[Paper Review] High order correctors and two-scale expansions in stochastic homogenization

Yu Gu|arXiv (Cornell University)|Jan 29, 2016

Advanced Mathematical Modeling in Engineering22 references3 citations

TL;DR

This paper establishes moment bounds for high-order correctors in stochastic homogenization of discrete elliptic equations with i.i.d. random coefficients. Under log-Sobolev conditions and dimensional constraints (d ≥ 2n−1 for gradients, d ≥ 2n+1 for fields), it proves uniform moment estimates for n-th order correctors and their gradients, enabling a rigorous two-scale expansion that identifies first and higher-order random fluctuations in a strong sense.

ABSTRACT

In this paper, we study high order correctors in stochastic homogenization. We consider elliptic equations in divergence form on $\mathbb{Z}^d$, with the random coefficients constructed from i.i.d. random variables. We prove moment bounds on the high order correctors and their gradients under dimensional constraints. It implies the existence of stationary correctors and stationary gradients in high dimensions. As an application, we prove a two-scale expansion of the solutions to the random PDE, which identifies the first and higher order random fluctuations in a strong sense.

Motivation & Objective

• To extend quantitative homogenization theory beyond first-order correctors by analyzing high-order correctors in random media.
• To establish uniform moment estimates for high-order correctors and their gradients under dimensional and mixing conditions.
• To derive a two-scale expansion of solutions that captures first and higher-order random fluctuations in a strong, pathwise sense.
• To show the existence of stationary correctors and gradients in high dimensions via moment bounds.

Proposed method

• Constructs n-th order correctors via formal two-scale expansion in the discrete setting with i.i.d. conductances satisfying log-Sobolev inequality.
• Uses regularized corrector equations involving the generator (λ + ∇*a∇)ϕ_ξ^λ = −∇*aξ to define high-order correctors ψ_n^λ.
• Applies log-Sobolev inequality to control moments of correctors and their gradients through spectral gap and concentration techniques.
• Employs discrete Green's function estimates and scaling arguments to compare solutions of discrete and continuous PDEs.
• Establishes convergence of discrete solutions to homogenized solutions via L2 norm estimates on differences involving Green's functions.
• Uses dominated convergence and exponential decay estimates to prove convergence of approximations in the limit ε → 0.

Experimental results

Research questions

• RQ1 Under what dimensional and mixing conditions do high-order correctors in stochastic homogenization admit uniform moment bounds?
• RQ2 Can the first and higher-order random fluctuations in solutions to random PDEs be identified via a two-scale expansion in a strong, pathwise sense?
• RQ3 What is the relationship between the existence of stationary correctors and the dimension d relative to the order n of the corrector?
• RQ4 How do high-order correctors explain the discrepancy between pointwise and weak large-scale fluctuations in random media?
• RQ5 Can the formal two-scale expansion be made rigorous using moment estimates and convergence arguments?

Key findings

• For n ≥ 2 and d ≥ 2n−1, the gradient of the n-th order corrector ψ_n^λ satisfies uniform L^p moment bounds independent of λ.
• For n ≥ 2 and d ≥ 2n+1, the n-th order corrector ψ_n^λ itself satisfies uniform L^p moment bounds independent of λ.
• These moment bounds imply the existence of stationary correctors and stationary gradients in high dimensions.
• A two-scale expansion of the solution to the random PDE is rigorously established, identifying first and higher-order random fluctuations in a strong sense.
• The convergence of discrete solutions to the homogenized solution is proven via L2 norm estimates and dominated convergence, relying on exponential decay of Green's functions.
• The results extend to general i.i.d. ensembles under a weaker log-Sobolev-type condition, though the paper focuses on the stronger version for simplicity.