Non Gaussian matrix-valued random fields for nonparametric probabilistic modeling of elliptic stochastic partial differential operators

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Non Gaussian matrix-valued random fields for nonparametric probabilistic modeling of elliptic

stochastic partial differential operators

Christian Soize

To cite this version:

Christian Soize. Non Gaussian matrix-valued random fields for nonparametric probabilistic modeling of elliptic stochastic partial differential operators. 9th ASCE Joint Speciality Conference on Prob- abilistic Mechanics and Structural Reliability, Sandia National Laboratory, Jul 2004, Albuquerque, New Mexico, United States. pp.1-6. �hal-00688125�

Conference on Probabilistic Mechanics and Structural Reliability, PMC 2004, Albuquerque, July 26-28, 2004

Non Gaussian Matrix-Valued Random Fields for Nonparametric Probabilistic Modeling of Elliptic Stochastic Partial Differential Operators

C. Soize

University of Marne-la-Vall´ee, Paris, France [email protected]

Abstract

This paper deals with the construction of a non Gaussian positive-definite matrix-valued random field whose mathematical properties allow elliptic stochastic partial differential operators to be modeled. Such a matrix- valued random field can directly be used for modeling random uncertainties in computational sciences with a stochastic model having a small number of parameters. For instance, in three-dimensional linear elasticity, the fourth-order elasticity tensor of a random non homogeneous anisotropic elastic material is constituted of 21 dependent random fields which have to be such that the positive-definiteness property of this fourth-order tensor be verified in a given probabilistic sense. If the usual parametric probabilistic approach is used, then the identification of such a probabilistic model by using experimental data seems to be difficult. The non Gaussian positive-definite matrix-valued random field presented in this paper allows such a probabilistic model of the fourth-order tensor-valued random field to be constructed and depends only of 4 scalar parameters: three spatial correlation lengths and one parameter allowing the level of the random fluctuations to be controlled. Such a model can directly be used in the stochastic finite element method.

1. Introduction

A great challenge is the construction of construct stochastic representations for uncer- tain parameters for which probabilistic data are known and can be identified by using experimental data. Such a probabilistic model is useful in computational sciences and in particular for stochastic finite elements (Kleiberet al., 1992; Ghanem and Spanos, 2003).

For instance, consider the following deterministic elliptic partial differential operatorA on a bounded open domainΩof ³, related to the three-dimensional linear elasticity for a non homogeneous anisotropic elastic material,

A u=−

3

X

i=1

eⁱ

3

X

j=1

∂

∂x_j

3

X

k,h=1

c_ijkh(x)ε_kh(u) , (1)

in which x = (x₁, x₂, x₃) ∈ Ω ⊂ ³, where e¹ = (1,0,0), e² = (0,1,0) and e³ = (0,0,1) are the vectors of the canonical basis of ³ and where x 7→ u(x) = (u₁(x), u₂(x), u₃(x)) is a twice differentiable function from Ω into ³. The second- order strain tensor is such thatε_kh(u) = (1/2) (∂u_k/∂x_h+∂u_h/∂x_k). The fourth-order elasticity tensor c_ijkh(x) has to verify the symmetry property c_ijkh(x) = c_jikh(x) = c_ijhk(x) = c_khij(x) and, for all symmetric second-order real tensors {z_ij}ij, has to verify the positive-definiteness property,P3

i,j,k,h=1c_ijkh(x)z_khz_ij ≥ c₀P3

i,j=1z_ij², in whichc₀ is a positive constant independent ofx. For a random medium, for all xfixed inΩ, tensor {c_ijkh(x)}ijkh is replaced by a fourth-order tensor-valued random variable {C_ijkh(x)}ijkh whose mean value is {c_ijkh(x)}ijkh and which has to verify the sym- metry and the positive-definiteness properties in a probabilistic sense which has to be defined. Nevertheless, for the random case, the deterministic constant c₀ (introduced above) cannot generally be justified from a probabilistic modeling point of view. Fi- nally, x 7→ {C_ijkh(x)}ijkh is a fourth-order tensor-valued random field indexed by Ω, constituted of 21 mutually dependent random fields and the stochastic partial differential operatorAassociated with operatorAwritten as

