Spectral measure and approximation of homogenized coefficients

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coefficients

Antoine Gloria, Jean-Christophe Mourrat

To cite this version:

Antoine Gloria, Jean-Christophe Mourrat. Spectral measure and approximation of homogenized coefficients. Probability Theory and Related Fields, Springer Verlag, 2012, 154, pp.287-326.

�10.1007/s00440-011-0370-7�. �inria-00510513v2�

ANTOINE GLORIA & JEAN-CHRISTOPHE MOURRAT

Abstract. This article deals with the numerical approximation of effective coefficients in stochastic homogenization of discrete linear elliptic equations. The originality of this work is the use of a well-knownabstract spectral representation formula to design and analyze effective andcomputableapproximations of the homogenized coefficients.

In particular, we show that information on the edge of the spectrum of the generator of the environment viewed by the particle projected on the local drift yields bounds on the approximation error, and conversely. Combined with results by Otto and the first author in low dimension, and results by the second author in high dimension, this allows us to prove that for any dimensiond≥2, there exists an explicit numerical strategy to approximate homogenized coefficients which converges at the rate of the central limit theorem.

Keywords: stochastic homogenization, spectral theory, ergodic theory, numerical method.

2010 Mathematics Subject Classification: 35B27, 37A30, 65C50, 65N99.

1. Introduction

We consider a discrete elliptic operator−∇^∗·A∇, where∇^∗·and∇are the discrete back- ward divergence and forward gradient, respectively. For all z ∈Z^d, A(z) is the diagonal matrix whose entries are the conductancesω(z, z+e_i) of the edges (z, z+e_i) starting atz, where{e_i}i∈{1,...,d} denotes the canonical basis of Z^d. The values of the conductances are random and their realizations are assumed to be independent and identically distributed.

In what follows, for any random variableX taking values in R^n×m (n, m ≥1) we denote by hXi the expectation of X in the underlying probability space componentwise. If X has bounded second moments componentwise, we denote by var [X] its variance, that is var [X] :=

|X− hXi |²

, where| · |is the Euclidian norm onR^n×m.

Provided that the conductances lie in a compact set of (0,+∞), standard homogenization results (see for instance [13, Theorem 4] and [12, Corollary 1]) ensure that there exists somedeterministic matrixA_hom such that the solution operator of the deterministic con- tinuous differential operator−∇ ·A_hom∇describes the large scale behavior of the solution operator of the random difference operator −∇^∗·A∇ almost surely. As a by-product of this homogenization result, one obtains a characterization of the homogenized coefficients A_hom: it is shown that for every direction ξ ∈ R^d, there exists a unique scalar field φ such that ∇φ is stationary, h∇φi= 0 (vanishing expectation), which solves the corrector equation

−∇^∗·A(ξ+∇φ) = 0 inZ^d, (1.1)

Date: May 13, 2011.

and normalized byφ(0) = 0. With this corrector, the homogenized coefficients A_hom can be characterized as

ξ·A_homξ = h(ξ+∇φ)·A(ξ+∇φ)i (1.2) (since the random fieldx7→(ξ+∇φ)·A(ξ+∇φ)(x) is stationary, the point at which the expectation is taken is irrelevant).

In this article, we are interested in numerical approximations of these effective coefficients A_hom. From the practical point of view, (1.2) is not of immediate interest since the corrector equation (1.1) has to be solved

• for every realization of the coefficients ω,

• on the wholeZ^d.

Ergodicity allows one to replace the expectation by a spatial average (on increasing do- mains) almost surely. To approximate φ, one usually uses φ_R, the unique solution to equation (1.1) on some large but finite domain Q_R = (−R/2, R/2)^d, completed by say periodic or homogeneous Dirichlet boundary conditions (see for instance [5]). Yet, the comparison of∇φ_R to ∇φis not obvious since ∇φ_R and ∇φare not “jointly stationary”.

In order to avoid this difficulty, Otto and the first author have used a somewhat different strategy, that proceeds in two steps. First,φis replaced by its standard regularization φ_µ, unique stationary solution to the modified corrector equation

µφ_µ− ∇^∗·A(ξ+∇φ_µ) = 0 inZ^d

for some smallµ >0 (see the original contribution by Papanicolaou and Varadhan in the continuous case [17], and the discrete counterpart by K¨unnemann [13]). Then,φ_µ itself is replaced byφ_µ,R, the unique weak solution to

µφµ,R− ∇^∗·A(ξ+∇φµ,R) = 0 inQR∩Z^d, φµ,R = 0 on Z^d\QR. The advantages are twofold:

• ∇φand ∇φ_µ are jointly stationary, which is of great help for the analysis,

• φµis accurately approximated byφµ,Ron domains of the formQL= (−L/2, L/2)^d provided that (R−L)√µ≫1, due to the exponential decay of the Green’s function associated with µ− ∇^∗·A∇inZ^d (see [8]), so that we only focus on φ_µ and not φ_µ,R from now on.

In particular, we may approximateA_hom by the following average ξ·A_µ,1,Lξ :=

Z

QL

(ξ+∇φ_µ)·A(ξ+∇φ_µ)χ_L(x)dx,

where χL is a smooth mask supported on QL and of mass one. Note that Aµ,1,L is a stationary random variable itself. Let us stress the fact that µwill be small (the pertur- bation is sent to zero) whereasL will be large (to apply the ergodic theorem). In [9], it is observed that theL²-norm of the error in probability takes the form

D ξ·A_µ,1,Lξ−ξ·A_homξ2E

= var [ξ·A_µ,1,Lξ] + ξ·(A_µ,1−A_hom)ξ2

, (1.3)

where

ξ·A_µ,1ξ := h(ξ+∇φ_µ)·A(ξ+∇φ_µ)i.

