca 1.2 SiO 3.2 :2.06H 2 O			10.5%	pH = 5
	C3FH6 Hydrotalcite Calcite	Ca 3 Fe 2 (OH) 12 Mg 4 Al 2 O 7 :10H 2 O CaCO 3	5.5% 5.4% 0.6% CSH0.8 Ferrihydrite Fe(OH) 3 Ca 0.8 SiO 2.8 :1.54H 2 O	Amorphous Silica, hydrated SiO 2 :0.5H 2 O

K 0.85 Mg 0.25 Al 2.35 Si 3.4 O 10 (OH) 2 0.1% 42.3% Quartz Mg-Montmorillonite-Na Na 0.33 Mg 3 Al 0.33 Si 3.67 O 10 (OH) 2 180 bar, 75°C, Salt = 25g/kg w 7 :10H 2 O C3FH6 Ca 3 Fe 2 (OH) 12 Al-Gel (Boehmite) AlO(OH)

Acknowledgements: This work has been supported by the French Research National Agency (ANR) through CO2 program (project INTERFACE, n° ANR-08-PCO2-006) Well -Reservoir -Cap rock INTERFACE


Emmanuel Vincent
email: emmanuel.vincent@inria.fr

Shoko Araki

Fabian Theis

Guido Nolte

Pau Bofill

Hiroshi Sawada

Alexey Ozerov

Vikrham Gowreesunker

Dominik Lutter

Ngoc Q K Duong

The Signal Separation Evaluation Campaign (2007-2010): Achievements and Remaining Challenges

Keywords: source separation, evaluation, audio, biomedical, resources

We present the outcomes of three recent evaluation campaigns in the field of audio and biomedical source separation. These campaigns have witnessed a boom in the range of applications of source separation systems in the last few years, as shown by the increasing number of datasets from 1 to 9 and the increasing number of submissions from 15 to 34. We first discuss their impact on the definition of a reference evaluation methodology, together with shared datasets and software. We then present the key results obtained over almost all datasets. We conclude by proposing directions for future research and evaluation, based in particular on the ideas raised during the related panel discussion at the Ninth International Conference on Latent Variable Analysis and Signal Separation (LVA/ICA 2010).

Introduction

In many areas of signal processing, e.g. telecommunication, chemistry, biology and audio, the observed signals result from the combination of several sources. Source separation is the general problem of characterizing the sources and estimating the source signals underlying a given mixture signal.

Early source separation techniques based on spatial filtering are now established: beamforming and time-frequency masking are employed in mobile phones and consumer audio systems to suppress environmental noise and enhance spatial rendering [START_REF]Microphone Arrays: Signal Processing Techniques and Applications[END_REF][START_REF]Blind speech separation[END_REF], while independent component analysis (ICA) is used for the extraction of specific signals from electroencephalogram (EEG), electrocardiogram (ECG) and functional magnetic resonance imaging (fMRI) data [START_REF]Handbook of Blind Source Separation, Independent Component Analysis and Applications[END_REF][START_REF] Albera | One decade of biomedical problems using ICA: a full comparative study[END_REF]. The emergence of more powerful source separation techniques in the last five years has led to a boom in the range of applications. Data that were thought as too difficult to separate can now be processed, as illustrated by companies providing commercial source separation services and software for real-world music data [START_REF] Jaureguiberry | Adaptation of source-specific dictionaries in non-negative matrix factorization for source separation[END_REF].

These advances have transformed source separation into a mainstream research topic, with dozens of new algorithms published every year 2 . Regular evaluation has become necessary to reveal the effects of different algorithm designs, specify a common evaluation methodology and promote new results in other research communities and in the industry. It is with these objectives in mind that several evaluation campaigns have been held in the last few years, including the 2007 Stereo Audio Source Separation Evaluation Campaign (SASSEC) [START_REF] Vincent | First Stereo Audio Source Separation Evaluation Campaign: Data, algorithms[END_REF] and the 2008 and 2010 Signal Separation Evaluation Campaigns (SiSEC) [START_REF] Vincent | The 2008 Signal Separation Evaluation Campaign: A community-based approach to large-scale evaluation[END_REF][START_REF] Araki | The 2010 Signal Separation Evaluation Campaign (SiSEC 2010): Audio source separation[END_REF][START_REF] Araki | The 2010 Signal Separation Evaluation Campaign (SiSEC 2010): Biomedical source separation[END_REF] run by the authors in conjunction with the 2007 and 2009 International Conferences on Independent Component Analysis and Signal Separation (ICA) and the 2010 International Conference on Latent Variable Analysis and Signal Separation (LVA/ICA).

While SASSEC was restricted to audio and fully specified by the organizers, the two SiSEC campaigns were open to all application areas and organized in a collaborative fashion. A few initial datasets, tasks and evaluation criteria were proposed by the organizers. Potential entrants were then invited to give their feedback and contribute additional specifications using collaborative software tools (wiki, mailing list). Although few people eventually took advantage of this opportunity, those who did contributed a large proportion of the evaluation materials. This resulted in an increasing number of datasets from 1 to 9 and an increasing number of submissions from 15 to 34 and in the extension of the campaign to the field of biomedical signal processing. The datasets and the corresponding number of submissions for each campaign are listed in Table 1. Detailed results are available from the websites of SASSEC3 and SiSEC 4 .

In this article, we uncover the general lessons learned from these three campaigns and outline the remaining challenges. Due to the nature of the datasets, we focus on audio and to a small extent on biomedical data. The structure of the rest of the article is as follows. In Section 2, we describe the reference evaluation methodology, including shared datasets and software. In Section 3, we present the key results obtained over almost all datasets. We conclude in Section 4 by proposing directions for future research and evaluation, based in particular on the ideas raised during the related panel discussion at LVA/ICA 2010.

Reference evaluation methodology and resources

The most important outcome of SASSEC and SiSEC is perhaps the definition of a reference methodology for the evaluation of source separation systems. In particular, it has been clarified that the general problem of source separation refers to several tasks that were not always distinguished in the past. The evaluation of a source separation system requires four ingredients that we describe in the following:

• a dataset,
• a task to be addressed,

• one or more evaluation criteria,

• ideally, one or more performance bounds.

Development datasets and evaluation software are available from the SiSEC website 4 . Readers are encouraged to use these resources for the evaluation of their own systems, in order to obtain performance figures that are both reproducible and comparable with the state of the art established by SiSEC.

Datasets

The datasets in Table 1 belong to two categories:

• application-oriented datasets,

• diagnosis-oriented datasets.

The application-oriented datasets "Professionally produced music recordings" and "Cancer microarray gene expression profiles" consist of real-world signals, in which all the challenges underlying source separation are faced at once. The other datasets were built artificially so as to face as few challenges as possible at a time. These challenges include under-determination, i.e. when the number of sources is larger than the number of mixture channels, and convolutive mixing, i.e. when the mixing process involves nontrivial filters as opposed to gains or pure delays. Both categories of datasets are needed: application-oriented datasets help assessing the remaining performance gap towards industrial applications, while diagnosis-oriented datasets help improving performance and robustness by combining the best solutions to individual challenges.

The characteristics of the mixtures within diagnosis-oriented datasets must be controlled as well as possible in order to quantify their difficulty. The main characteristics and the corresponding parameters to be specified are listed in Table 2. The SiSEC diagnosis-oriented audio datasets typically involve 2 to 5 different settings for each parameter of interest and as many multichannel test signals for each setting, so as to evaluate the effect of each setting on separation performance while favoring narrow confidence intervals on the average performance for each setting. This resulted in a total number of 27 to 84 test signals per dataset. By contrast, the number of test signals was limited to 5 for the application-oriented audio dataset and to a single test signal for the two biomedical datasets, for which the collection of ground truth data is notoriously harder.