A U=−

3

X

i

eⁱ

3

X

j=1

∂

∂x_j{

3

X

k,h=1

Cijkh(x)εkh(U)} . (2) It should be noted that the probability distribution of this fourth-order tensor-valued random field (that is to say the system of the marginal distributions) is required because the unknown solution of the stochastic boundary value problem is a nonlinear mapping of random field x 7→ {Cijkh(x)}^ijkh. If the usual parametric probabilistic approach is used, then the identification of this probability model by using experimental data seems to be difficult. This paper deals with a nonparametric construction of a random field such asx7→ {C_ijkh(x)}ijkh. For that, an ensemble of non Gaussian positive-definite matrix- valued random fields is constructed and studied which allows, for instance, the fourth-order tensor-valued random field x 7→ {C_ijkh(x)}ijkh to be modeled. Then, such a tensor- valued random field will depend only on 4 scalar parameters: three spatial correlation lengths and one parameter allowing the level of the random fluctuations to be controlled.

With such a model, the inverse problem related to the experimental identification seems to be more feasible.

The following algebraic notations are used. Let x = (x1, . . . , xn) be a vector in ⁿ. The Euclidean space ⁿ is equipped with the usual inner product (x,y) 7→<x,y>=

Pn

j=1xjyj and the associated norm kxk =< x,x>^1/2. Let ^!_n,m( ) be the set of all the (n×m) real matrices, ^!_n( ) = ^!_n,n( ) be the set of all the square (n×n) real matrices, ^!S_n( ) be the set of all the (n×n) real symmetric matrices and ^!+_n( ) be the set of all the (n×n) real symmetric positive-definite matrices. We then have

!

+n( ) ⊂ ^!Sn( ) ⊂ ^!n( ). We denote (i) the trace of the matrix [A] ∈ ^!n( ) as tr[A] = Pn

j=1[A]_jj; (ii) the transpose of [A] ∈ ^!n,m( ) as [A]^T ∈ ^!m,n( ); (iii) the operator norm of the matrix[A] ∈ ^!n,m( )as kAk = sup_kxk≤1k[A]xk, x ∈ ^m, which is such that k[A]xk ≤ kAk kxk, ∀x ∈ ^m, and ifm = n, then kAk = |λ_n|, in which |λ_n| is the largest modulus of the eigenvalues of [A]; (i^v) for [A] ∈ ^!n,m( ), we note kAk²F = tr{[A]^T[A]} = Pn

j=1

Pm

k=1[A]²_jk and for [A] in ^!_n( ), we have kAk ≤ kAkF ≤ √

nkAk.

2. Construction and properties of the ensembleSFG⁺ of homogeneous and normal- ized non Gaussian positive-definite matrix-valued random fields

2.1. Random fieldU as the germ of ensembleSFG⁺

Definition. Let d ≥ 1 be an integer. Let x 7→ U(x) be a second-order centered ho- mogeneous Gaussian random field, defined on probability space (Θ,T, P), indexed by

d, with values in . Let L₁, . . . , L_d be positive real numbers. Its autocorrelation func- tionR_U( ) = E{U(x+ )U(x)}, defined for all = (η₁, . . . , η_d)in ^d, is written as R_U( ) = ρ₁(η₁)×. . .×ρ_d(η_d)in which, for allj = 1, . . . , d, we haveρ_j(0) = 1and ρ_j(η_j) = 4L²_j/(π²η_j²) sin²(πη_j/(2L_j))forη_j 6= 0.

Properties. For all x in ^d, E{U(x)} = 0 and E{U(x)²} = 1. The random field U is mean-square continuous on ^d. Let k = (k₁, . . . , k_d) be a point in ^d and let dk = dk₁. . . dk_dbe the Lebesgue measure. Then, there is a power spectral density functionk7→

S_U(k)from ^d into ⁺, integrable, such that∀ ∈ ^d,R_U( ) = R

de^{i< ,k>}S_U(k)dk which can be written as S_U(k) = s₁(k₁)×. . .×s_d(k_d)in which, for all j = 1, . . . , d, the functionk_j 7→s_j(k_j)from into ⁺is defined bys_j(k_j) = (L_j/π)q(k_jL_j/π). The functionτ 7→ q(τ)is continuous from into ⁺, has a compact support[−1,1] and is such that q(0) = 1, q(−τ) = q(τ) and q(τ) = 1−τ for τ ∈ [0,1]. This means that

SU has a compact support. IntroducingL^U_j as the spatial correlation length relative to coordinate x_j and defined byL^U_j = R+∞

0 |R_U(0, . . . ,0, η_j,0, . . . ,0)|dη_j, it can easily be deduced that L^U_j = L_j. Consequently, parameters L₁, . . . , L_d represent the spatial correlation lengths of random fieldU.

