Mean Field and Gaussian Approximation for Partial Differential Equations with Random Coefficients

Abstract

Abstract. After discussing in several contexts how mean field and fluctuation approximations arise and can be used, we give a simple method by which the analysis of the approximations can be carried out.

Fluctuation of Solutions to Linear Elliptic Equations with Noisy Diffusion Coefficients

“…In the case when the perturbation is in the zero-order term, that is for [3], and more recently for d ≤ 3 by Bal in [1]. In their works, they also provide a precise description of the limit law, which we don't do in this article.…”

Random integrals and correctors in homogenization

“…Such formulas are not available in higher space dimensions for elliptic equations of the form −∇ · [a( x ε )∇u ε ] = f . However, in simpler situations, such as for problems of the form [−Δ + λ + q( x ε )]u ε = f [7] or for more general perturbations of elliptic operators such as [−∇ · a(x)∇ + q 0 (x) + q( x ε )]u ε = f [1] (with smooth, deterministic functions a and q 0 and random ergodic processes q with strong mixing properties), the correctors to homogenization exhibit similar behaviors to those presented in Theorem 2.6. We are tempted to believe that the main qualitative conclusion of our paper will remain valid in higher space dimensions, namely that long-memory effects will trigger large random corrections to homogenization, at least for the problems considered in [1,7].…”

Anomaly Detection in Random Heterogeneous Media

“…The ideas presented in the third part of this work are motivated by Nolen and Papanicolaou's paper [130], where, in the case of Schrödinger operators with random potentials, closed-form expressions were employed in order to quantify both the uncertainty due to fine scale fluctuations in the coefficients of the forward problem as well as its propagation to the inverse parameter identification problem. More precisely, they modeled the effect on the solution to the forward problem using a central limit theorem from [49]. Unfortunately, analogous theoretical results for divergence form operators are only available in one spatial dimension at this point.…”