Anthony Nouy
email: anthony.nouy@univ-nantes.fr

A generalized spectral decomposition technique to solve a class of linear stochastic partial differential equations

Keywords: Computational Stochastic Mechanics, Stochastic Finite Element, Spectral Decomposition, Karhunen-Loève, Stochastic Partial Differential Equations

We propose a new robust technique for solving a class of linear stochastic partial differential equations. The solution is approximated by a series of terms, each of which being the product of a scalar stochastic function by a deterministic function. None of these functions are fixed a priori but determined by solving a problem which can be interpreted as an "extended" eigenvalue problem. This technique generalizes the classical spectral decomposition, namely the Karhunen-Loève expansion. Ad-hoc iterative techniques to build the approximation, inspired by the power method for classical eigenproblems, then transform the problem into the resolution of a few uncoupled deterministic problems and stochastic equations. This method drastically reduces the calculation costs and memory requirements of classical resolution techniques used in the context of Galerkin stochastic finite element methods. Finally, this technique is particularly suitable to non-linear and evolution problems since it enables the construction of a relevant reduced basis of deterministic functions which can be efficiently reused for subsequent resolutions.

Introduction

Stochastic finite element methods have been recently proposed to solve stochastic partial differential equations and offer a significant tool to deal with the randomness which is inherent to mechanical systems. Non-intrusive techniques, such as Monte-Carlo simulation [START_REF] Caflisch | Monte carlo and quasi-monte carlo methods[END_REF][START_REF] Papadrakakis | Robust and efficient methods for stochastic finite element analysis using monte carlo simulation[END_REF], response surface method, projection [START_REF] Puig | Non-gaussian simulation using hermite polynomial expansion: convergences[END_REF] or regression [START_REF] Berveiller | Stochastic finite element: a non intrusive approach by regression[END_REF] methods, have the great advantage that they only require the use of a simple deterministic calculation code. Stochastic problems, whatever their complexity, can be solved without any further developments, as long as the associated deterministic code exists. However, they require a huge number of deterministic calculations, which leads to high computational costs. Galerkin-type methods [START_REF] Ghanem | Stochastic finite elements: a spectral approach[END_REF][START_REF] Deb | Solution of stochastic partial differential equations using galerkin finite element techniques[END_REF][START_REF] Matthies | Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations[END_REF], which differ from one another in the choice of the approximation space, systematically lead to a high precision solution which is explicit in terms of the basic random variables describing the uncertainties. However, they require the resolution of a huge system of equations. Ad-hoc Krylov-type iterative techniques have been proposed to make use of the sparsity of the system [START_REF] Ghanem | Numerical solution of spectral stochastic finite element systems[END_REF][START_REF] Pellissetti | Iterative solution of systems of linear equations arising in the context of stochastic finite elements[END_REF][START_REF] Keese | Hierarchical parallelisation for the solution of stochastic finite element equations[END_REF]. The difficulty to build efficient preconditioners and memory requirements induced by these techniques still limit their use to low stochastic dimensions when dealing with large scale applications. In this paper, we propose a new alternative resolution technique to solve stochastic problems, inspired by a resolution technique for solving evolution equations [START_REF] Ladevèze | Nonlinear Computational Structural Mechanics -New Approaches and Non-Incremental Methods of Calculation[END_REF][START_REF] Nouy | Multiscale computational strategy with time and space homogenization: a radial-type approximation technique for solving micro problems[END_REF]. The idea is to approximate the solution u by:

u ≈ M i=1 λ i U i ,
where the U i are deterministic functions and where the λ i are scalar stochastic functions (i.e. random variables). A decomposition of this type will be said optimal if the number of terms M is minimum for a given quality of approximation. The set of deterministic (resp. stochastic) functions can be then considered as an optimal deterministic (resp. stochastic) reduced basis. Here, neither the λ i nor the U i are fixed a priori. The key questions are then: how to define the "optimal" reduced basis and how to compute it? In fact, the obtained decomposition depends on what we mean by "optimal". If we knew the solution, the best approximation would classically be defined by minimizing the distance to the solution in a mean square sense. In this case, it simply leads to the classical spectral decomposition, namely the Karhunen-Loève expansion truncated at order M (see e.g. [START_REF] Ghanem | Stochastic finite elements: a spectral approach[END_REF]). The problem is that the solution, and a fortiori its correlation, are not known. In [START_REF] Matthies | Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations[END_REF], the authors propose to estimate the correlation of the solution by a Neumann expansion technique.

Having the correlation, the deterministic functions are simply obtained by solving a classical eigenproblem. The associated random variables are finally computed by solving the initial problem on the reduced basis of deterministic functions.

Here, we propose an intuitive and simple way to define the best decomposition, which generalizes the classical spectral decomposition. It leads to the resolution of a problem which can be interpreted as an "extended" eigenproblem. Ad-hoc iterative resolution techniques, inspired by classical techniques for solving eigenproblem, are then proposed. They transform the initial problem into the resolution of a few uncoupled deterministic problems and stochastic equations. This method then drastically reduces computational costs and memory requirements of classical resolution techniques.

The outline of the paper is as follows. In section 2, we introduce the abstract variational formulation of a class of linear stochastic partial differential equation, the discretization at the deterministic level and the stochastic modeling. In section 3, we describe the stochastic discretization and the classical Galerkin stochastic finite elements methods. Then, in section 4, we introduce the concept of generalized spectral decomposition. We will first focus on the case of symmetric problems before briefly introducing the case of non-symmetric problems. In section 5, we introduce power-type algorithms allowing to build the generalized spectral decomposition. Finally, in section 6, three examples will illustrate the efficiency of the proposed method.

2 Stochastic partial differential equation

Continuous problem

We first consider a deterministic partial differential equation which has the following variational formulation: find u ∈ V such that

a(u, v) = b(v) ∀v ∈ V, ( 1
)
where V is an appropriate space of admissible functions, a is a continuous bilinear form on V and b is a continous linear form on V.

In the stochastic context, a and b forms are random. We denote by (Θ, B, P ) the probability space, where Θ is the set of outcomes, B the σ-algebra of events and P the probability measure. We denote the bilinear and linear forms respectively by a(•, •; θ) and b(•; θ), which underlines their dependence on the outcome θ. The stochastic problem now consists in finding a stochastic process u which can be viewed as a random function with value in V. Function space V can sometimes depend on the outcome, for example when dealing with random geometry [START_REF] Nouy | X-SFEM, a computational technique based on X-FEM to deal with random shapes[END_REF]. Here, we consider that V does not depend on the outcome. The appropriate function space for u can then be chosen as the tensor product space V ⊗ S, where S is an ad-hoc function space for real-valued random functions. The weak formulation of the stochastic partial differential equation (see e.g. [START_REF] Babuška | Solving elliptic boundary value problems with uncertain coefficients by the finite element method: the stochastic formulation[END_REF][START_REF] Frauenfelder | Finite elements for elliptic problems with stochastic coefficients[END_REF][START_REF] Matthies | Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations[END_REF]) can then be written as follows: find u ∈ V ⊗ S such that

A(u, v) = B(v) ∀v ∈ V ⊗ S, (2)
where

A(u, v) = Θ a(u(θ), v(θ); θ)dP (θ) := E(a(u, v)), (3)
B(v) = Θ b(v(θ); θ)dP (θ) := E(b(v)). ( 4
)
E(•) denotes the mathematical expectation. In this article, we focus on the case of a linear elliptic stochastic partial differential equation. The bilinear form A is continuous on (V ⊗ S) × (V ⊗ S) and (V ⊗ S)-coercive, and the linear form B is continuous on V ⊗ S. We consider that S = L 2 (Θ, dP ) is an ad-hoc choice, such that [START_REF] Babuška | Solving elliptic boundary value problems with uncertain coefficients by the finite element method: the stochastic formulation[END_REF]).

V ⊗ S = V ⊗ L 2 (Θ, dP ) ∼ = L 2 (Θ, dP ; V) (see e.g.

Discretization at the deterministic level

The discretization at the deterministic level consists in searching an approximation of the solution of problem (1) under the form

u = n i=1 ϕ i u i (θ), (5)
where u i ∈ S and where {ϕ i } n i=1 is a basis of a finite dimensional space V n ⊂ V. We will denote by u = (u 1 , ..., u n ) T ∈ R n ⊗ S the random vector of unknowns representing the approximate solution. It is the solution of the following semidiscretized problem: find u ∈ R n ⊗ S such that

E v T Au = E v T b ∀v ∈ R n ⊗ S, (6)
where

A : Θ → R n×n and b : Θ → R n are such that ∀u, v ∈ V n , we have P -almost surely a(u, v; θ) = v T A(θ)u, (7) b(v; θ) = v T b(θ). (8)
Random matrix A and random vector b inherit respectively from continuity and coercivity properties of bilinear form A and continuity property of linear form B. Then, a solution in R n ⊗ S of problem (6) exists and is unique.

We will then denote by

• the classical norm on R n ⊗ S ∼ = L 2 (Θ, dP ; R n ), defined by u 2 = E(u T u). (9)
We will denote by ((u, v)) the associated scalar product.

Stochastic modeling

We consider that the probabilistic content of the problem is represented by a finite set of random variables ξ = (ξ 1 , ..., ξ m ) : Θ -→ R m . This is naturally the case when A and B forms in (2) only depend on a finite set of parameters which are random variables. When these parameters are stochastic fields, an approximation step can allow their expression as functions of a finite set of random variables. For example, in the case of second-order random fields, such an approximation can consist of the truncated Karhunen-Loève expansion [START_REF] Ghanem | Stochastic finite elements: a spectral approach[END_REF]. However, in the case of non-Gaussian random fields, the probabilistic characterization of the obtained finite set of random variables is not trivial. Some recent works try to answer this question by using truncated polynomial chaos expansions of the stochastic field, the coefficients of the expansions being constructed by the resolution of an adapted inverse problem (see e.g. [START_REF] Ghanem | On the construction and analysis of stochastic models: characterization and propagation of the errors associated with limited data[END_REF][START_REF] Desceliers | Maximum likelihood estimation of stochastic chaos representations from experimental data[END_REF]).

Those techniques lead to a description of the probabilistic content by a finite set of independent Gaussian random variables. Let us note that an approximation is done on the bilinear form A. A particular care must then be taken on the truncation in order to keep coercivity properties of the approximate bilinear form [START_REF] Babuška | On solving elliptic stochastic partial differential equations[END_REF][START_REF] Frauenfelder | Finite elements for elliptic problems with stochastic coefficients[END_REF] and then to ensure existence and uniqueness properties for the approximate model.