Representation of the random field U adapted to its numerical simulation. The spatial discretization of this random field will directly be related to the spatial discretization of the elliptic stochastic partial differential operator for which germ U will be used. In general, the problem is setted on an arbitrary bounded domain Ω of ^d and the finite element method is utilized. Consequently, U has to be simulated in N given points x¹, . . . ,x^N inΩ⊂ ^d(for instance, located in the integrating points of the finite elements of the finite element mesh of domain Ω). We then have to simulate realizations of the random vectorU = (U(x¹), . . . , U(x^N)). A first representation adapted to a large value ofN is based on the usual numerical simulation of homogeneous Gaussian vector-valued random field U constructed with the stochastic integral representation of homogeneous stochastic fields. A second representation adapted to a small or moderate value of N consists in writing U = [LU]^TV in which V = (V1, . . . , VN) is an ^N-valued random variable whose componentsV₁, . . . , V_N areN independent normalized Gaussian random variables (E{Vj} = 0 and E{V_j²} = 1 forj = 1, . . . , N) and where[LU] is the upper real triangular matrix corresponding to the Chowlesky factorization[CU] = [LU]^T[LU]of the covariance matrix[CU]in^!+_N( )such that[CU]_ij =R_U(xⁱ−x^j).

2.2. EnsembleSFG⁺

Defining the family of functions{u 7→h(α, u)}α>0. Letαbe a positive real number. The functionu 7→h(α, u)from into]0,+∞[is such thatΓ_α =h(α, U)is a gamma random variable with parameterαwhileU is a normalized Gaussian random variable (E{U}= 0 and E{U²} = 1). Consequently, for all u in , we have h(α, u) = F_Γ⁻¹_α(F_U(u)) in whichu7→F_U(u) =P(U ≤u)is the cumulative distribution function of the normalized Gaussian random variableU. The functionp7→ F_Γ⁻¹_α(p)from]0,1[into]0,+∞[is the reciprocical function of the cumulative distribution functionγ 7→ FΓ^α(γ)from]0,+∞[ into]0,1[of the gamma random variableΓ_α with parameterα, which is such that, for all γ in ⁺,FΓ^α(γ) =Rγ

0 1

Γ(α)t^α−1e^−tdtin whichΓ(α)is the gamma function.

Defining the ensembleSFG⁺ of the random fieldx 7→ [Gn(x)]. The ensemble SFG⁺ is defined as the set of all the random fields x7→ [G_n(x)], defined on the probability space (Θ,T, P), indexed by ^d whered ≥ 1 is a fixed integer, with values in ^!+_n( ) where n≥2is another fixed integer, and defined as follows: (i) Let{U_jj^′(x),x∈ ^d}1≤j≤j^′≤n

ben(n+ 1)/2independent copies of the random field{U(x),x∈ ^d}. Consequently, for 1 ≤ j ≤ j^′ ≤ n, we have E{U_jj^′(x)} = 0and E{U_jj^′(x)²} = 1and the random field x7→Ujj^′(x)is completely defined. (ii) Letδbe the real number, independent ofxandn, such that0 < δ <p

(n+ 1)(n+ 5)⁻¹ < 1. This parameter will allow the dispersion of the random field to be controlled. (iii) For all x in ^d, [G_n(x)] = [L_n(x)]^T [L_n(x)]

in which[Ln(x)]is the upper (n×n)real triangular random matrix defined as follows.

The n(n+ 1)/2 random fields x 7→ [L_n(x)]_jj^′ for 1 ≤ j ≤ j^′ ≤ n, are independent.

Forj < j^′, the real-valued random fieldx 7→ [Ln(x)]jj^′, indexed by ^d, is defined by [L_n(x)]_jj^′ = σ_nU_jj^′(x)in whichσ_n is such thatσ_n = δ(n+ 1)^−1/2. For j = j^′, the positive-valued random field x 7→ [Ln(x)]jj, indexed by ^d, is defined by[Ln(x)]jj = σ_np

2h(α_j, U_jj(x))in which, forj = 1, . . . , n,α_j = (n+ 1)/(2δ²) + (1−j)/2.