The first term of the r. h. s. of (1.3) is stochastic in nature: it measures the fluctuations of A_µ,1,L around its mean value. It is called the random error. The second term is a

2

deterministic error related to the fact that we have modified the corrector equation by a zero-order term. It is refered to as thesystematic error, and will be the main focus of this work.

In [9], it is proved that the random error depends on the dimension: there exists q de- pending only on the ellipticity constantsα, β such that

var [A_µ,1,L]^1/2 .

(L⁻¹+µ) ln^qµ if d= 2,

L^−d/2(1 +µL) if d >2, (1.4) where.stands for≤up to some constant depending only onα, β andd. In particular, in the regimeµL.1, the random error has the scaling of the central limit theorem (in other words the energy density of the approximate corrector behaves as if it were independent from site to site). The systematic error has been identified in [10]. It also depends on the dimension for d < 5, but saturates at d = 5: there exists q depending only on the ellipticity constants α, β such that

|A_µ,1−A_hom| .

µln^qµ⁻¹ if d= 2, µ^3/2 if d= 3, µ²lnµ⁻¹ if d= 4, µ² if d >4.

(1.5)

These two estimates are optimal (up to some possible logarithmic corrections ford= 2).

In order to useφ_µ,R as a proxy for φ_µ on Q_L, at first order we may take µ⁻¹∼L² ∼R². Hence, the random error dominates up to d = 8, so that the convergence rate of the numerical strategy is optimal (it coincides with the central limit theorem scaling, which is an upper bound):

D ξ·A_L⁻2,1,Lξ−ξ·A_homξ2E1/2

.

L⁻¹ln^qL if d= 2, L^−d/2 if 2< d≤8.

Yet, for d >8, the systematic error dominates and the numerical strategy is not optimal any longer:

D

ξ·A_L⁻²_,1,Lξ−ξ·A_homξ2E1/2

. L⁻⁴ if 8< d.

The aim of this paper is to introduce new formulas for the approximation ofA_hom using the modified correctorφ_µ(possibly with differentµ’s) in order to reduce thesystematic er- ror. In early and seminal papers on stochastic homogenization (for instance [17] and [11]) spectral analysis has been used to prove uniqueness of correctors, and devise a spectral representation formula forA_hom. In particular, another point of view on the homogeniza- tion of discrete elliptic equations with random i. i. d. coefficients consists in studying a random walk in the random environment described by these i. i. d. conductances on Z^d. Adopting this point of view, Kipnis and Varadhan [11] have proved the convergence of the random walk in the diffusive scaling to a Brownian motion with covariance matrix 2A_hom for the annealed law (that is to say, after averaging over the random environment). This result has then been extended in [6] to general integrable conductances. Convergence for almost every environment has been obtained for uniformly elliptic conductances in [18], and has recently been extended to more general conductances in [2, 15, 3, 14, 1].

Denoting by −L the generator of the environment viewed by the particle, and by ed

its spectral measure projected on the local drift d= ∇^∗·Aξ (see Section 3), Kipnis and

Varadhan [11] have shown that

ξ·A_homξ = hξ·Aξi − Z

R₊

1

λded(λ),

whereR₊= [0,+∞). As noticed by the second author in [16],A_µ,1 can also be written in terms of the spectral measureed(see Section 2 for details):

ξ·A_µ,1ξ = hξ·Aξi − Z

R₊

λ+ 2µ

(µ+λ)²ded(λ)

= ξ·A_homξ+µ² Z

R₊

1

λ(µ+λ)²ded(λ).

The key idea of the present paper is to use this spectral representation in order to design approximations ofA_hom at an abstract level first, and then go back to physical space and obtain formulas in terms of the modified correctors φ_µ. We shall actually introduce, for every integer k ≥ 1, an approximation A_µ,k of A_hom defined in terms of φ_µ, . . . , φ₂k−1µ, and prove that, up to logarithmic corrections, the difference |A_hom−A_µ,k|is bounded in our discrete stochastic setting by

µmin{2k,d/2} if d≤6,

µmin{2k,max(3,d/2−3)} if d >6, (1.6) (see Theorem 3 for a more precise statement). The systematic error associated with the new approximations can be made of a higher order than (1.5) as soon asd≥4. The proof of these estimates relies on the observation that the systematic error is controlled by the edge of the spectrumed((0, µ)). In turn, the systematic error also controls the edge of the spectrum (see Theorem 4 for a precise statement), so that estimating the systematic error is equivalent to quantifyinged((0, µ)).

As we shall also prove, the variance estimate (1.4) is unchanged if A_µ,1 is replaced by A_µ,k for all k ≥ 1. In particular, if we keep µ⁻¹ ∼ L², we obtain a numerical strategy whose convergence rate is optimal with respect to the central limit theorem scaling in the stochastic case, for any d ≥ 2. This improves and completes for d > 8 the series of papers [9, 10, 8] by Otto and the first author on quantitative estimates in stochastic homogenization of discrete elliptic equations. In turn, we also obtain “optimal” bounds oned((0, µ)) up tod= 6 (see Theorem 5), thus improving the corresponding results of the second author in [16].

Note however that the bounds (1.6) are not yet optimal: the systematic error is expected to behave asµmin{2k,d/2} in any dimension (up to logarithmic corrections), see also Remark 1 for the equivalent statement in terms of the edge of the spectrum. We wish to address this issue in a future work.

The article is organized as follows. Although the main focus of this work is on stochastic homogenization of discrete elliptic equations, we first describe the strategy on the elemen- tary case of periodic homogenization of continuous elliptic equations in Section 2 (this new strategy may indeed be valuable to numerical homogenization methods, see in particular [7] for related issues). We introduce the spectral decomposition formula for the homog- enized coefficients. The binomial formula then provides with natural approximations of the homogenized coefficients in terms of the associated spectral measure. We conclude

4

the section by rewriting these formulas in physical space using solutions to the modi- fied corrector equation, which yields newcomputable approximations of the homogenized coefficients. In particular, this generalizes the method introduced in [7] and makes the systematic error decay arbitrarily fast. Some numerical tests displayed in Appendix B illustrate the sharpness of the analysis.