After this stochastic modeling step, a random function f can then be rewritten as a function of the basic random variables f (ξ) and the stochastic problem (2) can be equivalently formulated on the finite dimensional probability space (Θ (m) , B (m) , P (m) ), where Θ (m) = Range(ξ) is a subset of R m , B m is the associated Borel σ-algebra and P (m) is the image probability measure (cf. [START_REF]Stochastic Differential Equations. An Introduction with Applications[END_REF][START_REF] Babuška | Solving elliptic boundary value problems with uncertain coefficients by the finite element method: the stochastic formulation[END_REF]).

In the following, for the sake of simplicity, we will still denote by (Θ, B, P ) the new finite dimensional probability space (Θ (m) , B (m) , P (m) ).

3 Classical stochastic finite element methods

Stochastic discretization

Classical stochastic finite element methods introduce a finite dimensional subspace S P ⊂ S for the approximation at the stochastic level. Several choices have been proposed for building approximation basis; spectral approaches (polynomial chaos [START_REF] Wiener | The homogeneous chaos[END_REF][START_REF] Ghanem | Stochastic finite elements: a spectral approach[END_REF], generalized chaos [START_REF] Xiu | The Wiener-Askey polynomial chaos for stochastic differential equations[END_REF]) classically use orthogonal polynomial basis and show exponential convergence rates [START_REF] Xiu | The Wiener-Askey polynomial chaos for stochastic differential equations[END_REF] in the case of quite smooth solutions. For the case of non-smooth solutions, other approximation techniques have been introduced (Wiener-Haar chaos [START_REF] Le Maître | Uncertainty propagation using Wiener-Haar expansions[END_REF], finite elements [START_REF] Deb | Solution of stochastic partial differential equations using galerkin finite element techniques[END_REF]). The approximation space is defined as follows:

S P = {v ∈ S ; v(θ) = α∈I P H α (θ)v α , v α ∈ R}, (10)
where {H α } α∈I is a basis of S and where I P is a subset of I with cardinal P .

If random variables ξ i are independent, S is a tensor product space S 1 ⊗ . . . ⊗ S m . Each dimension can be independently discretized; denoting by α = (α 1 , . . . , α m ) ∈ N m a multi-index, we can write

H α (θ) = h 1 α 1 (ξ 1 ) . . . h m αm (ξ m ), where h i α i ∈ S i .
In practice, we will use an orthonormal basis, i.e.

E(H α H β ) = δ αβ = δ α 1 β 1 ...δ αmβm .
In the case of mutually dependent random variables ξ i , it is possible to use generalized "non-polynomial" chaos expansions [START_REF] Soize | Physical systems with random uncertainties: chaos representations with arbitrary probability measure[END_REF] or more classically, to change the basic random variables by using an adapted mapping of the ξ i into independent Gaussian random variables. Below, we present a classical way of defining and computing an approximation of the solution of problem (6).

Galerkin approximation at the stochastic level

Classical Galerkin approximation of problem ( 6) is obtained by replacing the function space V n ⊗ S by the approximation space V n ⊗ S P . The problem is then to find u ∈ R n ⊗ S P such that,

E v T Au = E v T b ∀v ∈ R n ⊗ S P , (11)
which leads to the following system of equations:

β∈I P E(AH α H β )u β = E(H α b) ∀α ∈ I P . ( 12
)
Denoting the solution by a block vector u, whose block α is defined by (u) α = u α , system (12) can be written

Au = b, (13)
where A and b are respectively a block matrix and a block vector. Their blocks are defined by (A) αβ = E(AH α H β ) and (b) α = E(H α b).

Krylov-type iterative techniques

In practice, system [START_REF] Nouy | X-SFEM, a computational technique based on X-FEM to deal with random shapes[END_REF] can not be solved by a direct resolution technique. Indeed, memory requirements and computational costs of assembling and solving this huge system of size P × n become prohibitive for large-scale engineering problems. To avoid assembling and to take part of the sparsity of this system, we classically use a Krylov-type iterative resolution technique [START_REF] Ghanem | Numerical solution of spectral stochastic finite element systems[END_REF][START_REF] Pellissetti | Iterative solution of systems of linear equations arising in the context of stochastic finite elements[END_REF] such as preconditioned conjugate gradient (PCG) for symmetric problems, conjugate gradient square (CGS), etc. The resolution then only necessitates matrix-vector products which can be eventually parallelized [START_REF] Keese | Hierarchical parallelisation for the solution of stochastic finite element equations[END_REF]. The preconditioner M is classically taken as a block-diagonal matrix, each block being the inverse of the mean value of matrix A, i.e. (M) αβ = E(A) -1 δ αβ . The reason for this choice is that the preconditioner is quasi-optimal when A has low variance terms. As the variance of matrix A increases, the preconditioner M becomes less and less optimal. Iterative techniques then require a large number of iterations, which can drastically increase computational costs. Memory capacities required by these techniques can also be significant. Indeed, when dealing with such a huge system, reorthogonalization of the generated Krylov subspace is necessary, which implies the storage of this subspace. For example, if one considers a problem with n = 10 5 and P = 5000, storing the solution as double-precision floating-point numbers requires around 4 Gigabytes. Storing a Krylov subspace of dimension only 10 then requires 40 Gigabytes! 4 Generalized spectral decomposition

Principle

Here, we try to find an approximation of the solution of problem [START_REF] Ladevèze | Nonlinear Computational Structural Mechanics -New Approaches and Non-Incremental Methods of Calculation[END_REF] under the form

u(θ) ≈ M i=1 λ i (θ)U i , ( 14
)
where λ i ∈ S P are scalar random variables and U i ∈ R n are deterministic vectors. A decomposition of this type will be said optimal if the number of terms M is minimum for a given quality of approximation. The set of deterministic vectors (resp. stochastic functions) would then be considered as an optimal deterministic (resp. stochastic) reduced basis. Neither the λ i nor the U i are fixed a priori. The key questions are then: how to define an "optimal" decomposition and how to compute it? The answer is of course related to what we mean by "optimal".

When we know the stochastic vector u, a natural way to define the "best" approximation of the form ( 14) is to minimize the distance between the approximation and the solution:

u - M i=1 λ i U i 2 = min U 1 ,...,U M ∈R n λ 1 ,...,λ M ∈S P u - M i=1 λ i U i 2 , ( 15
)
where • denotes the classical L 2 -norm defined in [START_REF] Pellissetti | Iterative solution of systems of linear equations arising in the context of stochastic finite elements[END_REF]. It is well known that the obtained approximation is the classical Karhunen-Loève expansion truncated at order M (cf. [START_REF] Ghanem | Stochastic finite elements: a spectral approach[END_REF]). Vectors U i are the M rightmost eigenvectors of E(uu T ), which is the correlation matrix of u, and can be characterized by

p(U i ) = min V ∈V n-i+1 max U∈V p(U), ( 16
)
with p(U) = U T E(uu T )U U T U ,
where p is the classical Rayleigh quotient and V k is the set of all k-dimensional subspaces of R n . The associated λ i are then obtained by

λ i = (U T i U i ) -1 U T i u.
In this section, we introduce an extension of this principle in order to build a decomposition of the solution of problem [START_REF] Ladevèze | Nonlinear Computational Structural Mechanics -New Approaches and Non-Incremental Methods of Calculation[END_REF] without knowing this solution a priori. After an introduction of several notations and comments, we fill first focus on the case where operator A is symmetric before treating the general case.

Remark 1 In this article, we consider that the approximation space V n ⊗S P is given. The approximate solution u is then considered as our reference solution.

For details on convergence properties of the approximation and estimation of errors with respect to the exact solution of [START_REF] Papadrakakis | Robust and efficient methods for stochastic finite element analysis using monte carlo simulation[END_REF], see e.g. [START_REF] Babuška | Solving elliptic boundary value problems with uncertain coefficients by the finite element method: the stochastic formulation[END_REF][START_REF] Frauenfelder | Finite elements for elliptic problems with stochastic coefficients[END_REF][START_REF] Oden | Theory and methodology for estimation and control of errors due to modeling, approximation, and uncertainty[END_REF][START_REF] Ladevèze | Verification of stochastic models in uncertain environments using the constitutive relation error method[END_REF].

Preliminaries and notations

In the following, we will denote by W = (U 1 . . . U M ) ∈ R n×M the matrix whose columns are the deterministic vectors and by Λ = (λ 1 . . . λ M ) T ∈ R M ⊗ S P the stochastic vector whose components are the stochastic functions. The approximation of order M (14) will then be rewritten

u (M ) = M i=1 U i λ i = WΛ. (17)
When neither Λ nor W are fixed, approximation [START_REF] Desceliers | Maximum likelihood estimation of stochastic chaos representations from experimental data[END_REF] is not defined uniquely. In another words, there are infinitely many choices of stochastic functions and deterministic vectors leading to the same approximation. Indeed, for any invertible matrix P ∈ R M ×M , we clearly have

(WP)(P -1 Λ) = WΛ. (18)
The couple composed by matrix (WP) and stochastic vector (P -1 Λ) then yield the same approximation as the couple composed by matrix (W) and stochastic vector (Λ). Therefore, without loss of generality, we can for example impose orthogonality or orthonormality conditions on the λ i or the U i for the definition of the best approximation.

Finally, if the U i (resp. the λ i ) were not linearly independent, it would be clearly possible to obtain a new decomposition of type ( 17) with a lower number of terms by using linear combinations of the U i (resp. the λ i ).

Remark 2 Random functions {λ i } M i=1 are said "linearly independent" if they span a M -dimensional linear subspace of S P . In the finite dimensional space S P , the λ i can be identified with vectors λ i ∈ R P whose components are the coefficients λ i,α of the λ i on the basis {H α } α∈I P of S P . The property "random variables {λ i } M i=1 are linearly independent" is then equivalent to "vectors {λ i } M i=1 are linearly independent", or "the rank of matrix (λ 1 . . . λ M ) ∈ R P ×M is M ". Of course, it does not mean that random functions are statistically independent.

We will suppose that [START_REF] Desceliers | Maximum likelihood estimation of stochastic chaos representations from experimental data[END_REF] is the optimal decomposition which means that the U i (resp. the λ i ) are linearly independent. We will denote by G n,M = {W = (U 1 . . . U M ) ∈ R n×M ; rank(W) = M } the set of full rank matrices and by

G P,M = {Λ = (λ 1 . . . λ M ) T ∈ R M ⊗ S P ; dim(span({λ i } M i=1
)) = M } the set of linearly independent stochastic functions 1 .

