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.

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. You'll find a German language general audience introduction into partical aspects of our work under [8].

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 \(\mathbb{Z}^d\) with random bond conductivities. They proved that in the homogenization limit an effective conductivity emerges described by the homogenization formula

\(\xi \cdot a_\text{hom}\,\xi = \big\langle (\xi + \nabla \varphi) \cdot a(\xi + \nabla \varphi)\big\rangle \)  for all  \(\xi \in \mathbb{R}^d \).   (1)

In the formula,

• \(a_\text{hom}\) denotes the effective conductivity - a symmetric, deterministic matrix
• \(a(x)\) denotes the bond conductivities - a random field on \(\mathbb{Z}^d\) of diagonal matrices that is assumed to be stationary and ergodic,
• \(\langle\cdot\rangle\) denotes the ensemble average or expected value.

The homogenization formula involves a corrector function \(\varphi=\varphi(a,x)\) which is defined as a solution to the corrector problem

\(-\nabla \cdot a(x) \big( \xi + \nabla \varphi(a,x) \big) = 0\) ,   \(\nabla\varphi\) stationary and \(\langle \nabla \varphi \rangle =0\).   (2)

Despite its simplicity, the homogenization formula has to be approximated in practice, since (2) has to be solved

• on the whole lattice \( \mathbb{Z}^d \) and
• for almost every realization of the coefficients \(a(x)\).

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

  \(\lim\limits_{L \to \infty}L^{-d} \sum\limits_{x\in\left(\left[0,L\right) \cap \mathbb{Z}\right)^d } \big( \xi + \nabla \varphi(a,x) \big) \cdot a(x) \big( \xi + \nabla \varphi(a,x) \big)\) .   (3)

  Notice that (3) only involves a single realization of the coefficients.
• For stationary and periodic coefficients the infinite domain \(\mathbb{Z}^d\) 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 \(a^L (x)\) (with large period \(L\)) are constructed by introducing an \(L\)-periodic ensemble \( \langle \cdot \rangle_L \) - 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 \(a^{L}_\text{hom} \) is defined via space averaging in the spirit of (3) with \(a(x)\) replaced by \(a^L (x)\) and \( \nabla \varphi (a,x) \) replaced by \( \nabla \varphi (a^L,x) \).

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 \(L \to \infty \) (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 [2] - [4] 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 a discrete medium and independent identically distributed random coefficients, and partial results for correlated coefficients. In particular in [4], we show that the overall approximation error for i.i.d. coefficients decays with the rate of the central limit theorem, i.e.

 \(\left< \left| a_{hom,N}^L - a_{hom} \right|^2 \right>^{1/2} \le C\left( \frac{1}{\sqrt{N}}L^{-d/2} + L^{-d} \ln^d L\right)\),

where \(a_{hom,N}^L\) is an average of \(a_{hom}^L\) evaluated for \(N\) independent realizations of \(a\) according to \(\left< \cdot \right>_L\). 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 [2] 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 \( \langle a^L_\text{hom} \rangle \) to \(a_\text{hom}\).

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 [2] for i.i.d. coefficients and in [4] for more general statistics, we prove that the random error has the critical scaling \(L^{-d/2}\).

The systematic error is analyzed in [3] and [4]. We observe that the systematic error is of lower order and decays almost double as fast as the random error.

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 \(N\).

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 \(L^2\)-probability space that allows to estimate the variance of a random variable by mean of its "vertical" gradient. As shown in [4], 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 [4] 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 [2] and [4] 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 [3] and [Glo]. As shown in [4], our quantitative methods yield optimal estimates on the spectral exponents in any dimension.

Motivated by the work of Armstrong and Smart [AS], we have recently focused on questions of regularity for random elliptic operators [12]. Following the philosophy of Avellaneda and Lin [AL], who in the periodic homogenization lifted the regularity theory of the homogenized limit to the heterogeneous situation, we obtain \(C^{1,\alpha}\) estimates on large scales. Moreover, only assuming stationarity and ergodicity of the coefficient fields we show that for almost every \(a\) the dimension of \(a\)-harmonic functions with subquadratic growth is \(d+1\) (i.e. it is the same as in the constant coefficient case). We have extended this result to \(a\)-harmonic functions with larger growth by constructing higher-order correctors in order to get \(C^{k,\alpha}\) estimates [15].