We turn to the core of this article in Section 3: the stochastic homogenization of discrete elliptic equations. We first recall the spectral decomposition of the generator of the en- vironment viewed by the particle. The algebra is the same as in the continuous periodic case, so that the formulas we obtain in Section 2 adapt mutatis mutandis to the discrete stochastic case. Yet, the error analysis is more subtle. We show that the asymptotic behavior of the systematic error is driven by the behavior of the edge of the spectrum of the generator. Using results of [16] in high dimension, and results in the spirit of [10]

(see Lemma 5 and Appendix A) in low dimension, we obtain estimates on the edge of this spectrum, which show that the systematic error is effectively reduced in high dimensions (although our bounds are not optimal whend >6). We then note that the variance esti- mates derived in [9] also hold for these approximations, thus concluding the error analysis of the numerical strategy.

We will make use of the following notation:

• d≥2 is the dimension;

• In the discrete case, R

Zddx denotes the sum over x ∈ Z^d, and R

Ddx denotes the sum over x∈Z^d such thatx∈D,D open subset ofR^d;

• h·i is the average in the periodic case, and the expectation in the stochastic case;

• var [·] is the variance in the stochastic case;

• . and & stand for ≤ and ≥up to a multiplicative constant which only depends on the dimensiond and the constantsα, β (the ellipticity constants of the matrix A, see Definitions 1 and 5) if not otherwise stated;

• when both. and& hold, we simply write∼;

• we use≫ instead of & when the multiplicative constant is (much) larger than 1;

• (e1, . . . ,e_d) denotes the canonical basis of R^d.

2. The continuous periodic case

Definition 1. LetA:R^d→ Md(R) be aQ= (0,1)^d-periodic symmetric diffusion matrix which is uniformly continuous and coercive with constantsβ ≥α >0: for almost allx∈Q and allξ ∈R^d,|Aξ| ≤β|ξ|and ξ·Aξ ≥α|ξ|². The associated homogenized matrix A_hom is characterized for allξ ∈R^d by

ξ·A_homξ = h(ξ+∇φ)·A(ξ+∇φ)i

= Z

Q

(ξ+∇φ)·A(ξ+∇φ)dx,

whereh·idenotes the average on the periodic cellQ, andφis the uniqueQ-periodic weak solution to

−∇ ·A(ξ+∇φ) = 0 (2.1)

with zero averagehφi= 0.

Let us define E : H_per¹ (Q)×H_per¹ (Q) → R,(ψ, χ) 7→ R

Q∇ψ·A∇χdx the Dirichlet form associated withA. We (abusively) refer to the quadratic formψ→ E(ψ, ψ) as theDirichlet form as well. One may write the homogenized matrix as

ξ·A_homξ = hξ·Aξi − E(φ, φ). (2.2) Indeed, the weak formulation of (2.1) implies

Z

Q∇φ·A(ξ+∇φ)dx = 0, (2.3)

and therefore Z

Q

ξ·A∇φdx = Z

Q∇φ·Aξdx ^(2.3)= − Z

Q∇φ·A∇φdx = −E(φ, φ).

The objective of this section is to use a spectral decomposition to design approximations forE(φ, φ).

2.1. Spectral decomposition.

Definition 2. Let A be as in Definition 1. We let L⁻¹ denote the inverse of the el- liptic operator L = −∇ ·A∇ with periodic boundary conditions on L²₀(Q) = {v ∈ L²(Q)|R

Qv(x)dx = 0}. It is a well-defined compact operator by generalized Poincar´e’s inequality, Riesz’s, and Rellich’s theorems.

By Hilbert-Schmidt’s theorem, there exist an orthonormal basis {ψ_i}i>0 of L²₀(Q) and positive eigenvalues {λ_i}i>0 (in increasing order) such that for all i > 0, L⁻¹ψ_i = _λ¹

iψ_i. By definition, ψ_i ∈ H_per¹ (Q). Setting ψ₀ ≡ 1 and λ₀ = 0, one may then characterize H_per¹ (Q) as

H_per¹ (Q) = (

u=X

i∈N

α_iψ_i

X

i∈N

(1 +λ_i)α²_i <∞ )

.

By Riesz’s representation theorem, this also implies for the dualH_per⁻¹(Q) ofH_per¹ (Q):

H_per⁻¹(Q) = (

f =X

i∈N

β_iψ_i

X

i∈N

β_i² 1 +λi

<∞ )

. (2.4)

Hence, for all f ∈ H_per⁻¹(Q) such that hf,1iHper⁻¹,H_per¹ = 0, the unique weak solution u ∈ H_per¹ (Q) to

−∇ ·A∇u = f is given by

u = X

i∈N\{0}

hf, ψ_ii_Hper⁻¹_,H¹_per λ_i ψ_i.

For all f ∈H_per⁻¹(Q) such thathf,1i_Hper⁻¹_,Hper¹ = 0, we define the spectral measure e_f of L projected on f by

e_f = X

i∈N

hf, ψ_ii_Hper⁻¹_,Hper¹ δ_λi, (2.5) 6

whereδ_λi is the Dirac mass onλi. The above characterizations ofH¹ andH⁻¹ then allow us to give a mathematical meaning to the formal functional calculus

hf,Ψ(L)(f)i_Hper⁻¹_,Hper¹ = Z

R₊

Ψ(λ)de_f(λ),

for every continuous function Ψ : [0,+∞)→Rsuch that λΨ(λ) . 1 asλ→ ∞.