Case of a symmetric operator A

In this section, we consider the particular case where the continuous coercive bilinear form A is also symmetric. Therefore, random matrix A, which inherits from properties of A, is also symmetric. The right-hand side b is a stochastic vector.

Definition of the best approximation

The discretized problem ( 11) is equivalent to the following minimization problem:

J (u) = min v∈R n ⊗S P J (v), ( 19
)
where

J (v) = E( 1 2 v T Av -v T b). (20)
A natural definition for the approximation follows:

Definition 3 The best approximation of order M is defined by

J ( M i=1 λ i U i ) = min U 1 ,...,U M ∈R n λ 1 ,...,λ M ∈S P J ( M i=1 λ i U i ), ( 21
)
which can be equivalently written in a matrix form:

J (WΛ) = min W∈R n×M Λ∈R M ⊗S P J (WΛ). ( 22
)

Properties of the approximation

On one hand, the stationarity conditions of J (WΛ) with respect to Λ writes:

∀Λ * ∈ R M ⊗ S P , E(Λ * T (W T AW)Λ) = E(Λ * T W T b). (23)
This is clearly the way we could have naturally defined the best stochastic functions associated with known deterministic vectors.

On the other hand, the stationarity conditions with respect to W writes:

∀W * ∈ R n×M , E(Λ T (W * T AW)Λ) = E(Λ T W * T b). (24)
This is still the way we could have naturally defined the best deterministic vectors associated with known stochastic functions. For example, if we impose the λ i to be the basis functions of S P , namely the H α , equation ( 24) would simply yields the classical solution of system [START_REF] Ladevèze | Nonlinear Computational Structural Mechanics -New Approaches and Non-Incremental Methods of Calculation[END_REF], i.e. W = (. . . u α . . .). However, the resulting approximation is the less optimal approximation since it has the maximum number of terms M = P .

Here, we ask the best approximation for simultaneously verifying equations ( 23) and [START_REF] Oden | Theory and methodology for estimation and control of errors due to modeling, approximation, and uncertainty[END_REF]. The following proposition gives fundamental properties allowing to better understand the meaning of the approximation and to develop computational resolution techniques.

Proposition 4 The best approximation defined by definition 3 is characterized by:

• W maximizes on the set of full rank matrices G n,M the functional R(W) defined by

R(W) = T race(R(W)), ( 25
) with R(W) = E((W T AW) -1 (W T bb T W)).
• The stochastic functions are obtained by

Λ = (W T AW) -1 W T b. ( 26
)
Moreover, the best approximation verifies

J (WΛ) = - 1 2 R(W). ( 27
)
Proof. Equation ( 23) yields relation [START_REF] Sameh | The trace minimization method for the symmetric generalized eigenvalue problem[END_REF]. Then, we have

J (ΛW) = 1 2 E(Λ T W T AWΛ) -E(Λ T W T b) = - 1 2 E(Λ T W T b) = - 1 2 E(b T W(W T AW) -1 W T b) = - 1 2 T race(E((W T AW) -1 (W T bb T W))) = - 1 2 R(W).
The best deterministic vectors are then such that W ∈ G n,M maximizes R(W). 2

Remark 5 In equation [START_REF] Sameh | The trace minimization method for the symmetric generalized eigenvalue problem[END_REF], we use an abuse of notation. Indeed, the quantity (W T AW) -1 W T b does not necessarily belongs to R M ⊗ S P . In fact, Λ ∈ R M ⊗ S P must be interpreted as the solution of problem [START_REF] Soize | Physical systems with random uncertainties: chaos representations with arbitrary probability measure[END_REF]. This abuse of notation is also made in the definition of functional R(W). In fact, it should be interpreted as follows:

R(W) = T race(E(Λ T b T W))
where Λ is the solution of [START_REF] Soize | Physical systems with random uncertainties: chaos representations with arbitrary probability measure[END_REF].

For all invertible matrix

P ∈ R M ×M , R(WP) = P -1 R(W)P, (28)
and

R(WP) = R(W). ( 29
)
Equation ( 29) is a property of homogeneity. In another words, functional R takes the same value for all matrices whose column vectors span a given Mdimensional linear subspace of R n . This is related to the non-uniqueness of the best solution W (cf. section (4.2)). For the decomposition to be defined uniquely, we could impose orthonormality conditions on W. For example, denoting by M a symmetric definite positive matrix, the optimization problem on R could be defined on the space of M-orthogonal matrices

G * n,M = {W ∈ G n,M ; W T MW = I M }, where I M is the identity matrix on R M . The set G * n,M
is called a Stiefel manifold (cf. [START_REF] Sameh | The trace minimization method for the symmetric generalized eigenvalue problem[END_REF]).

Case of deterministic operator A

In order to interpret the approximation defined in proposition 4, let us consider the particular case where operator A is deterministic. In this case, we have

R(W) = (W T AW) -1 (W T E(bb T )W). ( 30
)
The stationarity condition of R(W) = T race(R(W)) writes

AWR(W) = E(bb T )W, (31)
which is a classical generalized eigenproblem written in a matrix form. R(W) (resp. R(W)) is the associated matrix (resp. scalar) Rayleigh quotient. Therefore, the best W characterized in proposition 4 is such that its column vectors span the rightmost M -dimensional eigenspace of the generalized eigenproblem (31). A particular choice for the columns of W = (U 1 . . . U M ) consists of the M rightmost eigenvectors. The U i are A-orthogonal and the associated λ i are characterized by

λ i = (U T i AU i ) -1 U T i b. (32) The couples (U i , λ i ) also verify J ( M i=1 λ i U i ) = - 1 2 M i=1 R(U i ). (33)
E(bb T ) is the correlation matrix of random vector b. The spectral decomposition can then simply be interpreted as a truncated Karhunen-Loève expansion of b in the metric induced by A. Here, we emphasize that the approximation is not unique. Infinitely many choices of W yield the same approximation as the particular choice consisting of eigenvectors.

Remark 6 If we choose for the U i the eigenvectors of the generalized eigenproblem (31), which are A-orthogonal, the associated stochastic functions are orthogonal.We have then simultaneous orthogonality properties for both the stochastic functions and the deterministic vectors. In general, this property can not be verified in the case of a stochastic operator A.

Interpretation and comments

Regarding the results of the previous section, functional R(W) (resp. R(W)), defined in proposition 4, will be called a "generalized" matrix (resp. scalar) Rayleigh quotient. In particular, R has the homogeneity property (29) of a classical Rayleigh quotient. The best approximation obtained will also be called a generalized spectral decomposition, although there is no associated classical eigenproblem.

One could think that since the obtained decomposition is not the Karhunen-Loève expansion, it is not the optimal decomposition. In fact, the obtained decomposition is optimal with respect to the optimality criterium introduced in definition 3. Properties of random matrix A, inherited from the continuous coercive bilinear form A, allow to define the following norm on R n ⊗ S P :

v 2 A = E v T Av . ( 34
)
We can then easily show that the decomposition characterized by proposition 4 verifies

u -WΛ 2 A = u 2 A -WΛ 2 A = u 2 A -R(W). ( 35
)
The obtained decomposition is then optimal with respect to the A-norm (34) while the direct Karhunen-Loève expansion of the solution is optimal with respect to the L 2 -norm [START_REF] Pellissetti | Iterative solution of systems of linear equations arising in the context of stochastic finite elements[END_REF].

As a last comment, let us mention that it is also possible to characterize the best approximation by defining an optimization problem on the stochastic functions. We can show that the following proposition 7 is equivalent to proposition 4. For this new proposition, we use the following notations: we denote by

E(Λ ⊗ A ⊗ Λ) ∈ R M ×n×n×M the four indices matrix such that (E(Λ ⊗ A ⊗ Λ)) ijkl = E(λ i A jk λ l ). We denote by E(b ⊗ Λ) ∈ R n×M the matrix such that (E(b ⊗ Λ)) ij = E(b i λ j ).
We then define the operation ":" as follows: for W, W * ∈ R n×M ,

W * T : E(Λ ⊗ A ⊗ Λ) : W = i,j,k,l W * ji E(λ i A jk λ l )W kl = E(Λ T W * T AWΛ).

Proposition 7 The best approximation defined by definition 3 is characterized by:

• Λ maximizes on G P,M the functional R(Λ) defined by

R(Λ) = E(Λ ⊗ b) : E(Λ ⊗ A ⊗ Λ) -1 : E(b ⊗ Λ). ( 36
)
• The deterministic vectors are obtained by

W T = E(Λ ⊗ A ⊗ Λ) -1 : E(b ⊗ Λ). ( 37
)
Moreover, the best approximation verifies

J (WΛ) = - 1 2 R(Λ).
(38)

Another definition of the approximation

The best approximation can also be naturally defined by the following optimization problems, which define the couples (λ i , U i ) one after the other:

Definition 8
The best approximation of order M can be defined recursively:

for i = 1, ..., M J (λ i U i ) = min U∈R n ,λ∈S P J (λU + i-1 j=1 λ j U j ) (39)
Following the proof of proposition 4, we obtain the following characterization of the new obtained approximation.

Proposition 9

The approximation defined in definition 8 can be charaterized by2

• U i maximizes the generalized Rayleigh quotient

R i (U) = E(U T b i (U T AU) -1 b T i U) with b i = b - i-1 j=1 Aλ j U j • λ i = (U T i AU i ) -1 U T i b i .
Moreover, the best approximation verifies

J ( M i=1 λ i U i ) = - 1 2 M i=1 R i (U i ). ( 40
)
In the case of a deterministic operator, we can easily show that definitions 8 and 3 yield the same approximation. However, these definitions do not match in general in the case of a stochastic operator. Definition 3 then clearly yields a better approximation as definition 8. The latter definition can however be interesting from a computational point of view, as we will see in section 5.

Case of a non-symmetric operator

For the case of non-symmetric operator, there is no direct minimization problem associated with the variational formulation [START_REF] Deb | Solution of stochastic partial differential equations using galerkin finite element techniques[END_REF]. A possible and natural way to define an approximation WΛ is to write the two orthogonality criteria [START_REF] Soize | Physical systems with random uncertainties: chaos representations with arbitrary probability measure[END_REF] and [START_REF] Oden | Theory and methodology for estimation and control of errors due to modeling, approximation, and uncertainty[END_REF] which can be obtained by introducing in [START_REF] Deb | Solution of stochastic partial differential equations using galerkin finite element techniques[END_REF] test functions of the form v = WΛ * +W * Λ. In the case of a non-symmetric deterministic operator A, we can easily show that it leads to a classical non-symmetric generalized eigenproblem. Therefore, the approximation still has a full meaning but no characterization as the one in proposition 4 can be derived.