Tasks and ground truth

For any data, the mixing process can always be formulated as follows [START_REF] Cardoso | Multidimensional independent component analysis[END_REF]. Denoting by J and I the number of sources and channels, each channel x i (t), 1 ≤ i ≤ I, of the mixture signal can be expressed as

x i (t) = J j=1 s img ij (t) (1)
where s img ij (t) is the spatial image of source j, 1 ≤ j ≤ J, on channel i, that is the contribution of this source to the observed mixture in this channel. This formulation does not make any assumption on the sources, e.g. several distant sound sources may be considered as a single background noise source.

Under the assumption that source j is a point source emitting in a single spatial location, its spatial image can be further decomposed as

s img ij (t) = τ a ij (t -τ, τ )s j (t -τ ) (2)
where s j (t) is a single-channel source signal and a ij (t, τ ) the time-varying mixing filter from source j to channel i. In the case of audio, this assumption is typically valid for speakers and small musical instruments, but not for large instruments (piano, drums) and diffuse background noise. The estimation of the mixing filters often relies on the localization of the source, expressed by its Direction-of-Arrival (DoA) θ j (t). Finally, in many applications, one is not interested in the source signals or the source spatial image signals themselves but in some of their features F(s j ) or F(s img ij ). Example features include cepstral features and speech transcription in the context of noisy automatic speech recognition or the indices t of the nonzero source coefficients corresponding to active genes in the context of microarray data analysis [START_REF] Vincent | A tentative typology of audio source separation tasks[END_REF][START_REF] Schachtner | Knowledge-based gene expression classification via matrix factorization[END_REF].

Based on the above formulation, the problem of source separation has been decomposed into six tasks listed in Table 3: source counting, source spatial image estimation 5 and source feature extraction, which always make sense, and source localization, mixing system estimation and source signal estimation, which make sense for point sources only. Each task corresponds to a distinct quantity to be estimated.

Evaluation consists of comparing the estimated quantity with the ground truth according to one or more criteria. The way to obtain the ground truth data depends whether the dataset consists of synthetic or recorded mixtures. Ground truth data are typically available for all tasks in the former case but not in the latter. One popular technique for the acquisition of ground truth data for live audio recordings consists of separately recording each source in turn, thus yielding ground truth source spatial image signals, and summing them to obtain the mixture signal [START_REF] Schobben | Evaluation of blind signal separation methods[END_REF]. This approach cannot be used for real-world biomedical datasets, for which the sources cannot be switched off. When feasible, the ground truth is then specified by experts.

Evaluation criteria

The evaluation criteria for source counting and source localization are straightforward.

Regarding the evaluation of mixing system estimation, several established criteria such as the Amari Performance Index (PI) or the Inter-Symbol Interference (ISI) have been proposed for over-determined mixing systems [START_REF] Mansour | A survey of the performance indexes of ICA algorithms[END_REF][START_REF] Lambert | Difficulty measures and figures of merit for source separation[END_REF] and widely applied to biomedical data. A more general Mixing Error Ratio (MER) criterion applicable to all mixing systems has been introduced in [START_REF] Vincent | The 2008 Signal Separation Evaluation Campaign: A community-based approach to large-scale evaluation[END_REF]. For instantaneous mixtures, the estimated mixing gains a ij for a given source j are decomposed as

a ij = a coll ij + a orth ij (3)
where a coll ij and a orth ij are respectively collinear and orthogonal to the true vector of mixing gains a ij , 1 ≤ i ≤ I, and are computed by least squares projection. Accuracy is then assessed via the following ratio in decibels (dB)

MER j = 10 log 10 I i=1 a coll 2 ij I i=1 a orth 2 ij . (4)
More generally, for time-varying convolutive mixtures, the accuracy of estimated mixing filters for source j is assessed by computing the MER in each frequency bin ν between a ij (t, ν) and a ij (t, ν) and averaging it over frequency and time. When the sources are estimated in arbitrary order, the order is selected that leads to the largest average MER. This criterion has been little used so far, however, so that general agreed-upon criteria remain to be found. Several evaluation criteria have also been proposed for source signal estimation and source spatial image estimation. Early criteria applied to biomedical data or toy audio data [START_REF] Schobben | Evaluation of blind signal separation methods[END_REF][START_REF] Mansour | A survey of the performance indexes of ICA algorithms[END_REF][START_REF] Yılmaz | Blind separation of speech mixtures via timefrequency masking[END_REF] were restricted to linear unmixing or binary time-frequency masking and required knowledge of the unmixing filters or the time-frequency masks. More recently, a family of criteria has been proposed that applies to all mixtures and algorithms [START_REF] Vincent | Performance measurement in blind audio source separation[END_REF][START_REF] Vincent | First Stereo Audio Source Separation Evaluation Campaign: Data, algorithms[END_REF]. In the case of source spatial image estimation, the criteria derive from the decomposition of an estimated source image ŝimg ij (t) as [START_REF] Vincent | First Stereo Audio Source Separation Evaluation Campaign: Data, algorithms[END_REF] ŝimg

ij (t) = s img ij (t) + e spat ij (t) + e interf ij (t) + e artif ij (t) (5)
where s img ij (t) is the true source image and e spat ij (t), e interf ij (t) and e artif ij (t) are distinct error components representing spatial (or filtering) distortion, interference and artifacts. This decomposition is motivated by the distinction between signal from the target source, residual noise from the other sources and extraneous noise introduced by the algorithm 6 , corresponding to the signals s img ij (t) + e spat ij (t), e interf ij (t) and e artif ij (t) respectively. Spatial distortion and interference components are expressed as filtered versions of the true source images, computed by least-squares projection of the estimated source image onto the corresponding signal subspaces

e spat ij (t) = P L j (ŝ img ij )(t) -s img ij (t) (6)
e interf ij (t) = P L all (ŝ img ij )(t) -P L j (ŝ img ij )(t) (7)
e artif ij (t) = ŝimg ij (t) -P L all (ŝ img ij )(t) (8)
where P L j is the least-squares projector onto the subspace spanned by

s img kj (t- τ ), 1 ≤ k ≤ I, 0 ≤ τ ≤ L -1, P L
all is the least-squares projector onto the subspace spanned by

s img kl (t -τ ), 1 ≤ k ≤ I, 1 ≤ l ≤ J, 0 ≤ τ ≤ L -1.
The length L of the distortion filter is equal to 1 tap for EEG, ECG or fMRI and is typically set to 32 ms in an audio context. The amount of spatial distortion, interference and artifacts is then measured by three energy ratios expressed in decibels (dB): the source Image to Spatial distortion Ratio (ISR), the Signal to Interference Ratio (SIR) and the Signal to Artifacts Ratio (SAR) ISR j = 10 log 10

I i=1 t s img ij (t) 2 I i=1 t e spat ij (t) 2 (9)
SIR j = 10 log 10

I i=1 t (s img ij (t) + e spat ij (t)) 2 I i=1 t e interf ij (t) 2 (10)
SAR j = 10 log 10

I i=1 t (s img ij (t) + e spat ij (t) + e interf ij (t)) 2 I i=1 t e artif ij (t) 2 . ( 11
)
The total error is also measured by the Signal to Distortion Ratio (SDR) SDR j = 10 log 10

I i=1 t s img ij (t) 2 I i=1 t (e spat ij (t) + e interf ij (t) + e artif ij (t)) 2 (12)
In the case of source signal estimation, similar criteria can be defined by grouping the first two terms in (5) [START_REF] Vincent | Performance measurement in blind audio source separation[END_REF]. Indeed, the source signals can only be estimated up to arbitrary filtering, which should not be taken into account in the SDR. Similarly, when the sources are estimated in arbitrary order, the order is selected that leads to the largest average SIR. In the specific case of audio, improved auditory-motivated variants of these criteria termed Target-related Perceptual Score (TPS), Interference-related Perceptual Score (IPS), Artifact-related Perceptual Score (APS) and Overall Perceptual Score (OPS) have also been employed [START_REF] Emiya | Subjective and objective quality assessment of audio source separation[END_REF]. Finally, the evaluation criteria related to source feature extraction are highly specific to the considered features. For example, noisy automatic speech recognition may be evaluated in terms of Word Error Rate (WER) while the detection of the indices t of the nonzero source coefficients in the context of microarray data analysis may be evaluated by counting the number of significantly detected indices using appropriate statistical tests, as detailed in Section 3.2.