We are now in position to express the Dirichlet form of the corrector φ in terms of the spectral measure projected on the “local drift”

d:=∇ ·Aξ ∈H_per⁻¹(Q). (2.6)

In particular,

E(φ, φ) = hLφ, φi_Hper⁻¹_,Hper¹

=

LL⁻¹d,L⁻¹d

Hper⁻¹,H¹_per

= Z

R₊

1

λded(λ). (2.7)

Let us then turn to the approximation ofA_hom used in [7], that is

ξ·A_µξ := h(ξ+∇φ_µ)·A(ξ+∇φ_µ)i, (2.8) whereφ_µ∈H_per¹ (Q) is the unique weak solution to the modified corrector equation

µφ_µ− ∇ ·A(ξ+∇φ_µ) = 0, (2.9) that we more compactly write as (µ+L)φ_µ = d. In this case, the weak formulation of the equation implies

Z

Q∇φ_µ·A(ξ+∇φ_µ)dx = −µ Z

Q

φ²_µdx, (2.10)

so that the defining formula forA_µ turns into

ξ·A_µξ = hξ·Aξi − E(φ_µ, φ_µ)−2µ φ²_µ

. (2.11)

Proceeding as above, we rewrite the last two terms of the r. h. s. of (2.11) as E(φ_µ, φ_µ) =

L(µ+L)⁻¹d,(µ+L)⁻¹d

Hper⁻¹,H¹_per

=

d,L(µ+L)⁻²d

Hper⁻¹,H¹_per

= Z

R₊

λ

(µ+λ)²ded(λ), (2.12)

and

φ²_µ

=

(µ+L)⁻¹d,(µ+L)⁻¹d

L²,L²

=

d,(µ+L)⁻²d

Hper⁻¹,H¹_per

= Z

R₊

1

(µ+λ)²ded(λ). (2.13)

The combination of (2.2), (2.7), (2.11), (2.12) & (2.13) allows us to express the difference betweenA_µ and A_hom in terms of the spectral measure of L projected on the local drift d, as observed in [16, Addendum]:

ξ·(A_µ−A_hom)ξ = E(φ, φ)− E(φ_µ, φ_µ)−2µ φ²_µ

= Z

R₊

1

λ− λ

(µ+λ)² − 2µ (µ+λ)²

ded(λ)

= Z

R₊

µ²

λ(µ+λ)²ded(λ).

Not only does this identity suggest that |A_µ −A_hom| ∼ µ² (as proved by a different approach in [7]), but it also gives a strategy to construct approximations of A_hom at any order. In particular, for allk∈N, we write

ξ·A_homξ = hξ·Aξi − Z

R₊

(µ+λ)^2k

λ(µ+λ)^2kded(λ)

= hξ·Aξi − Z

R₊

(µ+λ)^2k−µ^2k

λ(µ+λ)^2k ded(λ)− Z

R₊

µ^2k

λ(µ+λ)^2kded(λ) and set

ξ·A˜_µ,kξ := hξ·Aξi − Z

R₊

(µ+λ)^2k−µ^2k λ(µ+λ)^2k ded(λ)

= hξ·Aξi −

2k−1

X

j=1

2k j

Z

R₊

µ^jλ^2k−1−j

(µ+λ)^2k ded(λ). (2.14) Note that the only operator which has to be inverted to compute ˜A_µ,k is indeed µ+L, and notL, as desired. In addition, this definition implies

ξ·( ˜A_µ,k−A_hom)ξ = Z

R₊

µ^2k

λ(µ+λ)^2kded(λ), which suggests that the error is now of orderµ^2k.

In view of formula (2.14), the effective computation of (µ+L)^−kd is needed in practice to obtain ˜A_µ,k. This is a big handicap for the numerical method since the numerical inversion ofµ+Lhas to be iterated ktimes, which dramatically magnifies the numerical error. Fortunately, one may use a sligthly different approximation of A_hom which avoids this drawback, as shown in the following subsection.

2.2. Abstract approximations. Let us first introduce functionsd_µ,k, which are defined as linear combinations of φ_µ, . . . , φ₂^k−1µ (and therefore easily computable) and will serve as substitutes for (µ+L)^−kd.

Definition 3. Let A and Q, L, and d be as in Definition 1, Definition 2, and (2.6), respectively. For allµ >0, the sequence of functionsd_µ,k∈H_per¹ (Q) is defined by its first term

d_µ,1 =φ_µ= (µ+L)⁻¹d, c₁= 1, (2.15) 8

and by the induction rule

d_µ,k+1 =c_kµ⁻¹(d_µ,k−d_2µ,k), c_k+1 = 2

c_k + 1 −1

. (2.16)

Defined this way, the functionsd_µ,k satisfy the following fundamental properties:

Proposition 1. Let d_µ,k be as in Definition 3, then for all µ >0 andk≥1, we have

d_µ,k+1 = (µ+L)⁻¹d_2µ,k, (2.17)

Ld_µ,k+1 = (1 +c_k)d_2µ,k−c_kd_µ,k. (2.18) Proof. Identity (2.18) is a direct consequence of (2.16) & (2.17), and we only need to prove the latter. We proceed by induction. Let us first check that it is indeed true fork = 1.

By definition of d_µ,1, we have

(µ+L)d_µ,1 =d, (2.19)

and as a consequence,

(µ+L)d_2µ,1 =d−µd_2µ,1. (2.20) Combining (2.19) and (2.20), one obtains :

(µ+L)(d_µ,1−d_2µ,1) =µd_2µ,1,

from which it follows thatd_µ,2 = (µ+L)⁻¹d_2µ,1. Let us now assume that (2.17) is satisfied at levelk≥1. Similarly, we have

(µ+L)d_µ,k+1 =d_2µ,k, (µ+L)d_2µ,k+1=d_4µ,k−µd_2µ,k+1.

Using these equalities, together with the definition (2.16) ofd_µ,k+1, we are led to (µ+L)(d_µ,k+1−d_2µ,k+1) =d_2µ,k−d_4µ,k+µd_2µ,k+1 =µ

2 c_k + 1

d_2µ,k+1,

and thus,d_µ,k+2 = (µ+L)⁻¹d_2µ,k+1.