Another idea consists in reformulating the problem as a minimization problem in a least-square sense. Let us denote by M ∈ R n×n a symmetric positive definite matrix which defines the following scalar product on R n ⊗ S P :

((U, V)) M = E(U T MV). (41)
We will denote by • M the associated norm. For example, if the expected value of the symmetric part of A is positive definite, we can take M = ( 1 2 E(A+ A T )) -1 . We can then formulate the problem as a minimization problem of the norm of the residual:

E(u) = min v∈R n ⊗S P E(v) ( 42
)
where

E(v) = b -Av 2 M .
The previous theoretical results associated with symmetric operators can then be still applied by replacing operator A by A T MA and right hand side b by A T Mb. Of course, the approximate solution of this problem is not in general the same as for problem [START_REF] Ladevèze | Nonlinear Computational Structural Mechanics -New Approaches and Non-Incremental Methods of Calculation[END_REF] when A is random.

Although this symmetrization is well adapted to the discretized formulation (finite dimensional framework), it is not easy to transpose into the continuous framework. Indeed, the introduction of adjoint operator is non trivial, essentially due to the treatment of boundary conditions. Moreover, ad-hoc function spaces need for more regularity after symmetrization and the building of approximation spaces is then more difficult.

However, we will see that this symmetrization does not seem necessary since algorithms which are given in the following section also gives satisfactory results when directly applied to non-symmetric problems. Without the symmetry, we do not have an optimality criterium as in proposition 4. It is then more difficult to judge the quality of the approximation. Of course, it is possible to evaluate the norm of the residual of equation [START_REF] Ladevèze | Nonlinear Computational Structural Mechanics -New Approaches and Non-Incremental Methods of Calculation[END_REF] but this increases computational costs.

5 Power-type algorithms for the construction of the spectral decomposition

Description of the algorithm

Here, we propose a first algorithm to build the spectral decomposition defined in definition 8. The couples (λ i , U i ) are built one after the other. Each couple must solve a coupled minimization problem of the functional J . A natural and simple idea to build the approximation is to minimize J alternatively on the stochastic function and on the deterministic vector. Algorithm 1, named (P-GSD) for Power-type Generalized Spectral Decomposition, follows this idea. Some trivial computational improvements are straightforward (redundancy of several operations).

Algorithm 1 Power-type Generalized Spectral Decomposition (P-GSD)

1: u := 0 , b := b 2: for i = 1 to M do 3:

λ i := λ 0 , R (0)
i := 0 4:

for k = 1 to k max do 5:

U i := E(Aλ 2 i ) -1 E( bλ i ) 6: U i := U i / U i M 7: λ i = (U T i AU i ) -1 U T i b 8: R (k) i := E(U T i bλ i ) 9: γ := R (k) i -R (k-1) i /R (k) i 10:
if γ < γ stop then

u := u + λ i U i 15: b := b -Aλ i U i 16:
Compute error indicator (i) 17:

if (i) < stop then 18: break 19:
end if 20: end for Let us explain this algorithm. Step 1 is the initialization step of the solution and the residual. The loop beginning at step 2 corresponds to the recursive building of couples (λ i , U i ). At step 3, we initialize the stochastic function λ i = λ 0 ∈ S P . We will simply always take λ 0,α = 1, ∀α ∈ I P . We will see that this very simple initialization gives satisfactory results in practice. The loop beginning at step 4 is the alternate minimization procedure. For a fixed stochastic function λ i , the stationarity condition with respect to U i writes E(Aλ 2 i )U i = E( bλ i ). It has to be noticed that it is a simple deterministic problem whose resolution is relatively cheap. This system is solved at step 5 and U i is normalized at step 6. In the symmetric case, we simply take M = E(A) for the definition of the norm. Then, for a fixed U i , the stationarity condition with respect to λ i writes

E(λ * U T i AU i λ i ) = E(λ * U T i b) ∀λ * ∈ S P . ( 43
)
This system is solved at step 7. From step 8 to step 12, we introduce a stopping criterium based on the convergence of the quantity R (k)

i . This quantity corresponds to the generalized Rayleigh quotient in the case where operator A is symmetric (see definition 8). At steps 14 and 15, we reactualize the solution and the residual. From 16 to 19, we introduce a stopping criterium. The error indicator can be the residual error

(i) res = b -A i j=1 λ j U j b . ( 44
)
Regarding the results of proposition 9, we can also use the following error indicator, based on the generalized Rayleigh quotient evaluation:

(i) ray = R i (U i ) i j=1 R j (U j ) . ( 45
)
This last indicator has the advantage to be cheaper to compute. Indicators (44) and (45) will be compared in section 6.

Remark 10 Computing the matrix and right-hand side of the deterministic problem (step 5) requires the computation of quantities such as E(Aλ i λ j ) or E(bλ i ). This kind of computations are classical within the context of stochastic finite element methods. Let us consider that the random vector b is decomposed as follows

: b = M b k=1 b k (θ)b k , with b k (θ) = α∈I P b k,α H α (θ) ∈ S P .

Then, due to orthonormality property of the basis functions H α , E(bλ

i ) = M b k=1 b k α∈I P b k,α λ i,α .

Let us now consider that the random matrix writes as follows: A

= M A k=1 a k (θ)A k , where the a k are random variables. Then E(Aλ i λ j ) = M A k=1 A k E(a k λ i λ j ).
In practice, one pre-computes and stores the matrices

∆ (k) whose components are (∆ (k) ) αβ = E(a k H α H β ) and such that E(a k λ i λ j ) = α,β∈I P (∆ (k) ) αβ λ i,α λ j,β . If the a k are decomposed on the stochas- tic basis, i.e. a k (θ) = γ a k,γ H γ (θ), then (∆ (k) ) αβ = γ a k,γ E(H γ H α H β ),
where E(H γ H α H β ) only depends on the chosen basis functions {H α }.

Interpretation and comments

In the case of deterministic operator, iteration k of the alternate minimization stage consists in reactualizing the deterministic vector in the following way: U i ← A -1 E( bb T )U i /η, where η is a normalizing scalar. The proposed algorithm is then equivalent to a classical power method to solve the associated generalized eigenproblem AU i = ηE( bb T )U i . It classically converges toward the rightmost eigenvector. We will see in examples that this algorithm is also efficient in the case of stochastic operator. In practice, the alternate minimization procedure (steps 4 to 13) converges very fast. We will then classically limit the number of iterations k max to 3 or 4. In the case of deterministic non-symmetric operator, we know that the proposed power algorithm is not always convergent. If the maximum amplitude eigenvalue is complex, the generalized eigenproblem admits 2 complex conjugate eigenvalues. Vector U i does not converge in this case. However, it tends to stay in the subspace generated by the real and complex parts of the associated complex eigenvectors and the obtained couple (λ i , U i ) still happens to be pertinent. We will see in examples that this algorithm also gives satisfactory results in the general case of random eventually non-symmetric operators.

Power-type algorithm with updating

Let us suppose we have built an approximation M i=1 λ i U i = WΛ. We have seen that optimality depends on the way we define the "best" decomposition. Indeed, definitions 3 and 8 match only in the case where operator A is deterministic. Once we have obtained such a decomposition, it can then be interesting to update the decomposition. A natural way to do this is to fix the deterministic vectors U i and to compute new stochastic functions λ i ∈ S P by using a Galerkin orthogonality criterium [START_REF] Soize | Physical systems with random uncertainties: chaos representations with arbitrary probability measure[END_REF]. The updating can then be formulated: find

Λ ∈ R M ⊗ S P such that ∀Λ * ∈ R M ⊗ S P , E(Λ * T (W T AW)Λ) = E(Λ * T W T b). ( 46
)
To obtain the solution Λ(θ) = α∈I P Λ α H α (θ), we then have to solve a system of equations of size M × P . It is the same system as [START_REF] Nouy | Multiscale computational strategy with time and space homogenization: a radial-type approximation technique for solving micro problems[END_REF] where we replace A by the reduced random matrix W T AW and b by the reduced random vector W T b. With an abuse of notation, we will denote by Λ = (W T AW) -1 W T b the solution of this updating. Algorithm 2, named (PU-GSD), is a modified version of Algorithm 1 where we introduce the updating of stochastic functions after the construction of each couple (λ i , U i ). With this algorithm, we try to compute the decomposition defined in definition 3.

Algorithm 2 Power-type Generalized Spectral Decomposition with Updating (PU-GSD)

1: u := 0 , b := b 2: for i = 1 to M do 3:
do step 3 to 13 of algorithm 1 to compute U i 4:

W := (U 1 ... U i ) 5: Λ := (W T AW) -1 W T b 6: u := WΛ 7: b = b -AWΛ 8:
Compute error indicator (i) 9:

if (i) < stop then 10: break 11:
end if 12: end for Regarding the results of proposition 4, we can here use an error criterium based on the evaluation of the generalized Rayleigh quotient

(i) ray = R(W (i) ) -R(W (i-1) ) R(W (i) ) , ( 47
)
where W (j) = (U 1 . . . U j ). We recall that at step 8, the generalized Rayleigh quotient can be simply computed as follows R(W) = T race(E(Λb T W)).

Examples

The following three examples illustrate the efficiency of the proposed method on model problems. In example 1, the method is applied to a classical stationary heat diffusion problem with random source terms and a conductivity parameter which is modeled by a random field. In example 2, we consider the same problem but with a conductivity parameter modeled by a random variable. It is a degenerate case of the previous example (simple form of the random matrix), for which the solution is exactly represented with a decomposition of order 2. It illustrates that the proposed algorithms allow to automatically construct this exact decomposition. Finally, example 3 illustrates that the proposed algorithms are still efficient in the non-symmetric case. As a first model problem, we consider a classical stationary heat diffusion problem defined on a L-shaped spatial domain Ω (see figure 1). We denote by u(x, θ) the temperature field. The normal flux g is imposed on a part ∂ 2 Ω of the boundary and the temperature is imposed to be zero on the complementary part ∂ 1 Ω. A volumic heat source f is also imposed on Ω. The space of admissible functions used in formulation ( 1) is

V = {v(x) ∈ H 1 (Ω) ; v = 0 on ∂ 1 Ω}.
In variational formulation (2), a and b forms are respectively

a(u, v; θ) = Ω κ∇u • ∇v dx, b(v; θ) = Ω f v dx + ∂ 2 Ω
gv ds, where κ(x, θ) is the conductivity parameter. At the space level, we use a classical finite element approximation. The mesh is composed by 1200 fournodes elements and 1281 nodes. Random matrix A and random vector b, respectively defined in ( 7) and ( 8), have the following components:

(A) ij = Ω κ∇ϕ i • ∇ϕ j dx (b) j = Ω f ϕ j dx + ∂ 2 Ω gϕ j ds,
where the ϕ i are the basis functions of the approximation space V n ⊂ V.