Basic properties. x 7→[G_n(x)]is a homogeneous second-order mean-square continuous

random field indexed by ^d with values in^!+_n( ). In addition, the trajectories of random field x 7→ [G_n(x)] are continuous from ^d into ^!+_n( ) almost surely. For all x ∈ ^d, we haveE{k[Gn(x)]k²F} < +∞andE{[Gn(x)]} = [In]. The parameterδ is such that δ = ₁

nE{k[Gn(x)]−[In]k²F} ^1/2 which shows that E{k[Gn(x)]k²F} = n(δ² + 1).

For all x fixed in ^d, the probability distribution on ^!+_n( ) of random matrix [G_n(x)]

is explicitly calculated in (Soize, 2001) and shows that, for all x in ^d, the random variables {[G_n(x)]_ij,1 ≤ i ≤ j ≤ n} are mutually dependent. The system of the marginal probability distributions of random fieldx7→[G_n(x)]is well defined but cannot be explicitly calculated. Random field x 7→ [G_n(x)] is non Gaussian. There exists a positive constant c₀ independent of n and independent of x, but depending on δ, such that E{k[G_n(x)]⁻¹k²} ≤ c₀ < +∞ for all n ≥ 2 and for all x ∈ ^d. We then have E{k[G_n(x)]⁻¹k²F} ≤ c_n < +∞ for all n ≥ 2and for all x ∈ ^d in which c_n = n c₀. It should be noted that, since [G_n(x)] belongs to^!+_n( )almost surely, then [G_n(x)]⁻¹ exists almost surely. However, since almost sure convergence does not yield mean-square convergence, the previous result cannot simply be deduced (see Soize, 2001).

Fundamental property. LetΩbe a bounded open domain of ^d and letΩ = Ω∪∂Ωbe its closure in which∂Ωis the boundary ofΩ. We then have

E

(sup_x∈Ωk[G_n(x)]⁻¹k)² =c²_G < +∞ , (3) in which sup is the supremum and where0< c_G <+∞is a finite positive constant.

Remark concerning the proof of Eq. (3). Let us consider the case d = 1 with Ω be a compact interval of . Since the stochastic process{kGn(x)⁻¹k,x ∈ Ω ⊂ }is not a continuous local martingal with respect to an increasing family of σ-fields, the follow- ing fundamental Doob maximal inequality (Doob, 1953) E

sup_x∈Ωk[Gn(x)]⁻¹k² ≤ 4E

k[Gn(x)]⁻¹k² cannot be used. In addition, we have to consider the non Gaussian random field cased ≥ 2. Consequently, there is no known result allowing a direct proof of Eq. (3) to be obtained and a complete proof of this fundamental result is given in (Soize, 2004).

3. Construction and properties of the ensemble SFE⁺ of non Gaussian positive- definite matrix-valued random fields

3.1. Definition of the ensembleSFE⁺

Letd ≥ 1 andn ≥ 2 be two fixed integers. LetΩ be an open (or closed) bounded (or not) domain of ^d (we can have Ω = ^d). Let x 7→ [a_n(x)] be a matrix-valued field from Ω into ^!+_n( ). Then, for all x fixed in Ω, there is an upper triangular invertible matrix [L_n(x)]in ^!_n( ) such that [a_n(x)] = [L_n(x)]^T [L_n(x)]. It is assumed that: (i) there is a real positive constant 0 < c₀ < +∞ independent of x such that, for all x in Ωand for ally ∈ ⁿ, <[a_n(x)]y,y> ≥ c₀kyk²; (ii) there is a real positive constant 0< c₁ <+∞independent ofxsuch that, for allxinΩ, we havek[L_n(x)]k ≤√c₁which yields<[a_n(x)]y,y> ≤ c₁kyk², for allyin ⁿ and for allxinΩ. Consequently, for all x in Ω, we havek[a_n(x)]k ≤ c₁ and k[a_n(x)]k^F ≤ √

n c₁. The ensemble SFE⁺ is then defined as the set of all the random fieldsx7→[A_n(x)], defined on probability space (Θ,T, P), indexed byΩ, with values in^!+_n( ), such that

∀x∈Ω , [A_n(x)] = [L_n(x)]^T [G_n(x)] [L_n(x)] , (4) in whichx7→[G_n(x)]is the random field in SFG⁺, defined on(Θ,T, P), indexed by ^d and with values in^!+_n( )(see Section 2.2).