In order to be consistent with (2.17), we setd_µ,0 =d for all µ >0.

We are now in position to define a suitable approximation ofA_hom. The idea is to use the identity

1

λ = (µ+λ)²(2µ+λ)². . .(2^k−1µ+λ)² λ(µ+λ)²(2µ+λ)². . .(2^k−1µ+λ)²

in (2.7), expand, and take advantage of Proposition 1 to efficiently compute terms of the form d_µ,k = (µ+L)⁻¹(2µ+L)⁻¹. . .(2^k−1µ+L)⁻¹d. This gives rise to the following (abstract) approximations ofA_hom, and systematic errors:

Theorem 1.LetAbe as in Definition 1, andA_hom be the associated homogenized diffusion matrix. For any fixedξ ∈R^d such that |ξ|= 1, we denote by edthe spectral measure (2.5) ofL=−∇·A∇projected on the local driftd=∇·Aξ. For allk∈N, we letP_k :R×R→R be the polynomial given by

P_k(µ, λ) = λ⁻¹

(µ+λ)²(2µ+λ)²· · ·(2^k−1µ+λ)²−2^k(k−1)µ^2k

, (2.21)

and for all µ >0, we define the approximation A_µ,k of A_hom by ξ·A_µ,kξ = hξ·Aξi −

Z

R₊

P_k(µ, λ)

(µ+λ)²(2µ+λ)²· · ·(2^k−1µ+λ)²ded(λ). (2.22)

Then the systematic error satisfies

0 ≤ ξ·(A_µ,k−A_hom)ξ ≤ 2^k(k−1)

|A|² 1

4απ² 2k

µ^2k. (2.23) Proof. Starting point is the identity

ξ·(A_µ,k−A_hom)ξ = Z

R₊

1

λ− P_k(µ, λ)

(λ+µ)²· · ·(λ+ 2^k−1µ)²

ded(λ)

= Z

R₊

2^k(k−1)µ^2k

λ(λ+µ)²· · ·(λ+ 2^k−1µ)² ded(λ),

which is a direct consequence of (2.2), (2.7), and (2.22). From this identity, and using Def- inition 2, we infer that the systematic error is smaller than and asymptotically equivalent to Cµ^2k (asµtends to 0), where C >0 is given by

C := 2^k(k−1) Z

R₊

1

λ^2k+1 ded(λ) = 2^k(k−1)X

i∈N

hd, ψii²_Hper⁻¹_,Hper¹

λ^2k+1_i . (2.24) In order to estimateC via (2.24), we compare the spectral gapλ₁ ofLto the spectral gap λ⁰₁ of−△ on H_per¹ (Q). By comparison of the two Dirichlet forms, we have

λ1 ≥ αλ⁰₁.

The spectrum of the Laplace operator onH_per¹ (Q) is explicitly known, and the spectral gap given byλ⁰₁ = 4π². Hence, recalling thathd,1i_Hper⁻¹_,Hper¹ = 0 and using the characterization (2.4) ofH_per⁻¹(Q), one may bound the r. h. s. of (2.24) by

C ≤ 2^k(k−1) 1

(αλ⁰₁)^2k X

i∈N

hd, ψ_ii²_Hper⁻¹_,Hper¹ λ_i

= 2^k(k−1)kdk²_Hper⁻¹ 1

4απ² 2k

≤ 2^k(k−1)

|A|² 1

4απ² 2k

,

as desired.

2.3. New formulas for the approximation of homogenized coefficients. In this subsection, we show how to rewrite the approximations A_µ,k of A_hom introduced in The- orem 1 in terms of the modified correctorsφµ, φ2µ, . . . , φ₂^k−1_µ. We proceed by induction.

Proposition 2. Let c_k be as in Definition 3. We define the sequence {a_k,i}k≥1,i∈{0,...,k−1}

bya_1,0 = 1 and the induction rules a_k+1,0 = c_ka_k,0,

a_k+1,i = c_ka_k,i−2^1−kc_ka_k,i−1 for i∈ {1, k−1}, a_k+1,k = −2^1−kc_ka_k,k−1.

10

Within the assumptions and notation of Theorem 1, the approximations A_µ,k of A_hom satisfy the formula: for all ξ∈R^d,

ξ·A_µ,kξ = h(ξ+∇φµ)·A(ξ+∇φµ)i +µ

k−1

X

i=0

η_k,iD φ²₂iµ

E+µ

k−1

X

i=0 k−1

X

j>i

ν_k,i,j

φ₂iµφ₂jµ

, (2.25) where the{φ₂ⁱ_µ}i∈Nare the modified correctors associated withξ through (2.9), and the co- efficients{η_k,i}k≥1,0≤i<k, and {ν_k,i,j}k≥2,0≤j<k,0≤i<j are defined by the initial valueη_1,0 = 0, and the induction rules

η_k+1,i = η_k,i+ (2^k(k−1)+i−2^k2+1)a²_k+1,i for i∈ {0, k−1}, η_k+1,k = −2^k2a²_k+1,k,

ν_k+1,i,k = 2^k(k−1)+i−3×2^k2

a_k+1,ia_k+1,k, ν_k+1,i,j = ν_k,i,j+ 2^k(k−1)(2ⁱ+ 2^j)−2^k2+2

a_k+1,ia_k+1,j for j∈ {0, k−1}. Note that{ν_k,i,j}k≥1,0≤j<k,0≤i<j does not require further initialization.

Proof. We proceed in four steps.