Stochastic modeling

The conductivity parameter is modeled by a random field. The following definition is taken from [START_REF] Desceliers | Maximum likelihood estimation of stochastic chaos representations from experimental data[END_REF], where it was used to illustrate a method for identification of non-Gaussian random fields. Here, this definition allows us to impose a given marginal distribution which simply ensures the ellipticity of the bilinear form. We take

κ(x, θ) = F -1 Γ δ • Φ(γ(x, θ)))
, where γ is a normalized Gaussian second-order random field such that E(γ(x, •)) = 0 and E(γ(x, •) 2 ) = 1. Function y → Φ(y) is the cumulative distribution function of a normalized Gaussian random variable and function y → F Γ δ (y) is the cumulative distribution function of a Gamma random variable:

F Γ δ (y) = y 0 δ Γ(δ) (δt) δ-1 e -δt dt, with Γ(δ) = ∞ 0 t δ-1 e -t dt.
With this definition, κ has a Gamma marginal distribution with unitary mean and standard deviation σ = 1/ √ δ. We take δ = 16 so that σ = 0.25. The Gaussian random field γ(x, θ) is here defined by

γ(x, θ) = 3 k=1 √ η k ξ k (θ) Ṽk (x), ( 48
) with Ṽk (x) = V k (x) 3 j=1 η j V j (x) 2
, where the ξ k ∈ N (0, 1)3 are independent normalized Gaussian random variables and where (η k , V k (x)) are the eigenpairs of the following homogeneous Fredholm equation of the second kind:

Ω exp(- x -y 2 L 2 )V k (y)dy = η k V k (x). (49)
Then, the random field γ corresponds to a rescaled truncated Karhunen-Loève expansion of a Gaussian random field with exponential square correlation function. We take L = 0.5. In practice, problem (49) can be approximated by using finite elements. Here, we use the same approximation as for the solution u. We then have to solve a classical algebraic eigenvalue problem (cf. [START_REF] Ghanem | Stochastic finite elements: a spectral approach[END_REF]). We use a classical technique to find its 3 rightmost eigenpairs.

Volumic heat source f and normal flux g are taken independent of the variable x: f (x, θ) = ξ 4 (θ) ∈ N (0.5, 0.2) and g(x, θ) = ξ 5 (θ) ∈ N (0, 0.2), where ξ 4 and ξ 5 are independent Gaussian random variables, also independent of {ξ i } 3 i=1 . The source of randomness is then represented by m = 5 independent Gaussian random variables. We use a polynomial chaos approximation of degree p = 6 at the stochastic level. The dimension of approximation space S P is then P = (m+p)! m!p! = 462. The random field κ is projected on S P : κ(x, θ) = α∈I P κ α (x)H α (θ) where space functions κ α = E(κH α ) are computed using Gauss-Hermite quadrature for the integration at the stochastic level. Matrix A can then be written A =

α∈I P A α H α with (A α ) ij = Ω κ α ∇ϕ i • ∇ϕ j dx.

Reference solution and error criteria

We denote by u (M ) the approximation of order M (14) obtained by the generalized spectral decomposition algorithms, namely the power-type algorithms (P-GSD) or (PU-GSD). We denote by u the reference solution, which is the solution of [START_REF] Ladevèze | Nonlinear Computational Structural Mechanics -New Approaches and Non-Incremental Methods of Calculation[END_REF]. To compute u, system ( 12) is solved with a preconditioned conjugate gradient (PCG) (see section 3.2), with a stopping tolerance of 10 -8 . We introduce the following error indicators to evaluate the quality of the approximation:

(M ) sol = u -u (M ) u , (M ) sol,A = u -u (M ) A u A . ( 50
)
where • is the classical L 2 -norm (9) and • A is the A-norm defined in (34).

The generalized spectral decomposition obtained by power-type algorithms will be also compared with the direct Karhunen-Loève spectral decomposition of the reference solution u, denoted by (Direct-SD).

Remark 11 Of course, approximations are first introduced by the spatial and stochastic discretizations. The error introduced by the stochastic discretization can be defined by the difference between our reference solution u and the solution of the semi-discretized problem [START_REF] Deb | Solution of stochastic partial differential equations using galerkin finite element techniques[END_REF]. For this example, we estimated this error by computing the discretized solution associated with a polynomial chaos of degree p = 8. The estimation of the relative error in L 2 -norm is 1.04 10 -2 . In this article, we don't focus on those errors and consider the reference solution as the solution of the fully discretized problem [START_REF] Ladevèze | Nonlinear Computational Structural Mechanics -New Approaches and Non-Incremental Methods of Calculation[END_REF]. However, this remark indicates that a very small tolerance for the resolution of the discretized problem [START_REF] Ladevèze | Nonlinear Computational Structural Mechanics -New Approaches and Non-Incremental Methods of Calculation[END_REF] is generally useless for engineering applications, for which the discretization error is often greater than 1%.

Convergence of power-type algorithms

(P-GSD) and (PU-GSD) have two parameters which are associated with the iterative search of each couple of functions (steps 3 to 13 of Algorithm 1): γ stop , which defines the stopping criterium, and k max , which defines the maximum number of iterations. Figures 2(a) and 2(b) show the error

(M )
sol with respect to the order M of the decomposition for different values of these parameters. On one hand, figure 2(a) shows that a relatively coarse convergence criterium (γ stop ≈ 0.1) do not affect the quality of the decomposition. On another hand, figure 2(b) shows that for a fixed stopping criterium γ stop = 0.05, a few iterations are sufficient, which is in fact related to the fast convergence of this iterative procedure. For the following tests, we will choose γ stop = 0.05 and k max = 3.

[Fig. 2 about here.] Figures 3(a) and 3(b) compare the convergence of (P-GSD), (PU-GSD) and (Direct-SD) with respect to the order of expansion M . We can observe that (PU-GSD) has almost the same convergence rate as (Direct-SD). This figure also shows the importance of the updating of stochastic functions. Indeed, without knowing the solution a priori, algorithm (PU-GSD) leads to a spectral decomposition of the solution which has quite the same quality as a direct Karhunen-Loève decomposition of the reference solution. We also verify, as mentioned in section 4.3.4, that (Direct-SD), when compared with (PU-GSD), gives a better approximation with respect to the L 2 -norm but a coarser approximation with respect to the A-norm. We can also notice than with a decomposition of order M = 4, the error is less than 1%.

[Fig. 3 which can be used to evaluate the convergence of algorithms (P-GSD) and (PU-GSD). res is the norm of the residual, defined in (44), and ray is the indicator based on the Rayleigh quotient, defined in (45) for (P-GSD) and in (47) for (PU-GSD). These error indicators are compared to error indicators sol and sol,A , defined in (50). We can see that all these indicators are equivalent. The great advantage of estimator ray is that it leads to very low computational costs.

[Fig. 5 about here.]

Analysis of the generalized spectral decomposition

Figures 6(a) and 6(b) show the first 8 deterministic vectors obtained respectively by (P-GSD) and (PU-GSD). We observe that these algorithms yield to the construction of quite relevent deterministic vectors. Vectors 1 and 2 take respectively into account the volumic load f and the surface load g. Subsequent vectors seem to take into account the fluctuations of the conductivity parameter. The superiority of (PU-GSD) appears clearly on this figure. Indeed, vectors 5 and 7 obtained by (P-GSD) are very similar to vectors 1 and 2. In fact, they can be interpreted as correction terms for the first two modes. Algorithm (PU-GSD) seems to capture these modes with the first two vectors and does not require any further correction.

[Fig. 6 about here.] Figure 7 shows the 8 first vectors computed by a direct Karhunen-Loève decomposition of the reference solution (Direct-SD). If we compare this figure with figure 6 (b), we can see that (PU-GSD) and (Direct-SD) lead to very similar decompositions. In fact, we can say that the proposed algorithm (PU-GSD) allows to obtain a spectral decomposition of the solution, which is very similar to the Karhunen-Loève expansion, without knowing the solution a priori.

[Fig. 7 about here.]

Calculation time and memory requirements

We now look at the computation time of the proposed algorithms. Algorithm (PCG) took 156 s to compute the reference solution. Figure 8 shows the error with respect to the calculation time for (P-GSD), (PU-GSD) and (PCG) algorithms. The convergence curves for (P-GSD) and (PU-GSD) correspond to a generalized spectral decomposition up to order M = 20. We observe that power-type algorithms lead to the same computational cost. (PU-GSD) converges faster with respect to the order M but the cost of one iteration is greater than for (P-GSD), due to the updating of stochastic functions. On this example, we can then conclude that (PU-GSD) is more efficient since it leads to a decomposition with a lower order M for the same accuracy and calculation time. We see that power-type algorithms are significatively superior to the classical (PCG) algorithm. To compute a decomposition of order M = 4, which leads to a relatively good accuracy, it only takes 3.5 s with (PU-GSD). The same accuracy is reached in 40 s with (PCG). The calculation time is then divided by 11 on this simple example.

[Fig. 8

about here.]

To store the spectral decomposition of order M , we need to store M × (P + n) floating-point numbers. For (PCG) we need to store n × P floating-point numbers. For an approximation of order M = 4, memory requirements are divided by around 85. In fact, the gain is much greater since (PCG) algorithm generally requires reorthogonalization of the search directions. In fact, the storage of a Krylov subspace of dimension η requires to store η × n × P floating-point numbers.

Example 2

We consider the same problem as in example 1. The linear form b(v; θ) is unchanged. The sources f and g are still defined by f (x, θ) = ξ 4 (θ) ∈ N (0.5, 0.2) and g(x, θ) = ξ 5 (θ) ∈ N (0, 0.2). But now, we suppose that the conductivity is a simple uniform random variable κ(x, θ) = ξ 1 ∈ U (0.7, 1.3)4 . Random variables {ξ 1 , ξ 4 , ξ 5 } are considered independent. We use a generalized polynomial chaos approximation of degree p = 8 at the stochastic level [START_REF] Xiu | The Wiener-Askey polynomial chaos for stochastic differential equations[END_REF]. We then use Legendre polynomials in the first stochastic dimension and Hermite polynomials in the two other stochastic dimensions. The dimension of approximation space S P is then P = (m+p)! m!p! = 165, where m = 3.