Baseline algorithms and performance bounds

In addition to quantifying the performance of the source separation sysunder test, it is recommended to evaluate some reference algorithms via the same criteria. Indeed, the performance of all systems varies a lot depending on the mixture signal, so that the difference of performance with respect to reference algorithms often provides a more robust indicator. Two categories of reference algorithms have been considered in SiSEC: baseline algorithms providing medium to poor performance and oracle estimators providing theoretical upper bounds on performance. A range of oracle estimators were defined in [START_REF] Vincent | Oracle estimators for the benchmarking of source separation algorithms[END_REF][START_REF] Wang | On ideal binary mask as the computational goal of auditory scene analysis[END_REF] for linear unmixing-based and time-frequency masking-based algorithms.

Key results

Audio source separation

The audio datasets of SASSEC and SiSEC attracted a total of 79 submissions, from which many useful conclusions can be drawn. We let readers refer to [START_REF] Vincent | First Stereo Audio Source Separation Evaluation Campaign: Data, algorithms[END_REF][START_REF] Vincent | The 2008 Signal Separation Evaluation Campaign: A community-based approach to large-scale evaluation[END_REF][START_REF] Araki | The 2010 Signal Separation Evaluation Campaign (SiSEC 2010): Audio source separation[END_REF] for the detailed performance of each system as a function of the mixture characteristics, and provide here a broader perspective over the field by focusing on the best systems on average. Furthermore, we concentrate on the source signal estimation and source spatial image estimation tasks, which are the only ones for which sufficient submissions are available.

Evolution of performance over the "Under-determined speech and music mixtures" dataset

We first analyze the evolution of performance over the only dataset that was considered within the three campaigns, that is the "Under-determined speech and music mixtures" dataset. Due to the evolution of the dataset itself, the source signals were different in SASSEC and SiSEC. In order to compare the results, we consider the same categories of mixtures in both cases, that is two 2-channel mixtures of 4 speech sources and two 2-channel mixtures of 3 music sources mixed in three different ways: instantaneous mixing, live recording with 250 ms reverberation time and 5 cm microphone spacing, and live recording with 250 ms reverberation time and 1 m microphone spacing. For each campaign and each mixing condition, we select the system leading to best average SDR over all sources and all mixtures 7 .

The resulting average SDR, ISR, SIR and SAR are reported in Table 4. The following observations can be made:

• The separation of instantaneous mixtures is close to be solved in 2010, with an average SDR of 14 dB, while that of live recordings remains more difficult, with an average SDR of 3 dB. • All performance criteria improved by 3 to 4 dB on instantaneous mixtures when replacing the Sparse Component Analysis (SCA) method in [START_REF] Vincent | Complex nonconvex l p norm minimization for underdetermined source separation[END_REF] by multichannel Nonnegative Matrix Factorization (NMF) [START_REF] Ozerov | Multichannel nonnegative matrix factorization in convolutive mixtures for audio source separation[END_REF] or by the flexible probabilistic modeling framework in [START_REF] Ozerov | A general flexible framework for the handling of prior information in audio source separation[END_REF]. These new methods are examples of the emerging variance modeling framework [START_REF] Vincent | Probabilistic modeling paradigms for audio source separation[END_REF] for audio source separation. This framework addresses some shortcomings of the conventional linear modeling framework [START_REF]Blind speech separation[END_REF] underlying ICA and SCA by enabling the exploitation of additional prior information about the source spectra.

• These new methods remain inferior to conventional SCA on live recordings, however, perhaps because of the omnipresence of local optima in the objective function and the need for more accurate initialization. The best current SDR on these live recordings [START_REF] Sawada | Underdetermined convolutive blind source separation via frequency bin-wise clustering and permutation alignment[END_REF] remains 6 dB below that of the binary masking oracle [START_REF] Vincent | Oracle estimators for the benchmarking of source separation algorithms[END_REF], which indicates that room is left for progress.

Current performance on the other audio datasets

In addition to the above dataset which was used for all campaigns, two datasets, namely "Professionally produced music recordings" and "Determined and over-determined mixtures", were used for the last two campaigns. The corresponding results do not reveal any performance increase, however, but a performance decrease instead, due to the fact that different methods were submitted in 2008 and 2010.

The current performance on these two datasets and on the remaining audio datasets is shown in Table 5. For each dataset, we select the method providing the best average SDR over all sources of all mixtures, except for the "Determined and over-determined mixtures" dataset for which we consider the SIR instead8 , and for the "Head-mounted microphone recordings" datasets for which the best method separated only two sources out of three. The following observations can be made:

• Not surprisingly, the best separation is achieved on noiseless overdetermined mixtures, with an average SIR of 14 dB for 5-channel recordings of 3 sources and 11 dB for 4-channel recordings of 2 sources.

The corresponding methods both rely on frequency-domain ICA, where the source signals estimated within each frequency bin are ordered based either on their spatial location [START_REF] Nesta | Multiple TDOA estimation by using a state coherence transform for solving the permutation problem in frequency-domain BSS[END_REF] or on the correlation of their temporal activity patterns [START_REF] Sawada | Measuring dependence of bin-wise separated signals for permutation alignment in frequency-domain BSS[END_REF].

• Similar performance is achieved over 2-channel noiseless mixtures of 2 sources, again by means of frequency-domain ICA [START_REF] Nesta | Convolutive BSS of short mixtures by ICA recursively regularized across frequencies[END_REF]. Note that the considered 2-channel 2-source mixtures were either short or dynamic, which shows that frequency-domain ICA can efficiently adapt to such situations [START_REF] Nesta | Convolutive BSS of short mixtures by ICA recursively regularized across frequencies[END_REF]. These methods result in significant filtering distortion of the source signals, however, as indicated by the lower SAR.

• Performance drops on 4-channel mixtures of 4 sources, for which the best 4-channel 2-source separation method [START_REF] Nesta | Multiple TDOA estimation by using a state coherence transform for solving the permutation problem in frequency-domain BSS[END_REF] achieves a SIR of 3 dB only, and on professionally produced music recordings, for which the best method [START_REF] Arberet | A tractable framework for estimating and combining spectral source models for audio source separation[END_REF] based on the aforementioned variance modeling framework provided a SIR of 9 dB. This suggests that performance does not depend so much whether the mixture is determined or overdetermined but rather on the number of sources itself, since a larger number of sources makes it more difficult to achieve accurate source localization, which is a prerequisite in most source separation methods.

• The presence of background noise appears even more detrimental. Indeed, the SIR decreases by 8 dB when replacing one of the sources within a 2-channel 2-source mixture by diffuse background noise, yielding a SIR as low as 4 dB. This appears due to the lack of accurate noise models, despite recent advances in this direction in [START_REF] Duong | Under-determined reverberant audio source separation using local observed covariance and auditory-motivated time-frequency representation[END_REF].

Finally, it must be emphasized that none of the above methods is truly blind. All methods assume prior knowledge of the number of sources and the category of mixing (instantaneous vs convolutive), and most submissions to the "Professionally produced music recordings" dataset even relied on manual parameter fixing or manual grouping of the sounds composing each source.