Step 1. Proof that for allk≥1,

ξ·A_µ,k+1ξ = ξ·A_µ,kξ−2^k(k−1)µ^2kE(d_µ,k+1,d_µ,k+1)−2^k2+1µ^2k+1

d²_µ,k+1

. (2.26) In order to prove (2.26), we first note that the polynomialsP_kdefined in (2.21) satisfy the identity

P_k+1(µ, λ) = (2^kµ+λ)²P_k(µ, λ) + 2^k(k−1)λµ^2k+ 2^k2+1µ^2k+1. (2.27) Hence, formula (2.22) implies

ξ·A_µ,k+1ξ = hξ·Aξi − Z

R₊

P_k+1(µ, λ)

(µ+λ)²(2µ+λ)²· · ·(2^kµ+λ)²ded(λ)

(2.27)

= hξ·Aξi − Z

R₊

(2^kµ+λ)²P_k(µ, λ)

(µ+λ)²(2µ+λ)²· · ·(2^kµ+λ)²ded(λ)

− Z

R₊

2^k(k−1)λµ^2k+ 2^k2+1µ^2k+1

(µ+λ)²(2µ+λ)²· · ·(2^kµ+λ)²ded(λ)

(2.22)

= ξ·A_µ,kξ− Z

R₊

2^k(k−1)λµ^2k+ 2^k2+1µ^2k+1

(µ+λ)²(2µ+λ)²· · ·(2^kµ+λ)²ded(λ).

From (2.17) in Proposition 1, we infer that (µ+L)⁻¹· · ·(2^kµ+L)⁻¹d =d_µ,k+1, so that the above identity turns into

ξ·A_µ,k+1ξ = ξ·A_µ,kξ−2^k(k−1)µ^2khLd_µ,k+1,d_µ,k+1i_Hper⁻¹_,Hper¹

−2^k2+1µ^2k+1hd_µ,k+1,d_µ,k+1i_Hper⁻¹_,Hper¹

= ξ·A_µ,kξ−2^k(k−1)µ^2kE(d_µ,k+1,d_µ,k+1)−2^k2+1µ^2k+1

d²_µ,k+1 , as desired.

Step 2. Proof that for allk≥1,

µ^k−1d_µ,k =

k−1

X

i=0

a_k,iφ₂iµ. (2.28)

We proceed by induction, and assume that (2.28) holds at step k. The induction rule (2.16) then yields at stepk+ 1

µ^kd_µ,k+1 = µ^k−1c_k(d_µ,k−d_2µ,k)

= c_kµ^k−1d_µ,k−c_k2^1−k(2µ)^k−1d_2µ,k

= c_k

k−1

X

i=0

a_k,iφ₂iµ−c_k2^1−k

k−1

X

i=0

a_k,iφ₂i+1µ

= c_ka_k,0φ_µ+ k−1

X

i=1

c_k(a_k,i−2^1−ka_k,i−1)φ₂iµ

−c_k2^1−ka_k,k−1φ₂kµ, so that µ^kd_µ,k+1 = P_k

i=0a_k,iφ₂ⁱ_µ, as desired. It remains to recall that d_µ,1 = φ_µ to conclude the proof of (2.28).

Note thata_2,0 = 1 anda_2,1 =−1. In particular,a_2,0+a_2,1 = 0 and the property

k−1

X

i=0

a_k,i = 0 (2.29)

follows by induction, for allk≥2.

Step 3. Proof that for alli, j≥1, E(φ₂ⁱ_µ, φ₂^j_µ) = hξ·Aξi −1

2

(ξ+∇φ₂ⁱ_µ)·A(ξ+∇φ₂ⁱ_µ) +

(ξ+∇φ₂^j_µ)·A(ξ+∇φ₂^j_µ)

−µ2ⁱ⁻¹

φ₂ⁱ_µ(φ₂ⁱ_µ+φ₂^j_µ)

−µ2^j−1

φ₂^j_µ(φ₂ⁱ_µ+φ₂^j_µ)

. (2.30) We can easily see that (2.30) holds when i =j. From (2.8) and (2.11), we have indeed that

E(φ₂iµ, φ₂iµ) = hξ·Aξi −

(ξ+∇φ₂iµ)·A(ξ+∇φ₂iµ)

−2ⁱ⁺¹µD φ²₂iµ

E, (2.31) as desired. For generali, j ∈N, we have, using (2.10) first forφ₂^j_µ and then forφ₂ⁱ_µ,

E(φ₂ⁱ_µ, φ₂^j_µ) =

∇φ₂ⁱ_µ·A∇φ₂^j_µ

=

∇φ₂iµ·A(ξ+∇φ₂jµ)

−

∇φ₂iµ·Aξ

(2.10)

= −2^jµ

φ₂ⁱ_µφ₂^j_µ

−

∇φ₂ⁱ_µ·A(ξ+∇φ₂ⁱ_µ) +

∇φ₂ⁱ_µ·A∇φ₂ⁱ_µ

(2.10)

= −2^jµ

φ₂ⁱ_µφ₂^j_µ

+ 2ⁱµD φ²₂iµ

E

+E(φ₂ⁱ_µ, φ₂ⁱ_µ)

(2.31)

= −2^jµ

φ₂iµφ₂jµ

−2ⁱµD φ²₂iµ

E

+hξ·Aξi −

(ξ+∇φ₂ⁱ_µ)·A(ξ+∇φ₂ⁱ_µ) . We conclude the proof of (2.30) by changing the roles of iandj.

12

Step 4. Proof of (2.25).

In view of (2.26), we have to estimate two terms. We begin with the Dirichlet form:

inserting (2.28) in the integral yields µ^2kE(d_µ,k+1,d_µ,k+1) =

k

X

i=0 k

X

j=0

a_k+1,ia_k+1,jE(φ₂iµ, φ₂jµ).