Convergence of power-type algorithms

Figure 9 shows the convergence with respect to the order of decomposition M for (P-GSD), (PU-GSD) and the Direct Karhunen-Loève decomposition of the solution (Direct-SD) . We observe that the three algorithms lead to the exact solution with a decomposition of order 2. That could have been expected The iterative search of the couples (λ i , U i ) converges very fast. We also observe that with a relatively coarse stagnation criterium (γ stop = 0.05) and only two iterations (k max = 2), we still obtain the exact numerical solution at order M = 2. Parameters γ stop and k max have a small influence on the obtained couples (U 1 , λ 1 ) and (U 2 , λ 2 ) but have no influence on the obtained decomposition of order M = 2. In fact, the iterative procedure converges in only 1 iteration for the construction of couple (U 2 , λ 2 ).

[Fig. 10 about here.]

Here, both power-type algorithms required a calculation time of 0.7 s while (PCG) required 4 s. The gain is still significant. However, comparing calculation time on this simple example is not so relevant. 2) is the same for the three algorithms. However, (P-GSD) and (PU-GSD) lead to deterministic vectors which are different from the ones obtained by (Direct-SD). It simply illustrates the fact that the decomposition is not unique. If we normalize all the deterministic vectors with the same norm and compare the vectors obtained by power-type algorithms and the ones obtained by (Direct-SD), we find a relative error of 2% for U 1 and 5% for U 2 . The difference is due to the fact that the first ones are the optimal vectors with respect to the A-norm and the second ones are the optimal vectors with respect to the L 2 -norm.

Analysis of the generalized spectral decomposition

[Fig. 11 about here.]

Stochastic functions λ 1 and λ 2 obtained by (P-GSD) and (PU-GSD) are also identical. Figures 12 and13 show respectively the probability density functions of these two stochastic functions and their joint probability density function.

[Fig. 12 about here.]

[Fig. 13 about here.]

Example 3

In this last example, we briefly illustrate the fact that power-type algorithms also lead to satisfactory performances in the non-symmetric case. We here consider the same domain and boundary conditions as in example 1. The linear form b(v; θ) is unchanged but we take the following non symmetric bilinear form:

a(u, v; θ) = Ω κ∇u • ∇v dx - Ω χ u • ∇v dx,
where χ = (χ 1 , χ 2 ) T . We consider that material parameters are uniform random variables:

κ(x, θ) = ξ 1 (θ) ∈ U (0.7, 1.3), χ 1 (x, θ) = ξ 2 (θ) ∈ U (5.5, 6.5), χ 2 (x, θ) = ξ 3 (θ) ∈ U (5.5, 6.5
). The five random variables {ξ i } 5 i=1 are independent. We use a generalized polynomial chaos approximation of degree p = 5. We then use Legendre polynomials in the first three stochastic dimensions and Hermite polynomials in the last two stochastic dimensions. We can notice that A still defines a norm A since its symmetric part is almost surely positive definite. The reference solution, which solves system [START_REF] Nouy | Multiscale computational strategy with time and space homogenization: a radial-type approximation technique for solving micro problems[END_REF], is computed with a preconditionned conjugate gradient square algorithm (PCGS) with a stopping tolerance of 10 -8 .

Convergence of power-type algorithms

We take γ stop = 0.05 and k max = 4 for the parameters of power-type algorithms. Figure 14(a) and 14(b) show the convergence of the spectral decomposition obtained by (P-GSD), (PU-GSD) and (Direct-SD). The three algorithms have quite the same convergence rate, which confirms the fact that power-type algorithms are also adapted to the non-symmetric case. The superiority of (PU-GSD) over (P-GSD) is not significant in this case.

[Fig. 14

about here.]

The resolution with (PCGS) took here 16 s to reach a relative error of 10 -8 and 2.8 s to reach a relative error of 10 -2 . The construction of a decomposition of order M = 5, which leads to a relative error lower than 10 -2 , took 1.9 s with (P-GSD) and 2.3 s with (PU-GSD). For the same precision, the gain obtained in terms of computational time is not significant on this small-scale example. However, the gain in terms of memory requirements is still significant. of the solution (Direct-SD). We can observe that (P-GSD) and (PU-GSD) lead to quite the same vectors but than these vectors are significatively different from those obtained by (Direct-SD) (except for the first two vectors). However, we must recall than the obtained approximations u (M ) have quite the same accuracy although the deterministic vectors and stochastic functions are different.

Analysis of the generalized spectral decomposition

Conclusion

We proposed a new method for solving stochastic partial differential equations based on a generalized spectral decomposition technique. The decomposition is the solution of a problem which can be interpreted as an "extended" generalized eigenproblem. This method allows to obtain a decomposition of the solution which is very similar to the one obtained by a classical truncated Karhunen-Loève expansion of the solution. Power-type algorithms have been proposed to solve this "extended" eigenproblem. The proposed resolution technique leads to significant computational savings compared to Krylov-type techniques which are classically used in the context of Galerkin Stochastic Finite Element methods. It also leads to a drastic reduction of memory requirements. It could then allows to deal with large-scale engineering problems and large stochastic dimensionality. Future works will be devoted to the development of more efficient algorithms to solve the "extended" eigenproblem, particularly for non-symmetric problems. We will also focus on the resolution of evolution equations and non-linear problems. This new method should be very efficient in these last cases since it enables the construction of a relevant reduced basis of deterministic functions which can be efficiently reused for subsequent resolutions.

6. 1 1 6. 1 . 1

 1111 Example Description of the problem [Fig. 1 about here.]

  Figures 4(a),4(b) and 4(c) show the marginal probability density functions (PDFs) of the approximation obtained by (PU-GSD) for different orders of expansion M . The three sub-figures correspond respectively to points P 1 , P 2

Figures 5 (

 5 Figures 5(a) and 5(b) compare error indicators ray and res which can be used to evaluate the convergence of algorithms (P-GSD) and (PU-GSD). res is the norm of the residual, defined in (44), and ray is the indicator based on the Rayleigh quotient, defined in (45) for (P-GSD) and in (47) for (PU-GSD). These error indicators are compared to error indicators sol and sol,A , defined in (50). We can see that all these indicators are equivalent. The great advantage of estimator ray is that it leads to very low computational costs.

  Figures 10(a) and10(b) show the influence of parameters γ stop and k max of power-type algorithms. The iterative search of the couples (λ i , U i ) converges very fast. We also observe that with a relatively coarse stagnation criterium (γ stop = 0.05) and only two iterations (k max = 2), we still obtain the exact numerical solution at order M = 2. Parameters γ stop and k max have a small influence on the obtained couples (U 1 , λ 1 ) and (U 2 , λ 2 ) but have no influence on the obtained decomposition of order M = 2. In fact, the iterative procedure converges in only 1 iteration for the construction of couple (U 2 , λ 2 ).

Figures 11 (

 11 Figures 11(a), 11(b) and 11(c) show the two deterministic vectors which are obtained respectively by (P-GSD), (PU-GSD) and (Direct-SD). The obtained approximation of order 2 u(2) is the same for the three algorithms. However, (P-GSD) and (PU-GSD) lead to deterministic vectors which are different from the ones obtained by (Direct-SD). It simply illustrates the fact that the decomposition is not unique. If we normalize all the deterministic vectors with the same norm and compare the vectors obtained by power-type algorithms and the ones obtained by (Direct-SD), we find a relative error of 2% for U 1 and 5% for U 2 . The difference is due to the fact that the first ones are the optimal vectors with respect to the A-norm and the second ones are the optimal vectors with respect to the L 2 -norm.

Figures 15 (

 15 Figures 15(a), 15(b) and 16 show the deterministic vectors obtained respectively by (P-GSD), (PU-GSD) and the direct Karhunen-Loève decomposition

[

  Fig. 15 about here.] [Fig. 16 about here.]

∂ 1 Ω ∂ 2 ΩFig. 1 .Fig. 2 .Fig. 3 .Fig. 5 .Fig. 8 .

 1212358 Fig. 1. Description of example 1: mesh and boundary conditions

Fig. 9 .Fig. 10 .Fig. 14 .

 91014 Fig. 9. Error in A-norm with respect to the order of decomposition M for (P-GSD), (PU-GSD) and (Direct-SD)

span({λ i } M i=1 ) denotes the linear subspace of S P which is spanned by the set of functions {λ i } M i=1

We use the same abuse of notation as in proposition 4, explained in Remark 5

N (µ, σ) denotes the set of Gaussian random variables with mean µ and standard deviation σ

U (a, b) is the set of uniform random variables with value in ]a, b[.

Acknowledgement

This work is supported by the French National Research Agency (ANR). The author would like to thank Professor Pierre Ladevèze for fruitful discussions which helped to initiate the method proposed in this article.

List of Figures


J Pillay

PLANAR GRAPH HOMOLOGIES AND THE 4-COLOUR THEOREM

We present herein an alternate representation of Planar Graphs and make use of this to offer insights into the 4-Colour Theorem.

Introduction.

Planar graph theory needs vary little in the way of introduction. The importance of a planar graph representation stems from its ability to simplify complex information. In a sense, by virtue of the nature of the representation, these dispose of unnecessary information and expose only the bare necessary on a platform capable of allowing one to make observations or inferences. Many algebras have been developed on the infrastructure provided by matrix algebra providing a platform for use in investigations surrounding chromatic numbers [B]. Similar representations such as the Kneser Graph are further investigated for studying Fractional-Chromatic-Numbers[S-U], here the study extends to general non-planar graphs and hyper Graphs. Recent trends in Graph Theory[1][V] extend into areas as diverse as Engineering, Electronics and even Management Sciences where the study of work delegation is of interest. At this point, an impression is that all available algebra has been exhausted on this one homology. We believe that the time is ripe for the study into the relation (which we believe exists) between a homology and the associated platform it generates. We propose alterations to the existing Planar-Graph-representation that enables one to see symmetry on a grand scale for specific use in chromatic number analysis.

1 Dedicated to my manager Ekaterina. Stakova for much support. I have never been an easy human to manage, however neither is Satyagraha (the principles by which we live.)

Key words and phrases. Graph Theory, Planar Graphs, 4-Colour Theorem.

The Theory

A planar Graph is a set (V, E), we offer the morphing (V, E) → (B, L) where B is a set of blocks and L a set of conditioned edges simply lines in this system.