Biomedical source separation

Fewer conclusions can be drawn from the biomedical source separation results in SiSEC 2010, due to the smaller number of submissions. We summarize here the results obtained over the microarray gene expression dataset.

In this context, each channel x i (t) of the mixture signal, called expression profile, measures the level of messenger ribonucleic acid (mRNA) corresponding to one gene t within one subject or experimental condition i. The expression profiles can be regarded as a linear instantaneous mixture of several cell signaling pathways or more generally biological processes [START_REF] Liebermeister | Linear modes of gene expression determined by independent component analysis[END_REF][START_REF] Lee | Application of independent component analysis to microarrays[END_REF]. Using source separation techniques, the estimated source signals can be interpreted as patterns reflecting active signaling pathways. In SiSEC mRNA was extracted from I = 189 invasive breast carcinomas, measured using Affymetrix U133A gene-chips and normalized via the robust multiarray average (RMA) algorithm. Non-expressed genes were filtered out, resulting in a total of T = 11815 expressed genes [START_REF] Araki | The 2010 Signal Separation Evaluation Campaign (SiSEC 2010): Biomedical source separation[END_REF]. The J = 10 ground truth signaling pathways were approximated as simple gene lists, taken from NETPATH 9 . The quality of the estimated pathways was evaluated by means of statistical tests [START_REF] Araki | The 2010 Signal Separation Evaluation Campaign (SiSEC 2010): Biomedical source separation[END_REF]. More precisely, for each source signal, the genes mapping to the distinct pathways were identified and p-values were calculated using Fisher's exact test. The Benjamini-Hochberg procedure was then used to correct for multiple testing and an estimated pathway was declared as enriched if its p-value was below 0.05. Finally, the total number of distinct enriched pathways was counted.

Two methods were submitted that both rely on some form of prior information, implemented either via matrix factorization using a graph model (GraDe) [START_REF] Blöchl | Second-order source separation based on prior knowledge realized in a graph model[END_REF] or via Network Component Analysis (NCA) [START_REF] Chen | Transcription factor activity estimation based on particle swarm optimization and fast network component analysis[END_REF]. For each of the 10 ground truth pathways, both methods found at least one matching pathway with a p-value below 0.05 according to Fisher's exact test. After discarding duplicate pathways, the number of correctly estimated pathways reduced to 7 and 5, respectively. Finally, after Benjamini-Hochberg correction, the number of enriched pathways was equal to 5 and 0, respectively. This shows that the GraDe approach clearly outperformed the NCA approach. We hypothesize that the better performance of GraDe arises from the inclusion of pathway information within the graph model.

Remaining challenges

To sum up, SASSEC and SiSEC have been instrumental in the definition of a clear evaluation methodology for audio and biomedical source separation and in the creation of data and software resources. The results support the emergence of source separation systems exploiting advanced source models accounting for the source spectra in the case of audio source separation [START_REF] Ozerov | Multichannel nonnegative matrix factorization in convolutive mixtures for audio source separation[END_REF][START_REF] Ozerov | A general flexible framework for the handling of prior information in audio source separation[END_REF][START_REF] Vincent | Probabilistic modeling paradigms for audio source separation[END_REF][START_REF] Arberet | A tractable framework for estimating and combining spectral source models for audio source separation[END_REF] or for signaling pathway information in the case of biomedical source separation [START_REF] Blöchl | Second-order source separation based on prior knowledge realized in a graph model[END_REF]. Nevertheless, more conventional methods based on frequency-domain ICA or SCA still perform best on live audio recordings of many sources and/or background noise [START_REF] Sawada | Underdetermined convolutive blind source separation via frequency bin-wise clustering and permutation alignment[END_REF][START_REF] Nesta | Multiple TDOA estimation by using a state coherence transform for solving the permutation problem in frequency-domain BSS[END_REF][START_REF] Sawada | Measuring dependence of bin-wise separated signals for permutation alignment in frequency-domain BSS[END_REF][START_REF] Nesta | Convolutive BSS of short mixtures by ICA recursively regularized across frequencies[END_REF].

Evaluation methodology

The biggest challenge regarding evaluation methodology consists of extending the methodology summarized in this article to other datasets, tasks and application domains. Up to 2010, SASSEC and SiSEC have mainly focused on audio source signal estimation and source spatial image estimation, which are perhaps not the most useful tasks in the real world, and left the other audio tasks [START_REF] Vincent | A tentative typology of audio source separation tasks[END_REF] aside. Recently, a comprehensive dataset has been created for the evaluation of audio source separation systems in terms of WER in the context of noise-robust speech recognition in a domestic environment [START_REF] Cooke | Monaural speech separation and recognition challenge[END_REF][START_REF] Christensen | The CHiME corpus: a resource and a challenge for computational hearing in multisource environments[END_REF]. Stereo to multichannel upmix [START_REF] Avendano | A frequency-domain approach to multichannel upmix[END_REF] is also a vibrant area of research to which advanced source separation systems could contribute and for which novel performance criteria are needed. Appropriate statistical confidence measures, tighter oracle performance bounds and advanced diagnosis procedures such as those in [START_REF] Conover | Practical Non-Parametric Statistics[END_REF][START_REF] Emiya | An investigation of discrete-state discriminant approaches to single-sensor source separation[END_REF][START_REF] Mandel | Evaluating source separation algorithms with reverberant speech[END_REF] are also needed to increase the insight that can be gained from evaluation. Finally, increased publicity and networking efforts should be made to promote source separation evaluations in the biomedical signal processing community, as well as in other communities, e.g. cosmology or telecommunications, where the proposed tasks and evaluation criteria might also apply. As the first trial in this direction, the biomedical part of SiSEC 2010 clearly had a limited scope.

Key challenges for future research

In addition to these methodological challenges, we identified three key challenges for future research in audio and biomedical source separation in light of the campaign results:

• the experimentation of advanced source models and mixing models including as much available information as possible, especially for complex sources such as nonstationary background noise or taking into account the wealth of prior information readily available in the biomedical context,

• the design of accurate source localization methods, which are required for parameter initialization of the mixing model, especially for short and/or dynamic mixtures,

• the development of model selection techniques enabling truly blind separation by automatically finding the number of sources and adapting the source models and the mixing model to the mixture at hand.

Although recent advances have been made in each of these directions [START_REF] Ozerov | A general flexible framework for the handling of prior information in audio source separation[END_REF][START_REF] Blöchl | Second-order source separation based on prior knowledge realized in a graph model[END_REF][START_REF] Nesta | Multiple TDOA estimation by using a state coherence transform for solving the permutation problem in frequency-domain BSS[END_REF][START_REF] Tan | Automatic relevance determination in nonnegative matrix factorization[END_REF], they remain to be fully developed, combined together and validated on real-world data.

Table 1 :

 1

		Number of submissions
	Datasets	SASSEC SiSEC SiSEC
		2007	2008	2010
	Audio
	Under-determined speech and music mixtures	15	15	6
	Professionally produced music recordings		9	3
	Determined and over-determined mixtures		6	4
	Head-mounted microphone recordings		3	2
	Short two-source two-microphone recordings			7
	Mixed speech and real-world background noise			6
	Determined mixtures under dynamic conditions			3
	Biomedical
	Cancer microarray gene expression profiles			2
	EEG data with dependent components			1

Number of submissions associated to each dataset for each of the considered evaluation campaigns. The "Head-mounted microphone recordings" dataset consisted of distinct but conceptually similar recordings in 2008 and 2010, called "Head-geometry mixtures of two speech sources in real environments" and "Over-determined speech and music mixtures for human-robot interaction" respectively.

