Homogenization and the need for quantitative methods
Many applications, such as porous media or composite materials, involve heterogeneous media described by partial differential equations with coefficients that randomly vary on a small scale. On macroscopic scales (large compared to the dimension of the heterogeneities) such media often show an effective behavior. Typically that behavior is simpler, since the complicated, random small scale structure of the media averages out on large scales, and in many cases the effective behavior can be described by a deterministic, macroscopic model with constant coefficients. This process of averaging is called homogenization. Mathematically, it means that the replacement of the original random equation by one with certain constant, deterministic coefficients is a valid approximation in the limit when the ratio between macro- and microscale tends to infinity. A qualitative homogenization result typically states that the solution of the initial model converges to the solution of the macro model, and provides a characterization of the macro model, e. g. by a homogenization formula for the homogenized coefficients.
Figure 1: The pictures show different microstructures of a two-phase material with a highly conducting phase (red) and a low conducting phase (blue). In all three examples the volume fractions of the phases are the same. However, (c) leads to an almost isotropic effective behavior, while (a) and (b) effectively behave highly anisotropic with different directions of high conductivity.
The relation between the microscopic properties of the medium and its effective ones is subtle. Figure 1 shows an example of a two-phase medium: the effective behavior not only depends on the volume fraction of the phases - it is highly sensitive to the geometry and spatial arrangement of the phases.
A qualitative homogenization result, as described above, is the starting point for a precise understanding of effective properties: it provides the object to analyze - the formula for the homogenized coefficients. However, in all non trivial cases the homogenization formula cannot be studied directly and one has to appeal to approximations. Therefore, it is an important and natural task to construct approximation schemes and to develop precise quantitative methods to evaluate their quality.
In a series of papers we investigated such questions in one of the simplest situations, namely for effective heat conduction in random media.
Effective heat conduction in random media
Kozlov [Koz1979], Papanicolaou and Varadhan [Pap1979] studied (steady) heat conduction in a randomly inhomogeneous conducting medium and obtained a qualitative homogenization result for stationary, ergodic conductivities. Kozlov [Koz1987] and Künnemann [Kue1983] considered the analogue problem in a discrete setting, namely for diffusion on the lattice Zd with random bond conductivities. They proved that in the homogenization limit an effective conductivity emerges described by the homogenization formula
ξ ⋅ ahom ξ = <(ξ+∇ φ) ⋅ a(ξ+ ∇ φ)> for all ξ∈Rd. (1)
In the formula,
- ahom denotes the effective conductivity - a symmetric, deterministic matrix
- a(x) denotes the bond conductivities - a random field on Zd of diagonal matrices that is assumed to be stationary and ergodic,
- <⋅> denotes the ensemble average or expected value.
The homogenization formula involves a corrector function φ=φ(a,x) which is defined as a solution to the corrector problem
-∇⋅ a(x)(ξ+∇φ(a,x))=0, ∇φ stationary and <∇φ>=0. (2)
Despite its simplicity, the homogenization formula has to be approximated in practice, since (2) has to be solved
- on the whole lattice Zd and
- for almost every realization of the coefficients a(x).
Figure 2: In the case of i.i.d. coefficients the construction of the L-periodic ensemble is easy: each realization of the random coefficients, cf. (a), is first restricted to a box of size L, cf. (b), and then extended by periodicity, cf. (c).
How to approximate the effective conductivity?
A natural answer is guided by the following observations:
- By ergodicity the ensemble average in (1) can be replaced by the spatial average
limL→∞ L-dΣx∈ ([0,L)∩Z)d(ξ+∇φ(a,x))⋅ a(x)(ξ+∇φ(a,x)). (3)Notice that (3) only involves a single realization of the coefficients.
- For stationary and periodic coefficients the infinite domain Zd of the corrector problem can be replaced by a discrete torus with finite size, and (2) can be computed path-wise.
The periodic approximation procedure exploits both observations. It consists of two steps:
- In a first periodization step, L-periodic coefficients aL(x) (with large period L) are constructed by introducing an L-periodic ensemble <⋅>L - a suitable, stationary probability measure on the set of periodic coefficient fields. The L-periodic ensemble can be seen as the mathematical version of the representative volume element in numerical schemes. See Figure 2 for a construction in the case of i.i.d. coefficients.
- Secondly, a periodic proxy aLhom is defined via space averaging in the spirit of (3) with a(x) replaced by aL(x) and ∇φ(a,x) replaced by ∇φ(aL,x).
The resulting periodic proxy is a random matrix - it is in general non-deterministic, since periodization typically destroys ergodicity. However, we expect that the periodic proxy converges to the homogenized coefficients for L →∞ (provided the L-periodic ensemble is chosen in the "right way"). Indeed, almost sure convergence can be shown by soft arguments; however, these arguments do not yield any rate of convergence.
In the series of articles  -  we developed quantitative methods that allow to estimate approximation errors. By now, we have a complete picture of the periodic approximation procedure in the case of independent identically distributed random coefficients and partial results for correlated coefficients. In particular in , we show that the overall approximation error for i.i.d. coefficients decays with the rate of the central limit theorem, i.e.
<|aLhom-ahom|2>1/2 ≤ C L-d/2.
