Effective approximations of solutions to highly oscillatory diffusion equations from coarse measurements

TL;DR

This paper presents a method to approximate effective diffusion coefficients in highly oscillatory systems using coarse measurements, based on a non-convex optimization approach grounded in homogenization theory, with demonstrated numerical success.

Contribution

It introduces a novel approach to recover effective coefficients from partial data, improving previous methods by providing a rigorous framework and practical algorithms.

Findings

Successful approximation of effective coefficients from coarse measurements.

Numerical illustrations demonstrating practical applicability.

Theoretical foundation based on homogenization theory.

Abstract

We approximate a diffusion equation with highly oscillatory coefficients with a diffusion equation with constant coefficients. The approach is put in action in contexts where only partial information (namely the global energy stored in the physical system) is available. While the reconstruction of the microstructure is known to be an ill-posed problem, we show that the reconstruction of effective coefficients is possible and this even with only some coarse information. The strategy we present takes the form of a non-convex optimization problem. Homogenization theory provides elements for a rigorous foundation of the approach. Some algorithmic aspects are discussed in details. We provide a comprehensive set of numerical illustrations that demonstrate the practical interest of our strategy. The present work improves on the earlier works [C. Le Bris, F. Legoll and S. Lemaire, COCV 2018; C. Le Bris, F. Legoll and K. Li, CRAS 2013].