3.2. Properties of the random fieldx7→[An(x)]

Basic properties. For allxinΩ,[A_n(x)]is a random matrix with values in ⁺_n(^!), the mean function is such thatx7→ E{[An(x)]}= [a_n(x)] ∈ ⁺n(^!)andE{k[An(x)]k²} ≤ E{k[A_n(x)]k²F} ≤ n c²₁E{k[G_n(x)]k²} ≤ n c²₁E{k[G_n(x)]k²F} < +∞ which proves thatx7→[An(x)]is a second-order random field onΩ. In general, since[a_n(x)]depends on x, then the random field{[A_n(x)],x∈Ω}is non homogeneous. Nevertheless, if[a_n(x)] = [a_n]is independent of x, then the random field{[A_n(x)] = [L_n]^T [G_n(x)] [L_n],x ∈ Ω} can be viewed as the restriction to Ω of a homogeneous random field indexed by ^!d. We haveE{k[A_n(x)]−[a_n(x)]k²F} = {δ²/(n+1)}{k[a_n(x)]k²F + (tr[a_n(x)])²}. The dispersion parameter, defined byδ_An(x) ={E{k[A_n(x)]−[a_n(x)]k²F}/k[a_n(x)]k²F}^1/2, is such thatδ_An(x) = (δ/√

n+1){1 + (tr[a_n(x)])²/tr{[a_n(x)]²}}^1/2.

Spatial correlation lengths for the homogeneous case. Then (see above),δ_An(x) =δ_An is independent ofx. Let = (η₁, . . . , η_d)7→r^An( )be the function defined from^!d into^! byr^An( ) =trE{([A_n(x+ )]−[a_n]) ([A_n(x)]−[a_n])}/E{k[A_n(x)]−[a_n]k²F}. We have r^An(0) = 1 andr^An(− ) = r^An( ). For allj = 1, . . . , d, the spatial correlation length L^A_jⁿ of the homogeneous random field x 7→ [A_n(x)] indexed by ^!d, relative to coordinatex_j, can then be defined byL^A_jⁿ =R+∞

0 |r^An(0, . . . ,0, η_j,0, . . . ,0)|dη_j. 4. Elliptic stochastic partial differential operator

The presentation is limited to the second-order stochastic differential operator defined by Eq. (2) on an open bounded domainΩ of^!3 whose boundary∂Ωis written as Γ₀∪Γ.

OnΓ₀, there is a zero Dirichlet boundary condition. We introduce the real Hilbert spaces H = (L²(Ω))³ and V = {u ∈ (H¹(Ω))³, u = 0 on Γ₀} whose inner products are denoted by <u,w>_H and<u,w>_V respectively, and where the associated norms are denoted by kukH and kukV respectively. Let ^" = L²(Θ, H) and ^# = L²(Θ, V)) be the real Hilbert spaces of all the second-order random variables θ 7→ {x 7→ U(x, θ)} defined on probability space (Θ,T, P), with values in H and V respectively, equipped with the inner products≪U,W≫ =E{<U,W>_H}and≪U,W≫^!=E{<U,W>_V} respectively, and where the associated norms are denoted bykUk andkUk^!respectively.

4.1. Weak formulation of the elliptic stochastic partial differential operator.

Letn= 6and let us introduce the new indicesI andJ belonging to{1, . . . ,6}such that I = (i, j)andJ = (k, h)with the following correspondence: 1 = (1,1),2 = (2,2),3 = (3,3),4 = (1,2),5 = (2,3) and 6 = (3,1). Thus, for all x in Ω, we introduce the matrix [a_n(x)] in ⁺_n(^!) such that [a_n(x)]_IJ = c_ijkh(x) and the random (n×n) real matrix[An(x)]such that[An(x)]IJ = Cijkh(x). A nonparametric probabilistic model of the random fourth-order elasticity tensorC_ijkh(x)consists in choosing the random field x7→[An(x)]in SFE⁺with the mean value[a_n(x)] =E{[An(x)]}. The weak formulation of the stochastic partial differential operator defined by Eq. (2) leads the random bilinear form(U,W)7→K(U,W)on^#×^#to be introduced, such that

K(U,W) = Z

Ω

<[A_n(x)]e(U(x)),e(W(x))> dx , (5) in whiche(u) = (ε₁₁(u), ε₂₂(u), ε₃₃(u),2ε₁₂(u),2ε₂₃(u),2ε₃₁(u)).