Table 2 :

 2 Main specifications of a diagnosis-oriented dataset.

Table 3 :

 3 Main tasks, ground truth and evaluation criteria. See Section 2.2 for the notations and Section 2.3 for the acronyms.

Table 4 :

 4 Evolution of the average performance of the best source spatial image estimation method over the "Under-determined speech and music mixtures" dataset compared to that of the binary masking oracle.

Table 5 :

 5 Average performance of the best source separation method over all audio datasets except the "Under-determined speech and music mixtures" dataset. Figures relate to the source spatial image estimation task when the ISR is reported and to the source signal estimation task otherwise.

According to the Google Scholar search engine, the annual number of publications containing the words "audio" and "source separation" has roughly doubled every 2 years from 14 in 1994 to 903 in 2007 and has stalled around 900 since then. The annual number of publications containing the words "microarray" and "source separation" has roughly doubled every 4 years from 14 in 2002 to 76 in 2010 and is still increasing.

http://sassec.gforge.inria.fr/

http://sisec.wiki.irisa.fr/

The task of estimating the subspace spanned by certain point sources, which was specified for the EEG dataset in SiSEC 2010, is formally equivalent to the estimation of the spatial image of these sources considered as a single diffuse source[START_REF] Cardoso | Multidimensional independent component analysis[END_REF].

In the context of audio, this extraneous noise is called "musical noise".

The choice of the best system depends on the eventual application scenario, since different applications may involve different mixture characteristics and different evaluation criteria. Our choice promotes versatile algorithms that were able to separate all sources within all mixtures of the dataset.

Due to the unavailability of the ground truth source signals in this dataset, the results of the source signal estimation task were evaluated with respect to the first channel of the spatial image of each source instead. Only the SIR criterion then makes sense according to the specification of the task in Section 2.2.

http://www.netpath.org

Acknowledgments

We would like to thank all the entrants and all the persons besides the authors who helped organizing SiSEC 2008 and 2010 by contributing datasets, code or part of their time (in alphabetical order): J. Anemüller, M. Durkovic, M. Dyrholm, V. Emiya, K. E. Hild II, N. Ito, H. Kayser, M. Kleinsteuber, Z. Koldovsky, O. Le Blouch, B. Lösch, F. Nesta, L. C. Parra, M. Rothbucher, H. Shen, P. Tichavsky, M. Vinyes Raso and J. Woodruff.

 [START_REF]Microphone Arrays: Signal Processing Techniques and Applications[END_REF]Alexey Ozerov was supported by the Quaero Programme, funded by OSEO.


est destinée au dépôt

This technical report is deprecated. Please refer to the following paper instead: http://hal.inria.fr/inria-00630985/


Margaux Lhommet
email: margaux.lhommet@hds.utc.fr

Domitile Lourdeaux
email: domitile.lourdeaux@hds.utc.fr

Taking into account emotions in mixed human/agent systems for improving collaborative work

Keywords: component, multi-agent systems, cognitive agent, emotion modeling I

Interacting with other persons implies having appropriate reactions to their actions. In the professional field, inappropriate responses can be very prejudicial. Many studies point out the importance of emotions in the construct of behavioral responses. This article proposes a model of emotional, social and cognitive Personal Assistant agent that can help its user to avoid conflicts.

INTRODUCTION

Transferring knowledge results from a number of exchanges among a large number of persons involved in collaborative work. The behavior of each individual is influenced by the behavior of the others. If we assume that the process is mediated by a system of "intelligent" agents, then it would be interesting to take into account the various emotional states of the different actors.

Until now, doing so has not been considered mainly because modeling and handling emotions is a difficult problem and there was a lack of models for doing so. This paper presents a way of implementing emotional, social and cognitive agents, which could be an interesting addition in the context of multicultural interactions.

II. MODELS OF EMOTION

A. Emotional contagion

The construct of group emotion and shared emotion among group members has been broadened in recent years. Barsade [START_REF] Barsade | The ripple effect: Emotional contagion in groups[END_REF] demonstrates the existence of emotional contagion among collaborative workers. She formulates some hypotheses that she validates with experiments: (1) There will be contagion of mood among group members. [START_REF] Le | Psychologie des foules[END_REF] Unpleasant emotions are more likely to lead to mood contagion than are pleasant emotions. (3) Positive emotional contagion will lead to greater cooperativeness, (4) to less group conflict, (5) and people will rate their own performance and that of others more highly.

From a broader point of view, emotional contagion between individuals is a long-time known phenomenon. In the first study concerning crowd and group behaviors published in 1895, Le Bon [START_REF] Le | Psychologie des foules[END_REF] (page 13) writes: "Contagion is a phenomenon of which it is easy to establish the presence, but that it is not easy to explain. [...] In a crowd every sentiment and act is contagious, and contagious to such a degree that an individual readily sacrifices his personal interest to the collective interest."

Since then, several psychologist and sociologist studies have been focused of this phenomenon. In 1992, Hatfield et al. [START_REF] Hatfield | Primitive emotional contagion[END_REF] (page 153) define emotional contagion as "the tendency to catch (experience/express) another person's emotions (his or her emotional appraisals, subjective feelings, expressions, patterned physiological processes, action tendencies, and instrumental behaviors)."

B. Emotions

To date, there has been little agreement on what is an emotion. Most theorists would agree that emotion is a mental experience that motivates a reaction toward the initial stimuli. Usually, emotions are defined by common terms such as Joy, Distress, Fear, Hope, Pride... Yet, we are here less interested in what emotions are than how they affect people's behavior. Since each person in the group is represented by her/his Personal Assistant (PA), emotions must be represented in a way that allows the PA to reason. Thus we focus on computational models of emotions.

The OCC emotional theory described by Ortony et al [START_REF] Ortony | The cognitive structure of emotions[END_REF] specifies how events, other agent's actions and objects are confronted to the agent's desires, to its moral and cultural norms and to its preferences to generate an emotional state. It is widely used in the field of emotional agents because it determines the cognitive appraisal process in a straightforward way that favors implementation.

Lazarus [START_REF] Lazarus | Emotion and adaptation[END_REF] considers a two-steps emotional theory. The first step (cognitive appraisal) computes the agent's capacity to deal with the consequences of an event. In the second step (coping), if the event is stressful, the agent chooses a way to react to improve its situation. This reaction can whether be an action in the environment (problem-coping) or a new evaluation of its mental state (emotional-coping) such as denial or culpability. This theory is very complete and therefore, requires extensive computation. For this reason, it is very complex to implement and rarely used.

C. Computational model of emotional contagion

Many studies investigating the contagion phenomenon have been carried out by biologist (such as contagious disease) and social scientists (such as rumors propagation and buyers or political tendencies). Dodds and Watts [START_REF] Dodds | A generalized model of social and biological contagion[END_REF] identify two groups of contagion models. Those of independent interaction, used by mathematicians to represent infectious diseases' propagation, do not consider individual characteristics nor repeated expositions. Thresholds models, used in social contagion, define conditions that have to be fulfilled for the contagion to occur. Finally, all those models classify individuals in two groups depending on the fact that an individual is contaminated (and then contagious) or not.

Those studies are not compatible with the emotionalcontagion definition given by psychologists and sociologists: in the case of emotions, individual characteristics have a major importance on contagion and an individual is at each moment contaminated and contagious.

III. AGENT MODEL

Each individual interacts with the group via her/his PA. This PA is a cognitive, emotive and social autonomous agent that communicates and collaborates with the other workers' PA.

Figure 1 represents the global architecture of the PA.

A. Personality

Since the PA represents an individual, it must know how its user may react to certain events. Thus each PA has a personality model that represents the one of its user.