Where a set of edges E n exist between the set of vertices e α , {e 1 , .., e n }, in our system, we collapse E n → l, for a single line l ∈ L. We introduce a concept of orientation on L in S = (B, L). Four orientations are made available to S L , {↑, ↓, ←, →}. These simply signify whether the arrow extends above, below or on either side of a series blocks b i .

Definition

Each series of blocks within a single row R j are allowed at most two line originator elements and orientations E 1j ↑, E 2j ↓ over a set of elements b i ∈ B, which of course can be the same element. A V-System is defined to be any system of the form:

V S := ∀j (b ij ∈ B, E 1j , E 2j )

Definition

A strategy is defined per block/originator-block as a choice of a set of three surrounding-lines of arbitrary orientations ∪l, per block b i limiting its choice, with the remaining orientation acting as an inhibitor of choice for an adjacent block instead. Making the choice for every E 1j ↑ per row R j ∈ V S to be an inhibitor, with V S surrounded by at most four lines (internal or external to V S ), one for each orientation of V S when treated as a single block {↑, ↓, ←, →}, is defined to be a (L, T ), (R, T ), strategy depending on whether we make the choice of blocks to be influenced left to right or the other way. One can easily see that multiple lines extending to a block can be replaced with a single line extending from the block through to the originating elements from which the lines initially extended.

Theorem Each V-System V S has one of four colouring strategies C : {(L, T ), (L, B), (R, T ), (R, B)} that solve them.

Proof

This easily follows from our preceding discussion.

Lemma

Each V-System V S can be collapsed to a block b V having the same conditioning as b i ∈ B ∈ S. When composed in this manner. Here again the strategies {(L, T ), (L, B), (R, T ), (R, B)} are availed, and when made use of again enable the further collapsing of ∪V S to a block. The ability to continue the process tells us that a 4-colour strategy always exists for any planar-graph.

Figure 1 .

 1 Figure 1. Morphism .

Figure 2 .

 2 Figure 2. V-System .

Figure 3 .

 3 Figure 3. Collapsing of V-Systems .


Usages du code-switching dans le théâtre breton contemporain

Mélanie Jouitteau UMR 5478, IKER, CNRS Tout phénomène de contact n'est pas du code-switching. Influence n'est pas interférence.

Il y a interférence, alternance de langues, code-switching « quand un sujet bilingue utilise dans une langue-cible L2, un trait phonétique, morphologique, lexical ou syntaxique caractéristique de la langue L1. » Kannas (1994: 252)  emprunts: Tu veux un petit caoua avant ton algèbre?  monolingues situés par l'influence de la langue de contact

(1) Phonologie

Pa gouezh ar mogerioù/Quand les murs tombent "Dans tous les ports tu peux… t'arrives lÔ, tu D' mets sur le port quand Y débarquent py t'apprends à parler aveG les gens, quoi." "Quand j'étais p'tit, très vite j'ai compris j'étais diZleGsique." "Moi j'étais pas dans la figure du paysan massif, c'est vrai en BreTAgne on peut être un peu b p ourrin des fois. Ah bah, à la camPAgne, c'est sur tu vas en trouver, du b p ourrin !"  Europanto [START_REF] Marani | Las adventures des inspector Cabillot[END_REF] [Je comprends mieux le prix des bijoux] LEILA: Priz ar braewidjoù.

[le prix des bijoux] ABIGAËLLE : Priz ar bravigoù. MARJANIK : Oui, bah en tout cas si en fait c'est de l'amour que tu vends, pas étonnant que ça coûte la peau des fesses. [mamie] Florence est encore un bébé face de pruneau.

De la même façon, dans Al liorzhour, la génération intermédiaire entre enfant et grand-parent est monolingue francophone. Ces indices mobilisent les connaissances sociolinguistiques (et intimes) de l'audience, pour annoncer un cadre prototypique de la passation de la langue bretonne au coeur du XX° siècle. Dans Métamorphose, la bordure de texte, en introduction et conclusion, est en néobreton typique XXI°, avec une influence marquée de français dans les marqueurs discursifs, ce qui renvoie à une réalité sociologique contemporaine.

Métamorphose, p1. Abeguile: Ya, mais on. Ket er mod peus graet. Mert', quoi.

[Oui, mais non. Pas de la façon que tu as faite. Merde, quoi.] Dav 'vo adplegañ pep tra goude.

[Il va falloir replier tout après]

Le coeur de l'histoire, le rêve, à l'espace et aux temporalités plus floues, où morts et vivants échangent plus librement, est breton monolingue sans influence particulière du français, en contraste aux passages breton/français mêlés.

Ce contraste linguistique souligne aussi la différence entre deux cadres énonciatifs.

3. … un tamm kengred ! Aet on faezh ganeoc'h ! [un peu de solidarité! J'en ai marre de vous !] LEILA : Whahou. Brezhoneg brav ganit.

[whahou. Tu parles bien Breton.] MARJANIK, laouen memestra : Da veg.

[contente quand même : notes de bas de page? traduction en une seule langue? laquelle? traduction monolingue en typographies différentes? traduction en regard ou blocs de texte séparés? traduction en deux langues de rapport minorisé équivalent?

(breton > français et français > anglais, avec distribution au Québec?) formes numériques paramétrisables? Une culture globale de l'écrit littéraire organise aussi une déperdition linguistique éditoriale. Jusqu'aux typifications locales ou dialectes sont communément gommés, dans l'idée qu'un bon texte est organisé autour d'une uniformité linguistique monolingue, monodialectale et monoregistre.

Dans Yerma tel que joué par Penn-ar-Bed, chaque comédien avait un breton dialectalement typifiable, standard, léonard ou cornouaillais, dont l'entremêlement était porteur de sens, en

  ket kat da ziskouezh un nebeut solidarité memes! neg êtes pas cap de montrer un peu solidarité même 'Vous n'êtes même pas capables de montrer un peu de solidarité !' Hagège (1996:239): un croisement involontaire entre deux langues. A grande échelle, l'interférence dénote l'acquisition incomplète d'une langue seconde.

	> En communication adulte entre deux bilingues aisés :
		marqueurs des facteurs personnels décidant du choix neutre pour un individu donné
	 ?	Genre des noms pour un locuteur non-natif loyauté envers la langue maternelle
		loyauté envers la langue non-maternelle choisie
	(5)	Tu as rangé où mon tasse? conformité à la langue associée au système scolaire	tchèque: ten hrnek
		idée de sa propre aptitude individuelle à maîtriser des codes linguistiques valorisés
	 ?	acquisition bilingue X représentations de l'interlocuteur
		signe de la confiance relationnelle entre les interlocuteurs
	(6)	Hein c'est moi boud dirag [être devant],
		Jeu du petit train, enfant 4 ans 1 mois, 8 mois d'exposition à la langue, Mermet (2006)
	Les révélateurs d'aisance linguistique ou de diglossie
	(7)	MARJANIK :		Privezioù Publik
	N' och >> phénomène associé à un jugement de valeur.
	Medane (2016): le problème d'interférence en français cassé, système fautif, erreurs
	l'hypercorrection, signe d'insécurité linguistique
	Medane (2016) : "l'analyse montre que les jeunes de plus de 20 ans et surtout ceux du niveau 2. Les révélateurs intimes
		universitaire commettent plus d'interférences dans leurs productions en « français
		cassé » que ceux des autres tranches d'âge et du niveau « terminale »."
	(8)	Les précieuses ridicules, Molière
		Madelon et Cathos refusent deux gentilhommes à marier et s'éprennent de deux valets
	(2)	[ilʝalədegutaʒdɑsəpeʝe] qui singent un parler précieux.
	(9)	'Il y a le dégoûtage dans ce pays.' (ennui, dégoût) vulnérabilité sociale > sensibilité à la pression normative [ilesyfʀi] 'Il est suffrit' > 'Il est meurtri.' femmes degré de conformité linguistique Français algérien, Medane (2016) non-diplômés
	Belazi & al. 1994) Les révélateurs d'aisance linguistique
	(3) expressions à doubles sens: richesse du message pour les multilingues: Je conseille de tous les gens pour arrêter ce phénomène catastrophique.
		'Je conseille à tous les gens d'arrêter ce phénomène catastrophique.'
	(10) On va voir une grande femme.	Français algérien, Medane (2016) = 'une vieille femme'
		Ne tombez pas les mots	= 'N'insultez pas'
	(4)	Le temps est couteau quand ne pas coupe il te coupe. couper la route = 'traverser la rue'
			Français algérien, Medane (2016) Français algérien, Medane (2016)
	(11) citations culturelles	(l'arabe est pro-drop & négation préverbale)
	situations d'apprentissage : chansons de Gainsbourg ou Presley dans Métamorphose
		chansons de Sardou ou Orelsan dans Privezioù Publik

-Las adventures des inspector Cabillot « In der story des morte des Diana er esse tropo viel unclaras chosas und confusio. Porqué Diana Mercedes was zo rapido roulante? […] Mucha estrange coincidenza, cinquo dags later, tambien Zuster Theresa uit Calcutta trespasse al Creator. Somechose kloppe nicht. » * dérivations morphologiques : Je vais au pressing * matériel fonctionnel seul : et, pour, lequel, qui… ni après lui (Functional Head Constraint,  Franponais (café chez la gare, restaurant remplir ici) Engrish (All your base are belong to us, Zero Wing) 2.1. marqueurs d'image personnelle dans la relation qui se place sur le terrain linguistique de qui? Privezioù Publik, senenn 1 ABIGAËLLE : Me a blij din gwerzhañ d'ar merc'hed. Gout a reont n'eo ket bravigoù 'ni eo a brenont, karantez ne laran ket. Ma c'hell an dra-se plijout dezho on-me kontan. O werzhañ plijadur n'eus ket mezh da gaout. [Moi j'aime vendre aux femmes. Elles savent que ce ne sont pas des bijoux qu'elles achètent. De l'amour, je dis pas. Si ça peut leur plaire, moi ça me va. Ya pas de honte à vendre du plaisir.] MARJANIK : Kompren a ran gwelloc'h le prix des bijoux.