We then appeal to (2.30) to turn this identity into µ^2kE(d_µ,k+1,d_µ,k+1)

= k

X

i=0

a_k+1,i 2

hξ·Aξi −

k

X

i=0 k

X

j>i

µ(2ⁱ+ 2^j)a_k+1,ia_k+1,j

φ₂iµφ₂jµ

−

k

X

i=0

µ2ⁱa²_k+1,iD φ²₂iµ

E

−

k

X

i=0

a_k+1,i k

X

j=0

a_k+1,j

2ⁱµD φ²₂iµ

E +

(ξ+∇φ₂ⁱ_µ)·A(ξ+∇φ₂ⁱ_µ)

. Taking into account (2.29), we finally have

µ^2kE(d_µ,k+1,d_µ,k+1)

= −

k

X

i=0 k

X

j>i

µ(2ⁱ+ 2^j)a_k+1,ia_k+1,j

φ₂ⁱ_µφ₂^j_µ

−

k

X

i=0

µ2ⁱa²_k+1,iD φ²₂iµ

E

. (2.32) We now turn to the last term of the r. h. s. of (2.26) and appeal to (2.28):

µ^2k+1

d²_µ,k+1

=

k

X

i=0

µa²_k+1,iD φ²₂iµ

E+ 2

k

X

i=0 k

X

j>i

µa_k+1,ia_k+1,j

φ₂ⁱ_µφ₂^j_µ

. (2.33) We then prove (2.25) by induction, recalling that

A_µ,1 = h(ξ+∇φ_µ)·A(ξ+∇φ_µ)i.

Let us assume that (2.25) holds at step k ≥ 1. Then, using (2.25) at step k ≥ 1, and (2.32) & (2.33), the identity (2.26) turns into

ξ·A_µ,k+1ξ ^(2.26)= ξ·A_µ,kξ−2^k(k−1)µ^2kE(d_µ,k+1,d_µ,k+1)−2^k2+1µ^2k+1

d²_µ,k+1

= h(ξ+∇φ_µ)·A(ξ+∇φ_µ)i +µ

k−1

X

i=0

η_k,iD φ²₂iµ

E +µ

k−1

X

i=0 k−1

X

j>i

ν_k,i,j

φ₂ⁱ_µφ₂^j_µ

+2^k(k−1)µ

k

X

i=0 k

X

j>i

(2ⁱ+ 2^j)a_k+1,ia_k+1,j

φ₂ⁱ_µφ₂^j_µ

+ 2^k(k−1)µ

k

X

i=0

2ⁱa²_k+1,iD φ²₂iµ

E

−2^k2+1µ

k

X

i=0

a²_k+1,iD φ²₂iµ

E−2^k2+2µ

k

X

i=0 k

X

j>i

a_k+1,ia_k+1,j

φ₂iµφ₂jµ

,

from which we deduce that (2.25) holds at stepk+ 1.

Proposition 2 yields the following formulas for the first four approximations ofA_hom: ξ·A_µ,1ξ = h(ξ+∇φ_µ)·A(ξ+∇φ_µ)i,

ξ·A_µ,2ξ = h(ξ+∇φ_µ)·A(ξ+∇φ_µ)i −3µ φ²_µ

−2µ φ²_2µ

+ 5µhφ_µφ_2µi, ξ·A_µ,3ξ = h(ξ+∇φ_µ)·A(ξ+∇φ_µ)i − 55

9 µ φ²_µ

−8µ φ²_2µ

− 4 9µ

φ²_4µ +41

3 µhφ_µφ_2µi − 22

9 µhφ_µφ_4µi+10

3 µhφ_2µφ_4µi, ξ·Aµ,4ξ = h(ξ+∇φµ)·A(ξ+∇φµ)i − 3655

441 µ φ²_µ

−128 9 µ

φ²_2µ

−16 9 µ

φ²_4µ

− 8 441µ

φ²_8µ

+1325

63 µhφ_µφ_2µi −370

63 µhφ_µφ_4µi+184

441µhφ_µφ_8µi +82

9 µhφ_2µφ_4µi −44

63µhφ_2µφ_8µi+20

63µhφ_4µφ_8µi. 2.4. Complete error estimate. In this subsection, we combine the approximation for- mulasA_µ,k with the filtering method used in [7]. The filters are defined as follows.

Definition 4. A functionχ: [−1,1]→R₊ is said to be a filter of order p≥0 if (i) χ∈C^p([−1,1])∩W^p+1,∞((−1,1)),

(ii) R1

−1χ(x)dx= 1,

(iii) χ^(k)(−1) =χ^(k)(1) = 0 for allk∈ {0, . . . , p−1}.

The associated maskχ_L: [−L, L]^d→R₊in dimension d≥1 is then defined for allL >0 by

χ_L(x) := L^−d

d

Y

i=1

χ(L⁻¹x_i), wherex= (x1, . . . , x_d)∈R^d.

Let now A and A_hom be as in Definition 1. For all k≥1, µ >0,p ≥0, and R≥L > 0, we define the approximationA_µ,k,R,L of A_hom as

ξ·A_µ,k,R,Lξ := hh(ξ+∇φ_µ,R)·A(ξ+∇φ_µ,R)iiL

+µ

k−1

X

i=0

η_k,ihhφ²₂iµ,RiiL+µ

k−1

X

i=0 k−1

X

j>i

ν_k,i,jhhφ₂ⁱ_µ,Rφ₂^j_µ,RiiL, (2.34) where the coefficientsη_k,iand ν_k,i,j are as in Proposition 2, the modified correctorsφ₂iµ,R

are the unique weak solutions inH₀¹(Q_R) to

2ⁱµφ₂ⁱ_µ,R− ∇ ·A(ξ+∇φ₂ⁱ_µ,R) = 0, andhh·ii^L denotes the average with maskχL:

hhhiiL :=

Z

Rd

h(x)χ_L(x)dx.