The two most important personality models are the threefactor model by Eysenck [START_REF] Eysenck | Biological dimensions of personality[END_REF] and the five-factor model (OCEAN) by Costa and McRae [START_REF] Costa | et Inc Psychological Assessment Resources, Revised NEO Personality Inventory (NEO PI-R) and Neo Five-Factor Inventory (NEO-FFI)[END_REF].

There are many questionnaires that allow to determine an user's personality in terms of OCEAN traits. This personality is static. Person A's personality is represented by the vector P(A) (1).

(1)

B. Relationship

Collaborative working requires people to respect certain behavioral rules such as hierarchy respect, social norms and cultural standards. PA thus needs to have knowledge about the social relationship between its user and other co-workers. This relationship may concern affinity, hierarchy links, social or cultural norms. PA uses this relationship to determine supposed impacts of other worker's actions on its user. By way of illustration, if the agent A sees the agent B doing something it judges blameworthy (like being insolent to a superior), agent A's appreciation of agent B will be negatively impacted (as should be the supposed relationship between the two users).

The relationship of agent A toward agent B at time t is represented by (2).

(2)

C. Emotions and mood

This paper uses the OCC emotional theory [START_REF] Ortony | The cognitive structure of emotions[END_REF] that specifies how events, other agent actions and objects are confronted to the agent's goals (desires), standards (moral acceptability of actions) and attitudes (appealing of objects) to generate emotions.

While OCC theory defines 22 emotions (11 positive emotions and their opposite), our model only considers two which are Joy and its opposite Distress. We then define the emotional state Well-Being that represents the aggregation of those two opposite emotions; a negative value represents Distress whereas a positive value represents Joy.

Moods represent the current emotional state of the PA (the user's supposed emotional state). It is an aggregation over time of the successive emotional states of the PA.

IV. EMOTIONAL MODULE

The emotional module is defined by two steps, the Event Appraisal and the Action Appraisal. They correspond to the OCC "Reactions to Events" and "Reactions to Agents " that respectively evaluates the impact of events and the impact of actions on the emotional state of the agent.

We draw a sharp distinction between the appraisal of events (Event Appraisal) and the appraisal of actions realized by other agents (Action Appraisal). In the first case, the module evaluates the impact of the event on the emotional state of the agent. In the second case, it evaluates the impact of the action on the relationship between the perceiver and the author of the action. Thus, an action is first evaluated by the Event Appraisal module to update the emotional state of the agent, then by the Action Appraisal module to update the relationship.

A. Event Appraisal

The OCC theory describes the processes used to appraise an event and issue an emotional state regarding to the agent's goals.

First, the desirability of the event is calculated. An event is said to be desirable (respectively undesirable) if it permits to increase (respectively decrease) the degree of achievement of one (or many) desire(s) of the agent.

Then, an emotion potential is computed for this event regarding its desirability (or undesirability).

It is not rare that many events occur at the same time, so those emotion potential are aggregated into a global emotional state that represents the instantaneous emotional state. Yet, the OCC theory provides no guidance about how to obtain a global emotional state from a set of potential emotions.

Egges [START_REF] Egges | Generic personality and emotion simulation for conversational agents[END_REF] proposes a model based on matrices that represent the influence of personality traits on the aggregation of potential emotions into an emotional state.

Finally, mood is updated regarding its previous value and the new emotional state.

B. Action Appraisal

Action Appraisal module evaluates an action done by another agent. The OCC theory [START_REF] Ortony | The cognitive structure of emotions[END_REF] (chapter 7) uses this evaluation to generate attribution emotions to the author of the action (such as anger or admiration) or to the perceiver itself (such as pride or shame). Since our model only defines Well-Being emotions, we directly use this appraisal to update the relationship between the perceiver and the actor.

Depending on the personality of the perceiver, the relationship may be impacted in different ways. For instance, an intransigent agent (with a low agreeableness value) will be less tolerant if an agent does an action its disapproves and the relationship between those two agents will be negatively impacted. In order to do so, we define a matrix that represents the impact of the personality on the evolution of the relationship.

C. Emotional contagion

So far, there is no emotional contagion model taking into account individual characteristics. However, researches and experiments have identified some individual differences that may influence emotional contagion. Hatfield et al. [START_REF] Hatfield | Emotional contagion[END_REF] point out that emotional contagion occurs between every individual, each being contaminated and contagious as the same time. For each individual, they also describe factors that influence the ability to infect others (Contagion Power or CP) and the susceptibility to be contaminated (Susceptibility to Contamination or SC).

An agent is more capable of transmitting its emotional state if it (1) feels strong emotions, (2) is able to express those strong emotions (facial, vocal, and/or postural expressions), ( 3) is relatively insensitive to the feeling of others when they are experiencing emotions incompatible with its own. On the other hand, an agent may be more susceptible to be contaminated by others' emotions if it (1) is attentive to others, (2) considers itself related and dependent on others, (3) reads other's emotional expressions, voices, gestures and postures and tends to mimic them, (4) is aware of its own emotional responses, ( 5) is related to the other agent (involved in love or power relationship). Indeed, those factors are generic and not directly usable. However, Hatfield et Therefore, given its user personality, the PA can compute the Power of Contagion and Susceptibility to Contagion at each time regarding who interacts in the collaborative workspace.

V. DISCUSSION

We propose a model of cognitive, social and emotional agent. This agent can be implemented as a Personal Assistant for co-workers sharing a collaborative workspace. In this context, the PA must know its user personality and initial emotional state. Literature provides a lot of psychological tests that can be implemented to collect those values. The PA can then interpret the actions occurring on the collaborative workspace and adapt its behavior. This application could be useful in three ways: Analysis: The system could compare the supposed emotional state of its user to its real one and then detect sudden change of mood. Simulation: The system could simulate the occurrence of an event and its impact in the co-workers of the collaborative workspace. Action: The system could act in many ways to avoid conflicts, such as tagging "bad mood" messages to alert the users or propose to reschedule meetings in case of a very tense situation.. Implementing emotions into Personal Assistants has not been considered yet, although many studies point out the importance of emotions in collaborate working. We provide a model of emotional, social and cognitive Personal Assistant that can act on the system and help the user to avoid conflicts.

Figure 1 .

 1 Figure 1. Architecture of the Personal Agent

  Table I describes with some adjectives each of the five factors of the OCEAN model.

TABLE I .

 I OCEAN PERSONALITY TRAITS

	Traits	Positive value	Negative value
	Openness to Experience	Curious, Imaginative, Individualist	Conventional
	Consciousness	Conscientious, Organized	Impulsive, Distracted
	Extroversion	Gregarious, Enthusiastic Not social
	Agreeableness	Friendly, Popular, Altruist,	Aggressive, Intransigent
	Neuroticism	Emotionally stable, Calm	Anxious, Angry, Self-conscious

  al. explain the behaviors associated with those factors well enough for us to propose a translation into our model of emotions (OCC), personality (OCEAN) and mood. Table II represents our interpretation of those descriptions.

TABLE II .

 II EMOTIONAL CONTAGION MODEL

	Behavior	Interpretation
	Power of Contagion
	Feel emotions	E-
	Express emotions	A+ (charismatic) E+, C-(impulsive)
	Insensitive to opposite emotions	*
	Susceptibility to Contagion
	Attentive to others	Mood +
	Dependent on others	A+, O+
	Read and mimic other's emotional state	A+ (friendly)
	Aware of emotions	O+ (sensitive)
	Emotionally reactive	E-
	Love, admiration, respect	Relationship +

*Could not be represented in our model

This research is financed by the Direction Générale de l'Armement, France.

We hope that the model we are developing will help doing experimental work on the impact of emotional states on the quality of the transfer of knowledge among participants engaged in cooperative activities, either co-located or through distance work.