The proof of the \(C^{1,\alpha}\) estimate for an \(a\)-harmonic function \(u\) is based on the Campanato-type iteration, where in each step we compare \(u\) with a solution of the homogenized equation with boundary data the same as \(u\). In order to control the error between \(u\) and the homogenized solution, besides using the scalar corrector \(\varphi\) we also introduce the vector potential for flux correction \(\sigma\) defined through

\begin{equation*} \nabla \cdot \sigma_i := a(e_i + \nabla \varphi_i) - \left< a(e_i + \nabla \varphi_i) \right>. \end{equation*}

In our error estimates we only require that the generalized corrector \((\varphi,\sigma)\) grows sublinearly from some scale on. More precisely, if \(r^*(a,0)\) is a minimal radius such that on all balls centered at the origin with radii larger than \(r^*(a,0)\) the corrector \((\varphi,\sigma)\) growths sublinearly, then for \(u\), any solution of the equation with coefficient field \(a\), we get the \(C^{1,\alpha}\) estimates. In this sense our proof splits into a deterministic part, where we give arguments for \(C^{1,\alpha}\) estimates for radii larger than \(r^*\), and a probabilistic part, where under some weak quantification of ergodicity we obtain stretched exponential moments for \(r^*\). The ergodicity is quantified through a version of the Spectral Gap inequality, which we modified by (almost dyadically) grouping terms on the right-hand side. This modification allows us to also consider models with non-integrable (long-range) correlations.

Using a duality argument borrowed from Avellaneda and Lin [AL], the work [12] implies estimates for the Green's function and its first and second derivative with optimal scaling away from the diagonal [18]. As a consequence of [16], where under a stronger assumption on the growth of the corrector we study the homogenization error for the solution of the equation with a compactly supported right-hand side, we obtain \(C^{2,\alpha}\) estimates on the annealed Green's function \(\left<G\right>\). The results [11,16,18] also hold in the system's case, where even the existence of \(G\) is not in general guaranteed. In [17] we showed that \(G\) exists almost surely and showed estimates for \(\left< |\nabla G| \right>\) and \(\left< |\nabla\nabla G| \right>\) only under the assumption of stationarity.

Further results

Appealing to the De Giorgi-Nash-Moser theory for scalar equations, in [11] we showed the existence of and obtained estimates for the corrector together with the optimal estimates on the random and systematic error. In the periodic setting and for systems, we estimated moments of the gradient of the corrector which we then used to obtain optimal bounds for the random error [19]. Some of the methods used in [19] we borrowed from [10], where the corrector itself (in the case of a single equation) is estimated. The fluctuations of the corrector around its mean were studied in [14], where we used the Helffer-Sjöstrand transformation to identify its correlation structure. Finally, in [9] we relaxed the assumption of uniform ellipticity by considering a toy model for percolation in \(\mathbb{Z}^d, d \ge 3\) for which we showed the existence of the stationary corrector \(\varphi\).
Recently, for the case of coefficient fields with finite range of dependence, we combined parabolic approach and several ideas from our previous works [4,12,16] to obtain bounds on the corrector, which are near optimal in the scaling and optimal in stochastic integrability [1].

• S. Armstrong and C. Smart. [AS]
  Quantitative stochastic homogenization of convex integral functionals.
  2014.
  Arxiv Preprint arxiv.org/abs/1406.0996
• M. Avellaneda and F.-H. Lin. [AL]
  Compactness methods in the theory of homogenization.
  Comm. Pure and Applied Math., 40(6):803--847, 1987.
• 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 \( \mathbb{Z}^d\) 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.

• Error bounds in stochastic homogenization (see PDF, 2.4 Mbyte)

• Peter Bella
• Julian Fischer
• Felix Otto