An optimal variance estimate in stochastic homogenization of discrete elliptic equations

By Antoine Gloria, Felix Otto. 2011

286 citations recorded by the source corpus.

We consider a discrete elliptic equation with random coefficients $A$, which (to fix ideas) are identically distributed and independent from grid point to grid point $x\in\mathbb{Z}^d$. On scales large w.\ r.\ t.\ the grid size (i.\ e.\ unity), the solution operator is known to behave like the solution operator of a (continuous) elliptic equation with constant deterministic coefficients. These symmetric ''homogenized'' coefficients $A_{hom}$ are characterized by % $$ \xi\cdot A_{hom}\xi\;=\;\langle\left((\xi+\nabla\phi)\cdot A(\xi+\nabla\phi)\right)(0)\rangle, \quad\xi\in\mathbb{R}^d, $$ % where the random field $\phi$ is the unique stationary solution of the ''corrector problem'' % $$ -\nabla\cdot A(\xi+\nabla\phi)\;=\;0 $$ % and $\langle\cdot\rangle$ denotes the ensemble average. \medskip It is known (''by ergodicity'') that the above ensemble average of the energy density $e=(\xi+\nabla\phi)\cdot A(\xi+\nabla\phi)$, which is a stationary random field, can be recovered by a system average. We quantify this by proving that the variance of a spatial average of $e$ on length scales $L$ is estimated as follows: % $$ {\rm var}\left[\sum_{x\in\mathbb{Z}^d}\eta_L(x)\,e(x)\right] \;\lesssim\;L^{-d}, $$ % where the averaging function (i.\ e.\ $\sum_{x\in\mathbb{Z}^d}\eta_L(x)=1$, ${\rm supp}\eta_L\subset[-L,L]^d$) has to be smooth in the sense that $|\nabla\eta_L|\lesssim L^{-1}$. In two space dimensions (i.\ e.\ $d=2$), there is a logarithmic correction. \medskip In other words, smooth averages of the energy density $e$ behave like as if $e$ would be independent from grid point to grid point (which it is not for $d>1$). This result is of practical significance, since it allows to estimate the error when numerically computing $A_{hom}$.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/e98e3c63dcb6cb2bb021a89dd954eeb40015c31a

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Citations in the corpus

Quantitative Stochastic Homogenization and Large-Scale Regularity (2017, background): “1, but with much weaker stochastic integrability, were proved several years earlier in [65, 66, 62, 67] for a certain class of random environments”
Annealed estimates on the Green function (2013, methodology): “The proof of estimate (31) in [12] in case of d = 2 is subtle and relied on an adaptation of [9].”
Quantitative estimates in stochastic homogenization for correlated coefficient fields (2019, background): “ant detail will only be used to avoid an arbitrarily small loss of stochastic integrability in critical cases, so that we can keep statements neat. The main progress over the sensitivity estimates of [22, 19, 24] are the following three lemmas — the only (speciﬁc) PDE ingredients of this work. The ﬁ”
Recent progress on the Random Conductance Model (2011, methodology): “Progress in the uniformly elliptic case has been achieved in recent work of Gloria and Otto [67] who have been able to prove that the corrector is in L(P) for all q < ∞, and thus a tight random variable, in all dimensions d ≥ 3.”
Solving parametric PDE problems with artificial neural networks (2017, background): “random coefficients, the effective conductance Aeff(a) concentrates with a standard deviation that scales like 1/ √ nd [5].”
A Regularity Theory for Random Elliptic Operators (2014, methodology): “In contrast to our framework based on nonlinear mixing conditions (as developed in [41, 43, 38] and the present paper), Armstrong and Smart quantify ergodicity in terms of linear mixing conditions.”
Invariance principle for the random conductance model in a degenerate ergodic environment (2013, background): “However, so far these estimates have only been derived for nicely mixing elliptic environments [26].”
A diagram-free approach to the stochastic estimates in regularity structures (2021, methodology): “In the use of Malliavin calculus, the spectral gap inequality, and annealed estimates, this paper is inspired by recent developments in quantitative stochastic homogenization [16, 11, 26].”
Quantitative stochastic homogenization and regularity theory of parabolic equations (2017, result): “nt interest in quantitative stochastic homogenization for elliptic equations, particularly in the case of linear, uniformly elliptic equations. This essentially began with the work of Gloria and Otto [19, 20], who proved the rst quantitative results which are optimal in the scaling of the parameter ”
Reduction of the resonance error---Part 1: Approximation of homogenized coefficients (2011, methodology): “[10, 11, 12] for related arguments), we have conducted the following series of tests.”
Variance decay for functionals of the environment viewed by the particle (2009, methodology): “As discussed in the introduction of [9], the function Rμd can be computed in practice.”
Quantitative Stochastic Homogenization of Elliptic Equations in Nondivergence Form (2013, background): “In recent and striking papers, Gloria and Otto [10] and Gloria, Neukamm and Otto [9] proved optimal error estimates for discrete elliptic equations in divergence form with i.i.d. coefficients, using a combination of regularity theory and concentration arguments.”
A higher-order large-scale regularity theory for random elliptic operators (2015, background): “Gloria and Otto [12] were the rst to obtain optimal estimates on the size of the homogenization error in the linear elliptic case, though with nonoptimal stochastic integrability.”
Boundary Estimates in Elliptic Homogenization (2015, methodology): “This allows us to use a general scheme for establishing Lipschitz estimates down to the scale ε, which was formulated recently in [4] and used for interior estimates in stochastic homogenization with random coefficients (also see [2, 1] as well as related work in [21, 22, 20, 19]).”
Mesoscopic Higher Regularity and Subadditivity in Elliptic Homogenization (2015, background): “A similar result was recently proved by Fischer and Otto [9], who developed a similar higher regularity theory in the general stationary ergodic setting.”
Quantitative stochastic homogenization of convex integral functionals (2014, methodology): “This escalated significantly with the work of Gloria and Otto [15, 16], who proved optimal quantitative bounds for the energy density of modified correctors and then that of Gloria Neukamm and Otto [14, 13], who proved optimal bounds for the error in homogenization.”
The additive structure of elliptic homogenization (2016, methodology): “This approach has been developed extensively by Gloria, Otto and their collaborators [19, 20, 23, 16, 15, 28], who proved optimal quantitative bounds on the scaling of the first-order correctors (including their sublinear growth and spatial averages of their energy density).”
Lipschitz Regularity for Elliptic Equations with Random Coefficients (2014, methodology): “[25] A. Gloria and F. Otto.”