Nicolas Combe

The inverted pendulum, interface phonons and optic Tamm states

Keywords: numbers: 68.65.Cd, 63.22.Np, 78.67.Pt

The propagation of waves in periodic media is related to the parametric oscillators. We transpose the possibility that a parametric pendulum oscillates in the vicinity of its unstable equilibrium positions to the case of waves in lossless unidimensional periodic media. This concept formally applies to any kind of wave. We apply and develop it to the case of phonons in realizable structures and evidence unconventional types of phonons. Discussing the case of electromagnetic waves, we show that our concept is related to optic Tamm states one but extends it to periodic Optic Tamm state.

Waves in periodic media have focused the interest of many scientists and have found numerous applications in different branches of physics: electromagnetic (EM) waves in photonic crystals [4], phonon (elastic wave) in crystalline solids or in phononic crystals [5], electron wave functions in crystalline solids [START_REF] Ashcroft | Solid State Physics[END_REF] or electronic superlattices [START_REF] Steslicka | [END_REF]. The periodicity of the media yields the existence of band gaps (BG) in which the amplitude of the wave exponentially varies, hence corresponding to unphysical states in infinite media. Outside of these gaps, waves are spatially periodic. In the presence of localized defects, localized modes, including surface or interface modes have been evidenced.

As exploited recently [8], the physic of the parametric oscillator is equivalent to that of the propagations of waves in a lossless unidimensional infinite periodic (LUIP) medium. In this manuscript, we transpose the striking possibility for an oscillator to oscillate in the vicinity of an unstable equilibrium position using a parametric excitation (in the inverted pendulum experiment, for instance) to the case of waves in LUIP media. We evidence unconventional types of waves. This concept is general and can formally apply to any kind of waves. Before exposing and applying this concept, we first explicit the equivalence between the parametric oscillator and the propagations of waves in LUIP media. Main features of the parametric oscillator infer from the fixed point stability of a parametrically and sinusoidally excited pendulum (for instance, see Fig. 1a) governed by the Mathieu equation:

d 2 φ d t2 + [η 0 + 2α cos(2 t)]φ = 0 (1)
with t = ω e t/2, η 0 = ±4ω 2 0 /ω 2 e and α = -2z 0 /l, ω 2 0 = g/l. where φ, l, g, ω e and z 0 are respectively the angle and length of the pendulum, the standard gravity and the excitation frequency and amplitude. η 0 is positive around the fixed point Φ = (φ, dφ d t ) = (0, 0) and negative around Φ = (π, 0). Depending on η 0 and α, solutions of Eq. (1) are either periodic or unbounded. Unbounded solutions, corresponding to the parametric resonance oscillate with an exponentially varying amplitude. Fig. 1b reports the phase diagram of Eq. (1): using the Floquet theory, it deduces from the eigenvalues of R π 0 with R t t0 the propagator of Eq. (1): Φ( t) = R t t0 Φ( t0 ). Remarkably, Fig. 1b evidences some periodic solutions in the η 0 < 0 region, corresponding to the inverted pendulum experiment [START_REF] Kapitza | Collected Papers[END_REF].

We consider now the propagation of waves in a LUIP medium and focus on EM and elastic waves to reveal the equivalence between the parametric oscillator and the propagation of waves in a LUIP medium. The time Fourier transform E(z, ω) of the electric field E(z, t) = E(z, ω)e iωt of an EM wave in a LUIP medium with normal incidence is solution of:

d 2 E dz 2 (z, ω) + p(z) E(z, ω) = 0 (2)
Where p(z) = ǫ(z)ω 2 c 2 and ǫ(z) and c are the relative dielectric permittivity and the light speed in vacuum. For a longitudinal elastic wave with normal incidence, the time Fourier transform of the projected wave equation of the phonon displacement field U (z, t) = U(z, ω)e iωt z writes using the linear elasticity theory:

C(z) d 2 U dz 2 (z, ω)+ dC dz (z) dU dz (z, ω)+ρ(z)ω 2 U(z, ω) = 0 (3)
where C and ρ(z) are the elastic coefficient c zzzz and the volumic mass. Actually, both equations Eq. (2) and Eq. (3) describe the same physic. A mathematical transformation of Eq. (3) evidences this point: setting U(z, ω) = Q(z, ω)u(z, ω) with Q(z, ω) satisfying 2C dQ dz + dC dz Q = 0, u(z, ω) is solution of:

d 2 u dz 2 (z, ω) + p(z)u(z, ω) = 0 (4)
Where

p(z) = ρω 2 C + 1 2 d 2 C dz 2 -1 4C 2 [ dC dz ]
2 is a generally positive quantity. Eq. ( 2) and ( 4) are actually equivalent. In a LUIP medium, the function p(z) is real and periodic (period λ) and Eq. (2)(Eq. ( 4)) is a Hill equation, the equation of a parametric oscillator. Considering a sinusoidal variation of p(z) or limiting the Fourier series of p(z) ≈ p 0 + p 1 cos( 2πz λ ), we recover the Mathieu equation

d 2 φ dz 2 + [η 0 + 2α cos(2z)]φ = 0 ( 5
)
Where

η 0 = p0λ 2 π 2 , α = p1λ 2 2π 2 , z = πz λ
, and φ designs either E(z, ω) from Eq. (2) or u(z, ω) from Eq. ( 4). Eq. (1) for the parametric oscillator and (5) for waves are equivalent: they have thus the same solutions [10]. Periodic solutions of Eq. ( 5) correspond to periodic waves whereas unbounded ones, to waves in the BGs.

In this study, we address the following questions: Do the periodic solutions in the phase space η 0 < 0 of Fig. 1b exist for waves in real, eventually advanced materials? if yes to what kind of waves do they correspond? p(z) in Eq. ( 2) is either positive (noticeably in dielectric media) or negative (noticeably in metals below their plasma frequency). The propagation of EM waves in a LUIP metallic system below the metals plasma frequencies seems a straightforward transposition of the inverted pendulum case: the negative values of the dielectric permittivity warrants a negative value of η 0 in Eq. ( 5). Studying the dispersion diagram of a theoretical LUIP metallic layered system, we have evidenced some periodic EM waves that propagate below the metals plasma frequencies. However, in real metals, p(z) in Eq. ( 2) is a complex quantity whose imaginary part is related to the absorption. A parametric excitation cannot affect the dissipation. Investigating bi-metallic structures based on real common pure metals [START_REF] Palik | Handbook of optical constants of solids[END_REF], the absorption dominates and screens the effect we wish to evidence.

To overcome the absorption drawback of metals, we investigate in the following weakly absorbing materials i.e. dielectric materials and neglect their absorption. The dielectric permittivity in Eq. ( 2) is thus real and positive i.e. a case equivalent to the propagation of the elastic waves: p(z) > 0 ∀z in Eq. ( 4).

Waves in the BGs of a LUIP medium oscillate with an exponentially varying amplitude: this amplitude hence varies as the angle of a free pendulum closed to its unstable equilibrium position. The amplitude of the EM or elastic waves in BGs behaves as a wave in a negative dielectric permittivity medium or in a hypothetical imaginary sound speed medium i.e. mimics a negative effective η 0 in Eq. ( 5) with φ the amplitude of the wave.