  Setu. Arruet eo-hi foll-tout. Na roue, na mestr, hag hormonoù e pep lec'h ! [Voilà. Elle est devenue complètement folle. Ni Dieu, ni maître, hormones partout !] LEILA, fulor o sevel enni [en colère]: Écoutez les filles, moi je vends des fringues.Toute la journée je vends des fringues. Vous savez pas comment c'est fatiguant.Pa gouezh ar mogerioù "J'avais l'impression que ce que je faisais ne rimait à rien, que, enfin il y avait des murs partout, et c'était un labyrinthe et puis il n'y avait pas d'issue. Il y avait rien, quoi. Du rien. J'habitais à Paris à z moment-là, ça se passait mal avec les gens, je ne pouvais pas rester, je suis partie." Al Liorzhour, texte orienté sur la temporalité, le passage des saisons et des générations, est ponctué d'une ritournelle de mots renvoyant à une temporalité cyclique, un temps non actualisé. Le temps de la Nature, au dessus des humains, est exprimé en breton, et est ici interrompu par le temps déictique, humain, exprimé en français. Bon alors, où en étais-je? J'étais venu chercher quelque chose. exemple de cadre énonciatif avec interlocuteur semi-absent (le français comme le discours politique ratent leur cible): Pa vezan fichet krann zo lod ne gredont ket lavarout grik. Met n'hallan ket mont-me ez naturel. Ret-mat eo e vefe ganin brozh, botoù uhel ha livaj-tout. E vefe kred din. Perak em bije-me d'en em dreuswiskañ pa vez ar merc'hed all 'barzh o bragoù jeans, ha blev berr ganto ? [Quand je suis tirée à quatre épingles, certains n'osent rien dire. Mais je ne peux pas y aller au naturel. Obligé que j'ai la robe, les talons et le maquillage complet. Que je sois crédible. Pourquoi je devrais me déguiser alors que les autres femmes y vont en jeans et cheveux cours ?] Pouffe, ça doit rester un choix. Je suis comme tout le monde, des fois ça m'amuse et des fois ça m'amuse pas. Alors, bon. J'y vais plus. 3.3. dater une interaction Les situations sociolinguistiques sont inscrites dans l'Histoire des locuteurs. Un cadre sociolinguistique de code-switching donné peut donc renvoyer précisément à une époque. Dans Al Liorzhour, le personnage de Joe contient deux temporalités différentes : le personnage de Joe, adulte, raconte son enfance (cf. encore). Dans ce passage, le grand-oncle Harry s'exprime en breton et son petit-neveu en français, sauf son usage du nom mamm-gozh, seul rescapé indemne dans la passation linguistique. Ce passage ancre la temporalité du personnage dans un cadre sociolinguistique connu, datable. J'ai dit non, c'est pas elle. Il a dit Hag ez eo [Si] J'ai dit c'est pas elle. C'est mamm-gozh.

	2.2. changement émotionnel
	> colère
				Privezioù Publik, senenn 1
	>temps, lieu< Joe: ABIGAËLLE : > surprise, choc Nevezamzer, Dihuniñ Hañv	[printemps] [se.réveiller] [été]	Al liorzhour, p.1
	Harry: Gortoz 'ta. N'eo ket ul louzaouenn. [Attends donc. C'est pas une herbe.] Joe: N'eo ket? [Non?] Harry: N'eo ket. Ur bizenn eo. Sell. [Non. C'est un poix. Regarde.] Joe: Et qu'est-ce que j'ai vu? Quelque chose vert. Une pousse. Les révélateurs interpersonnels 3.1. changement de cadre énonciatif [temps, espace, locuteur, interlocuteur] Al liorzhour p.18 p.24 3. > espace < An devezh-pad [toute la journée] Diskar-amzer [automne] Skeudoù ar serr-noz [ombres du crepuscule] Goañv [hiver] Noz [nuit] Noz hir ha teñval [nuit longue et sombre] Métamorphose, p.17 Damokle: Fanfares de tous pays, unissez-vous! L'impérialisme et tous les réactionnaires sont des tigres de papier, des tigres morts, des tigres de fromages de soya. 3.2. souligner un changement de rapport / public Privezioù Publik, prologue Joe: Puis j'ai entendu mamm-gozh [mamie] crier depuis la maison. Berc'hed: Al Liorzhour, p.12 Oncle Harry a dit Florence an hini eo. [C'est Florence]
	>interloc<
				Al liorzhour, p.20
	Joe:	Petra a raimp bremañ?

Harry: Ur babig eo, ur babig o c'harmiñ.

[C'est un bébé, un bébé en train de pleurer] Joe: Hag ez eo. Perenn dev tev eo.

[mais oui. Grosse poire est grosse.] On avait oublié le bébé. [Qu'est-ce qu'on fait maintenant?] Harry: Dastum ha debriñ [récolter et manger.] Joe: Nous voila donc à cueillir et cueillir. Il y en a tous les jours davantage

  Abigaelle est enfermée dans un box en train de pleurer, ses amies essaient de la consoler. Alice, non-brittophone, entre dans les toilettes pour femmes et se maquille. LEILA, gant un ton oc'h ober van da glask frealziñ Abigaëlle :[avec une prosodie qui fait semblant qu'elle est en train de consoler Abigaëlle] :Ma welfes an dra-se, pegen lous, n'he deus ket gwalc'het he daouarn. Gwir, pa laran dit. Ha livaj war-eeun. Ha bremañ 'zo titich e pep lec'h war he fas, pa n'eus ket gwashoc'h. Me 'lar dit ! Ha gouzout a rez piv eo ? Plac'h an ti bara. Ne zebrin ket mui bara-kornek james ken. [Si tu voyais ça, dégueu, elle s'est pas lavée les mains. En vrai, je te dis ! Et maquillage direct par là dessus. Maintenant elle a du pipi partout sur la face si c'est pas pire. Je te jure ! Et tu sais qui c'est ? La boulangère. Moi je mange plus jamais de croissants.] ALICE : Ça ira votre amie ? LEILA : Oui, oui, ça ira, merci, vous inquiétez pas, on gère. Allez embrasser votre petit bout de chou.

	Privezioù publik, arvest 2
	(14) la figure du correcteur
	Privezioù publik, arvest 2
	MARJANIK : Ah non alors, achu ! N'och ket kat da ziskouezh un tamm
	solidarité memes !
	[Ah non alors ! Vous n'êtes même pas capables de montrer un peu de
	solidarité !]
	LEILA : Kengred. [solidarité]
	MARJANIK :

4. marqueurs de rapports de pouvoir (12) la figure de l'interprète tricheur (Audiard Un prophète, King 2014, Homeland) (13) exclusion d'un tiers monoligue

  Prop, kwa. Mah bon. N'eo ket memestra. Me gav hirrrrr ma amzer, Marjanik, ma ouifes.[Propre, quoi. Mais bon. C'est pas pareil. Je m'ennuie, Marjanik, si tu savais.] Oh je m'ennuie, si tu savais comment je m'ennuie. MARJANIK, lentig [timidement]: T'as pas besoin de traduire, tu sais. Je parle un tammig, memestra [un peu, quand même]. C'est mieux en breton, il faut que j'apprenne. ABIGAËLLE : J'avais jamais travaillé dans une galerie marchande...4.Les révélateurs du monde d'écriture 4.1. marquage de la ou les langues neutres choix de langue out of the blue: la représentation du choix neutre pour soi ou pour l'autre. Bleunioù Gouez: le multilinguisme comme choix neutre Le mode d'emploi, les disdascalies, sont en français, espagnol ou breton. Le titre est breton sous-titré en espagnol. L'ortie, en code-switching intra-phrase, fait du méta. Elle dit ne rien comprendre à l'Art et au multilinguisme. La scène est résolue lorsqu'elle accepte d'être émue par le breton qu'elle révèle être sa langue d'origine (ou une de ses langues d'origine) :Bleunioù Gouez, p.24 ORTIGA à grand cris, afin d'attirer l'attention Un poco de respecto les ruego, Je ne fais pas que piquer y no solo soy util como abono. Quienes son ustedes para hablar de las plantas con tanta arrogancia? Nous sommes l'héritage d'une terre qui pleure, un jardin au beau milieu de l'oubli. [...] Ils récupèrent nos essences, nous enveniment de produits divers, puis nous commercialisent en gélules, pour plus de profits. Horror. Y ustedes que hacen aca, Teatro? Je n'y comprends rien à l'Art, sus vestuarios y artificios. No le entiendo nada a su multilingüismo. Multilingue ? Que voulez-vous donc dire, que prétendez-vous? L'ortie pleure. [...] PLAKOU Parfois, mon amie, il suffit simplement d'écouter et laisser reposer les mots. Le bien, le mal, le jour, la nuit, le chaud, le froid, tout cela cohabite en nous, la nature est partout, elle est en nous, et autour de nous. [...] Dans cette pièce de théâtre, nous n'avons pas toutes les vérités, c'est sûr. Par contre, nous avons l'héritage des anciens, qui nous ont offert leur sagesse, tel un présent.Dans le texte, on est au Chili ou en Bretagne, et tout le monde parle breton, français et espagnol. Un conquistador ancien parle français et mapuche. Trois policiers parlent français, breton et espagnol. Les plantes parlent aussi. l'indicible de Yerma (qui, de façon pratique, n'est donc pas littéralement indicible, ... mais seulement hispanisé). pied d'appel pour le passage vers/prose Métamorphose, p.4 Ar vamm: Serr anezhañ ba'n ti-pakajoù ma ne fell ket dit kas anezhañ ganit! Pokoù! [la mère : Laisse-le à la consigne si tu ne veux pas l'emmener ! Baisers !] , mise en présence de la langue minorisée question de l'accessibilité sans oser aborder la question du handicap linguistique. construction d'un tabou: les personnes à compréhension réduite non-valorisation du code-switching: les brittophones ont non seulement accès au breton, mais à toutes les richesses du bilinguisme/multilinguisme en soi.

	Ta gueule.] -Vous aviez rendez-vous à quelle heure? (15) refus du code > le choix neutre entre deux étrangers est le français. -Na c'hwi, da bet eur e oac'h da zont? > le choix neutre entre deux étrangers est le breton. Là un dard venimeux Là un socle trompeur plus loin ABIGAËLLE : (16) Deux personne silencieuses dans une salle d'attente: Une souche à demi-trempée Dans un liquide saumâtre 5. promotion ou effacement du bilinguisme 5.1. surtitrages, non sur-titrage et mélanges des publics le bilinguisme facilite l'accès des monolingues francophones au théâtre brittophone. Privezioù publik, arvest 1 surtitrages partiels, powerpoint/laser, allégés, iconographiques ou non-surtitrages casques audio susurrant des traductions > vivabilité financière > vivre ensemble5.2. effacement éditorial Le bilinguisme et le multilinguismes posent de réels challenges éditoriaux: JUANA Je m'appelle Juana. "Juanita"….hum Bleunioù gouez (17) Un témoignage d'ancien retentit. L'ortie s'émeut. Dentra cabro. Deus e-barzh paotr.
	ORTIGA
	Oh, va yezh karet, nag a envorioù. [ma langue aimée, que de souvenirs]