Pattern formation, energy landscapes, and scaling laws
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Felix Otto
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Katja Heid
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Stochastic Homogenization
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 twophase 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 twophase 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].
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 \(\mathbb{Z}^d\) 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 pathwise.
The periodic approximation procedure exploits both observations. It consists of two steps:
Figure 2: In the case of i.i.d. coefficients the construction of the Lperiodic 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).
 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 nondeterministic, 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.
Quantitative Analysis
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 Lperiodic 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 \(N\).
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 \(L^2\)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.
A regularity theory
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 higherorder 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 Campanatotype 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 righthand side. This modification allows us to also consider models with nonintegrable (longrange) 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 righthand 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 GiorgiNashMoser 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 HelfferSjöstrand 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].
References

Antoine Gloria and Felix Otto: The corrector in stochastic homogenization : optimal rates, stochastic integrability, and fluctuationsMISPreprint: 81/2015 ARXIV: http://arxiv.org/abs/1510.08290

Antoine Gloria and Felix Otto: An optimal variance estimate in stochastic homogenization of discrete elliptic equationsIn: The annals of probability, 39 (2011) 3, p. 779856DOI: 10.1214/10AOP571 LINK: http://webdoc.sub.gwdg.de/ebook/serien/e/sfb611/448.pdf

Antoine Gloria and Felix Otto: An optimal error estimate in stochastic homogenization of discrete elliptic equationsIn: The annals of applied probability, 22 (2012) 1, p. 128MISPreprint: 28/2010 DOI: 10.1214/10AAP745 ARXIV: http://de.arxiv.org/abs/1203.0908

Antoine Gloria ; Stefan Neukamm and Felix Otto: Quantification of ergodicity in stochastic homogenization : optimal bounds via spectral gap on Glauber dynamicsIn: Inventiones mathematicae, 199 (2015) 2, p. 455515MISPreprint: 91/2013 DOI: 10.1007/s002220140518z

Antoine Gloria ; Stefan Neukamm and Felix Otto: Quantification of ergodicity in stochastic homogenization : optimal bounds via spectral gap on Glauber dynamics  long versionMISPreprint: 3/2013

Daniel Marahrens and Felix Otto: Annealed estimates on the Green functionIn: Probability theory and related fields, 163 (2014) 34, p. 527573MISPreprint: 69/2012 DOI: 10.1007/s0044001405980 ARXIV: http://arxiv.org/abs/1304.4408

Daniel Marahrens and Felix Otto: On annealed elliptic Green's function estimatesIn: Mathematica bohemica, 140 (2015) 4, p. 489506MISPreprint: 13/2015 ARXIV: http://arxiv.org/abs/1401.2859 LINK: http://dml.cz/handle/10338.dmlcz/144465

Daniel Marahrens and Felix Otto: Effektive Beschreibung von heterogenen MedienIn: Jahrbuch der MaxPlanckGesellschaft, 2014 (2015), Forschungsbericht  MaxPlanckInstitut für Mathematik in den NaturwissenschaftenLINK: http://www.mpg.de/7775250/MPI_MIS_JB_2014

Agnes Lamacz ; Stefan Neukamm and Felix Otto: Moment bounds for the corrector in stochastic homogenization of a percolation modelIn: Electronic journal of probability, 20 (2015), 106MISPreprint: 63/2014 DOI: 10.1214/EJP.v203618 ARXIV: http://arxiv.org/abs/1406.5723

Antoine Gloria and Felix Otto: Quantitative estimates on the periodic approximation of the corrector in stochastic homogenizationIn: ESAIM / Proceedings, 48 (2015), p. 8097MISPreprint: 12/2015 DOI: 10.1051/proc/201448003 ARXIV: http://arxiv.org/abs/1409.1161

Antoine Gloria and Felix Otto: Quantitative results on the corrector equation in stochastic homogenizationIn: Journal of the European Mathematical Society, 19 (2017) 11, p. 34893548DOI: 10.4171/JEMS/745 ARXIV: http://arxiv.org/abs/1409.0801

Antoine Gloria ; Stefan Neukamm and Felix Otto: A regularity theory for random elliptic operators

Antoine Gloria ; Stefan Neukamm and Felix Otto: An optimal quantitative twoscale expansion in stochastic homogenization of discrete elliptic equationsIn: ESAIM / Mathematical modelling and numerical analysis, 48 (2014) 2, p. 325346MISPreprint: 41/2013 DOI: 10.1051/m2an/2013110

JeanChristophe Mourrat and Felix Otto: Correlation structure of the corrector in stochastic homogenizationIn: The annals of probability, 44 (2016) 5, p. 32073233MISPreprint: 11/2015 DOI: 10.1214/15AOP1045 ARXIV: http://arxiv.org/abs/1402.1924

Julian Fischer and Felix Otto: A higherorder largescale regularity theory for random elliptic operatorsIn: Communications in partial differential equations, 41 (2016) 7, p. 11081148DOI: 10.1080/03605302.2016.1179318 ARXIV: http://arxiv.org/abs/1503.07578

Peter Bella ; Arianna Giunti and Felix Otto: Quantitative stochastic homogenization : local control of homogenization error through correctorIn: Mathematics and materials / Mark J. Bowick... (eds.)Providence, RI : American mathematical society, 2017.  P. 301327(IAS/Park City mathematics series ; 23)DOI: 10.1090/pcms/023/07 ARXIV: http://arxiv.org/abs/1504.02487

Joseph G. Conlon ; Arianna Giunti and Felix Otto: Green's function for elliptic systems : existence and DelmotteDeuschel boundsIn: Calculus of variations and partial differential equations, 56 (2017) 6, 163MISPreprint: 34/2016 DOI: 10.1007/s0052601712550 ARXIV: http://arxiv.org/abs/1602.05625

Peter Bella and Arianna Giunti: Green's function for elliptic systems : moment boundsIn: Networks and heterogeneous media, 13 (2018) 1, p. 155176MISPreprint: 85/2015 DOI: 10.3934/nhm.2018007 ARXIV: http://arxiv.org/abs/1512.01029

Peter Bella and Felix Otto: Corrector estimates for elliptic systems with random periodic coefficientsIn: Multiscale modeling and simulation, 14 (2016) 4, p. 14341462MISPreprint: 100/2014 DOI: 10.1137/15M1037147 ARXIV: http://arxiv.org/abs/1409.5271

Felix Otto and Hendrik Weber: Hölder regularity for a nonlinear parabolic equation driven by spacetime white noiseMISPreprint: 31/2015 ARXIV: http://arxiv.org/abs/1505.00809
 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):803847, 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. NorthHolland, Amsterdam, 1981.
Presentations
 Error bounds in stochastic homogenization (see PDF, 2.4 Mbyte)