To model the inverted pendulum stabilization mechanism, we propose to use a medium displaying two periodicities: a first one that would create a BG (in the absence of the second periodicity) and mimic a negative effective η 0 , and a second one that parametrically creates some periodic solutions inside this BG. Eq. ( 5) is formally replaced by the following modified Mathieu equation:

dφ dz 2 + (η 0 + 2α cos(2z) + β cos(k e z + ϕ))φ = 0 (6)
Where η 0 is positive (since we assume p(z) > 0). cos(k e z) is the periodicity supposed to create some periodic solutions inside the BG induced by cos(2z)(with β = 0). Eq. ( 6) has been studied in the general case [START_REF] Davis | [END_REF]13]: actually, for some sets of parameters η 0 , α and β = 0, Eq. ( 6) has periodic solutions that would be unbounded if β = 0. However, rather than discussing on this theoretical equation, we propose to apply the previous concept to the propagation of phonons in an experimentally realizable structure. Such structure involves two periodicities: we propose to use a SuperSuperLattice (SSL) (see Fig. 2a): a L-periodic structure whose elementary unit cell is composed of two superlattices (SL) SL1 and SL2: respectively, 10 + x periods of a 5.65 nm (10 monolayers(ML))/2.26 nm (4ML) GaAs/AlAs SL (period L 1 ) with x = 0.5 and 10 periods of a 11.3 nm (20ML)/4.52 nm (8ML) GaAs/AlAs SL (period L 2 ) [14]. Considering wave vectors with normal incidence, Fig. 2b reports the dispersion diagrams of the SSL, SL1 and SL2 and, Fig. 2c a zoom in the frequency range 0.26 THz-0.34 THz, corresponding to an overlapping region of SL1 and SL2 BGs. These dispersion diagrams are calculated from the transfer matrix [15], a representation of the propagator of Eq. (3) in these systems. In Fig. 2b, the multiple foldings and the mini-BGs created by the periodicity L of the SSL appear. More interestingly, the SSL phase diagram shows some phonons (blue point curve in Fig. 2 c) referred in the following as interface acoustic phonons in the overlapping region of SL1 and SL2 BGs. Fig. 3 reports the displacement fields U(z, ω), solutions of Eq. (3) in the SSL for ν = 0.3012 THz, an interface acoustic phonons identified by an orange cross in Fig. 2c). Though in the BGs of both SL1 and SL2, this solution describes a periodic mode that thus propagate in an LUIP medium. U(z, ω) is a oscillating function whose envelop is alternatively exponentially increasing and decreasing. The Fourier analysis of this mode evidences three characteristic lengths: a short wavelength (roughly 2L 1 = L 2 ) oscillation exhibited in Fig. 3b, an intermediate length ≈ L, related to the exponential variation of the wave amplitude and, a long ≫ L (orange curve in Fig. 3a) characteristic length related to the Bloch wave vector of this mode reported in Fig. 2c.

These interface acoustic modes precisely correspond to some periodic solutions of the parametric oscillator in the phase space η 0 < 0 of Fig. 1b). We believe that the physical mechanism described here also leads to the existence of interface optic phonons [16] in SLs: the atomic potential, the fast periodicity (term cos(2z) in Eq. ( 6)) induces the BG between acoustic and optic phonons; the SL periodicity (term cos(k e z) in Eq. ( 6)) stands for the parametric stabilizing excitation. The common description of interface optic phonons involves a frequency dependent dielectric constant accounting for the fast periodicity [17].

Besides the interface acoustic modes and similarly to optic phonons, we also evidence in Fig. 2 some confined acoustic phonons. Their frequency belongs to only one of the BG of SL1 or SL2: for instance the mode at ν = 0.3104T Hz (green cross in Fig. 2c), in the BG of SL1 but ouside the BG of SL2. These confined acoustic phonons correspond to some periodic solutions of the parametric oscillator in the regions η 0 < 0 or η 0 > 0 of Fig. 1b). A detailed discussion will be reported elsewhere.

Beyond the existence of the interface acoustic modes, it is possible to fully control their frequencies from the engineering of the SSL, and more precisely by using fractional number of the periods of SL1 or SL2. We consider a SSL formed by 10 + x periods of SL1 and 10 periods of SL2 and vary x from 0 to 1. Fig. 4 shows a band frequency (blue point curve) in the dispersion diagram that contin-uously shifts from above to below the overlapping region of SL1 and SL2 BGs. Hence, by judiciously choosing the value of x, some interface or confined acoustic modes at arbitrarily frequencies can be produced.

As already mentioned the concept described here is totally general and can formally apply to any kind of waves. We now discuss this possibility for EM waves. The propagation of an EM wave in a structure approaching the one of a SSL is related to the existence of optic Tamm state(OTS). Kavokin et al. [18] have considered the propagation of EM waves across two different semi-infinite Photonic Crystals(PC) PC1 and PC2 with a common interface and with overlapping stop bands (or BGs). By assuring the continuity of the in-plane components of the electric and magnetic fields at the interface between the two PCs, they evidence some modes, called OTS in the overlapping region of both PC1 and PC2 stop bands. OTS have been experimentally evidenced recently [19].

Similarly to the work performed for phonons, we calculate(not shown) the phase diagram of a SuperPhotonic Crystal (SPC) whose unit cell is composed by (5+0.4) periods of a PC T a 2 O 5 /SiO 2 101.6 nm/149.2 nm and 5 periods of a PC Bi : Y IG/SiO 2 86 nm/138 nm [19] [20]. We exhibit a band of periodic modes of energy E ≃ 1.5eV in the overlapping region of BGs of both PCs. The corresponding modes, referred in the following as periodic OTS are similar to the interface acoustic phonons depicted in Fig. 3. Moreover, as expected, the transmission spectra of a finite SPC (with several unit cells) exhibits transmission peaks corresponding to these modes.

The concept described in this paper not only relates OTS to the physic of the parametric oscillator, but also extends the idea of Kavokin et al. showing the possibility to simultaneously assure the continuity conditions at all interfaces of an infinite succession of different finite PCs.

As a conclusion, we would like to underline the intrinsic relation between the propagation of wave in LUIP media and the physics of the parametric oscillator. Especially, we have shown how we can transpose the possibility to stabilize an inverse parametric pendulum to the case of waves. This concept is very general and can formally apply to any kind of waves: while it is well known that the periodicity induces some forbidden band frequency, a periodicity can also stabilize solutions that were expected to be unbounded. We have chosen here to develop this concept to the case of elastic waves evidencing some new classes of phonons in realizable structures. The case of EM waves has been discussed: we have evidenced the relation between the inverse parametric pendulum and the OTS and have extended their existence to periodic OTS. This work opens numerous possibilities of applications of this concept. Due to the high localized amplitude of interface type modes, exhibited devices (SSL or SPC) are expected to be useful in the investigation of nonlinear effects or in the improvements of electron-phonon or electron-photon couplings. Moreover, they can also be used to elaborate advanced interferential filters. Finally, numerous perspectives to this work can be thought about: one can consider to apply the present concept to any kind of waves: electronic, spin, capillary waves. . . While our study has focused on normal incident waves in lossless materials, an extension to oblique incidence and the investigation of the effect of absorption are required. Though we have studied an unidimensional system, the generalization of the described concept to 2 and 3-dimensional devices is worth being considered.

The author thanks J. Morillo, M. Benoit, A. Ponchet and J.R. Huntzinger for useful discussions.

  FIG. 1. a) Sketch of a parametric pendulum b) Phase diagram of Eq. (1). Periodic and unbounded solutions are in blue and dashed regions. A vertical red line points up η0 = 0.

FIG. 2

 2 FIG. 2. a)Sketch of the elementary cell of a SSL. The cyan and red backgrounds correspond to the SL1 and SL2 regions, dashed regions underline the stratified structure of each SL. b) and c) Dispersion diagram of the SSL (black), SL1 (cyan) and SL2 (red) between 0-0.6 THz and 0.26-0.34 THz, the xaxis reports the Block wave vector normalized by k f = π κ with κ = L(SSL), L1(SL1) or L2(SL2). In c), the blue point curve and orange and green crosses point up a SSL acoustic interface modes band and the SSL modes at ν = 0.3012 THz (see Fig. 3) and ν = 0.3104 THz.


Charles Blandin

Alexey Ozerov

Emmanuel Vincent

Multi-source TDOA estimation in reverberant