The combination of [7, Theorem 1] with Theorem 1 and Proposition 2 then yields 14

Theorem 2. Let d≥2, A and A_hom be as in Definition 1, k≥1, χ be a filter of order p ≥ 0, and A_µ,k,R,L be the approximation (2.34) of the homogenization matrix, where R² &µ⁻¹&R, R≥L∼R∼R−L. Then, there existsc >0 depending only on α, β and dsuch that we have

|A_µ,k,R,L−A_hom| . L^−(p+1)+µ^2k+µ^−1/4exp (−c√µ(R−L)), (2.35) where the multiplicative constant depends onk, next to α, β and d.

In order to illustrate Theorem 2, we provide the results of numerical tests in a periodic discrete case in Appendix B. They confirm the sharpness of the analysis.

3. The discrete stochastic case

We start this section by defining the discrete stochastic model we wish to consider.

3.1. Notation and preliminaries. We say thatx, yinZ^dare neighbors, and writex∼y, whenever |y−x| = 1. This relation turns Z^d into a graph, whose set of (non-oriented) edges we will denote by B. We now turn to the definition of the associated diffusion coefficients, and their statistics.

Definition 5 (environment). Let Ω = [α, β]^B. An elementω = (ω_e)_e∈^B of Ω is called an environment. With any edgee= (x, y)∈B, we associate the conductance ω_x,y :=ω_e (by constructionω_x,y =ω_y,x). Letν be a probability measure on [α, β]. We endow Ω with the product probability measureP=ν^⊗B. In other words, ifωis distributed according to the measureP, then (ω_e)_e∈^B are independent random variables of lawν. We denote by L²(Ω) the set of real square integrable functions on Ω for the measure P, and write h·i for the expectation associated withP.

In the framework of Definition 5, we can introduce a notion of stationarity.

Definition 6 (stationarity). For allz∈Z^d, we let θ_z : Ω→Ω be such that for all ω∈Ω and (x, y)∈B, (θz ω)x,y =ωx+z,y+z. This defines an additive action group {θz}z∈Z^d on Ω which preserves the measureP, and is ergodic forP.

We say that a functionf : Ω×Z^d→ Ris stationary if and only if for all x, z ∈ Z^d and P-almost everyω∈Ω,

f(x+z, ω) = f(x, θ_z ω).

In particular, with allf ∈L²(Ω), one may associate the stationary function (still denoted by f) Z^d×Ω → R,(x, ω) 7→ f(θ_x ω). In what follows we will not distinguish between f ∈L²(Ω) and its stationary extension on Z^d×Ω.

It remains to define the conductivity matrix onZ^d.

Definition 7(conductivity matrix). Let Ω,P, and{θ_z}z∈Zd be as in Definitions 5 and 6.

The stationary diffusion matrixA:Z^d×Ω→ Md(R) is defined by A(x, ω) = diag(ω_x,x+e_i, . . . , ω_x,x+e_d).

For each ω∈Ω, we may consider the discrete elliptic equation whose operator is

−∇^∗·A(·, ω)∇,

where∇and ∇^∗ are defined for allu:Z^d→Rby

∇u(x) :=







u(x+e₁)−u(x) ...

u(x+e_d)−u(x)





, ∇^∗u(x) :=







u(x)−u(x−e₁) ...

u(x)−u(x−e_d)





, (3.1) and the backward divergence is denoted by ∇^∗·, as usual. The standard stochastic ho- mogenization theory for such discrete elliptic operators (see for instance [13], [12]) en- sures that there exist homogeneous and deterministic coefficients A_hom such that the so- lution operator of the continuum differential operator −∇ ·A_hom∇ describes P-almost surely the large scale behavior of the solution operator of the discrete differential operator

−∇^∗ ·A(·, ω)∇. As for the periodic case, the definition of A_hom involves the so-called correctors φ : Z^d×Ω → R, which are solutions (in a sense made precise below) to the equations

−∇^∗·A(x, ω)(ξ+∇φ(x, ω)) = 0, x∈Z^d, (3.2) forξ∈R^d. The following lemma gives the existence and uniqueness of the correctorφ.

Lemma 1(corrector). Let Ω,P, {θ_z}z∈Zd, and A be as in Definitions 5, 6, and 7. Then, for all ξ ∈ R^d, there exists a unique measurable function φ : Z^d×Ω → R such that φ(0,·) ≡ 0, ∇φ is stationary, h∇φi = 0, and φ solves (3.2) P-almost surely. Moreover, the symmetric homogenized matrixA_hom is characterized by

ξ·A_homξ = h(ξ+∇φ)·A(ξ+∇φ)i. (3.3) As mentioned in the introduction, the standard proof of Lemma 1 makes use of the regu- larization of (3.2) by a zero-order termµ >0:

µφ_µ(x, ω)− ∇^∗·A(x, ω)(ξ+∇φ_µ(x, ω)) = 0, x∈Z^d. (3.4) Lemma 2(modified corrector). LetΩ,P,{θ_z}z∈Zd, andAbe as in Definitions 5, 6, and 7.

Then, for allµ >0andξ ∈R^d, there exists a unique stationary functionφµ∈L²(Ω)which solves (3.4) P-almost surely.

In order to proceed as in the periodic case and use a spectral approach, one needs to suitably define an elliptic operator on L²(Ω) (which is the stochastic counterpart to the space H_per¹ (Q) of Section 2). Stationarity is crucial here. Following [17], we introduce difference operators onL²(Ω): for allu∈L²(Ω), we set

Du(ω) :=







u(θe₁ω)−u(ω) ...

u(θe_dω)−u(ω)





, D^∗u(ω) :=







u(ω)−u(θ₋e₁ω) ...

u(ω)−u(θ₋e_dω)





. (3.5) We are in position to define the stochastic counterpart to the operator of Definition 2.

Definition 8. Let Ω, P, {θ_z}z∈Z^d, and A be as in Definitions 5, 6, and 7. We define L:L²(Ω)→L²(Ω) by

Lu(ω) = −D^∗·A(ω) Du(ω)

= X

z∼0

ω_0,z(u(ω)−u(θ_z ω)) where D and D^∗ are as in (3.5).

16