4.2. Ellipticity of the random bilinear form

Let(U,W)7→ K(U,W)be the bilinear form on^#×^#defined byK(U,W) =E{K(U,W)}. If the following property was introduced: for all x ∈ Ω and for all ^!n-valued random

variableYdefined on(Θ,T, P),<[An(x)]Y,Y>≥ ckYk²a.s in which0< c <+∞ is independent ofx, then the bilinear form(U,W)7→ K(U,W)on × would be coercive in (i.e. -elliptic) because, we would have K(U,U) ≥ cER

Ωke(U(x))k²dx ≥ c_KkUk² with 0 < c_K < +∞. This property, which is generally not coherent with the available information which can be deduced from objective data, does not hold for the random field x 7→ [A_n(x)]belonging to SFE⁺ and consequently, the usual analysis given above cannot presently be used. Another analysis has to be developed using the fundamental property defined by Eq. (3): it is proved (Soize, 2004) that, for all random field{x7→U(x)}in , we have

pE{K(U,U)²} ≥ c_KkUk² , (6) in whichc_K is a positive finite real constant. It should be noted that Eq. (6) differs from equationE{K(U,U)} ≥ cKkUk² due to the fact that the two positive-valued random variables sup_x∈Ωk[G_n(x)]⁻¹kandK(U,U)are dependent.

4.3. Existence and uniqueness of a weak second-order stochastic solution for a stochastic BVP

Letw7→f(w)be a given continuous linear form onV, that is to say such that|f(w)| ≤ c_fkwkV with0< c_f <+∞. Then, the following random problem: find a random field {x7→U(x)}in such that, for allW∈ ,K(U,W) =f(W)a.s , has a unique stochastic solution{x7→U(x)}in .

The proof can easily be constructed. From equationsK(U,W) = f(W)and |f(w)| ≤ c_fkwkV, we deduce that K(U,U) ≤ c_fkUkV and consequently, E{K(U,U)²} ≤ c²_fE{kUk²V}. Using Eq. (6) yields c²_KkUk⁴ ≤ c²_f kUk² which can be rewritten as kUk ≤c_U <+∞withc_U = c_f/c_K and yields the existence. Finally, the proof of the uniqueness is straightforward because, ifUandU^′ are two solutions in , for allWin , we haveK(U−U^′,W) = 0a.s and thusE{K(U−U^′,W)²} = 0. TakingW=U−U^′ and from Eq. (6) yieldkU−U^′k² = 0, i.e.,U=U^′ in .

5. Conclusions

We have presented the mathematical construction of a non Gaussian positive-definite (n×n)real matrix-valued random field, indexed by any domain of^!d, depending only on its mean function and on a smaller number of scalar parameters constituted of a dispersion parameter anddspatial correlation lengths. Such a random field is adapted to the inverse problem relative to the experimental identification. A fundamental mathematical property is proved and allows the ellipticity of stochastic partial differential operators to be obtained.

References

Doob, J.L. (1953),Stochastic Processes, John Wiley and Sons, New York (Wiley Classics Library Edition Published 1990).

Ghanem,R. and P.D. Spanos (2003), Stochastic Finite Elements: A spectral Approach, revised edition, Dover Publications, New York.

Kleiber, M., D.H. Tran and T.D. Hien (1992),The Stochastic Finite Element Method, John Wiley and Sons, New York.

Soize, C. (2001), "Maximum entropy approach for modeling random uncertainties in transient elastodynamics",J. Acoust. Soc. Amer.,109(5),1979-1996.

Soize, C. (2004), "Non Gaussian positive-definite matrix-valued random fields for elliptic stochastic partial differential operators", Computer Methods in Applied Mechanics and Engineering, submitted in March 2004.