This is optimal, as can be seen by a linearized analysis in the regime of vanishing ellipticity contrast. We expect the same behavior for correlated coefficients satisfying a spectral gap estimate (see below).
Decomposition of the approximation error.
In  we introduced a natural decomposition of the approximation error in
- a random error - the variance of the periodic proxy,
- a systematic error - the distance of <aLhom> to ahom.
The random error monitors the fluctuation of the periodic proxy around its average and originates in the lack of ergodicity of the L-periodic ensemble. In  for i.i.d. coefficients and in  for more general statistics, we prove that the random error has the critical scaling L-d/2.
The systematic error is analyzed in  and . We observe that the systematic error is of lower order and decays almost double as fast as the random error. A complete picture is presented in .
Empirical averaging and the size of the representative volume element.
An interesting difference between the random and systematic error is the following: in contrast to the systematic error, the random error can be reduced by empirical averaging, i.e. by computing the periodic proxy for N realizations and then considering the arithmetic mean. Due to the different scaling of both errors, better decay rates (for a prescribed number of degrees of freedom) can be achieved by combining empirical averaging and periodic approximation. Our detailed analysis yields a rule for the optimal ratio between L and N.
Elements of our method
In the following we briefly describe some ingredients and ideas of our method.
Spectral gap on Glauber dynamics
The statistics that we consider satisfy a spectral gap estimate - a Poincaré inequality in L2-probability space that allows to estimate the variance of a random variable by mean of its "vertical" gradient. As shown in , the spectral gap property is related to ergodicity - it allows to quantify the latter. In contrast to ergodicity, the spectral gap property is stable under periodization, e.g. the L-periodic ensemble associated to the periodization of i.i.d. coefficients satisfies a spectral gap estimate with a constant independent of the period L. In our articles the spectral gap estimate appears at several places; in particular, to estimate the variance of the periodic proxy by quartic moments of the corrector as primary step to bound higher moments of the corrector. Transition from physical space to probability space.
As in the seminal papers [Koz1979] and [Pap1979], stationarity allows to represent PDEs stated in physical space, such as the corrector problem (2), by equations in probability space. Based on that observation, we represent the corrector (and its gradient) by means of a parabolic equation in probability space.
Higher moments of the corrector
A main achievement of our analysis are estimates on higher moments of the corrector. We prove that any finite moment of the L-periodic corrector's gradient is bounded uniformly in L; similarly higher moments of the corrector itself are bounded uniformly in L - up to a logarithmic correction in dimension 2. In  we get these estimates by proving nonlinear decay estimates for the parabolic equation associated to the corrector problem.
Estimates on Green's functions
A central ingredient to quantify the decay of the mentioned parabolic equation are estimates on the parabolic Green's function which only depend on the ellipticity ratio of the coefficients. Our estimates are pointwise in time and weighted in space. They are obtained in  and  by appealing to elliptic and parabolic regularity theory.
Regularization and spectral exponents
In order to estimate certain contributions of the systematic error, we regularize the corrector equation (2) by adding a lower order term. Such a regularization is already utilized in the qualitative analysis in [Koz1979], [Pap1979]. In [Glo] it is shown that the error due to regularization is related to the spectral exponents of the elliptic operator that appears in the corrector equation. Optimal estimates on the spectral exponents for small dimensions are derived in  and [Glo]. As shown in , our quantitative methods yield optimal estimates on the spectral exponents in any dimension.
- Antoine Gloria and Felix Otto.
An optimal variance estimate in stochastic homogenization of discrete elliptic equations.
Ann. Probab. 39, 779-856, 2011.
download preprint (PDF, 489 kbyte)
- Antoine Gloria and Felix Otto.
An optimal error estimate in stochastic homogenization of discrete elliptic equations.
Ann. Appl. Probab. 22:1, 1-28, 2012.
- Antoine Gloria, Stefan Neukamm, and Felix Otto.
Quantication of ergodicity in stochastic homogenization: optimal bounds via spectral gap on Glauber dynamics.
- Antoine Gloria, Stefan Neukamm, and Felix Otto.
Approximation of effective coefficients by periodization in stochastic homogenization.
- Antoine Gloria and Felix Otto.
Optimal quantitative estimates in stochastic homogenization of linear elliptic equations.
- A. Gloria and J.-C. Mourrat. [Glo]
Spectral measure and approximation of homogenized coefficients.
To appear in Probab. Theory Related Fields.
- S.M. Kozlov. [Koz1979]
The averaging of random operators.
Mat. Sb. (N.S.), 109(151)(2):188202, 327, 1979.
- S.M. Kozlov. [Koz1987]
Averaging of difference schemes.
Math. USSR Sbornik, 57(2):351369, 1987.
- R. Künnemann. [Kue1983]
The diffusion limit for reversible jump processes on Zd with ergodic random bond conductivities.
Commun. Math. Phys., 90:2768, 1983.
- G.C. Papanicolaou and S.R.S. Varadhan. [Pap1979]
Boundary value problems with rapidly oscillating random coefficients.
In Random fields, Vol. I, II (Esztergom, 1979), volume 27 of Colloq. Math. Soc. Janos Bolyai, pages 835873. North-Holland, Amsterdam, 1981.
- Stefan Neukamm
- Felix Otto
- Error bounds in stochastic homogenization (see PDF, 2.4 Mbyte)