Low-rank approximation to heterogeneous elliptic problems

Abstract

© 2018 Society for Industrial and Applied Mathematics. In this work, we investigate the low-rank approximation of elliptic problems in heterogeneous media by means of Kolmogrov n-width and asymptotic expansion. This class of problems arises in many practical applications involving high-contrast media, and their efficient numerical approximation often relies crucially on certain low-rank structure of the solutions. We provide conditions on the permeability coefficient κ that ensure a favorable low-rank approximation. These conditions are expressed in terms of the distribution of the inclusions in the coefficient κ, e.g., the values, locations, and sizes of the heterogeneous regions. Further, we provide a new asymptotic analysis for high-contrast elliptic problems based on the perfect conductivity problem and layer potential techniques, which allows deriving new estimates on the spectral gap for such high-contrast problems. These results provide theoretical underpinnings for several multiscale model reduction algorithms.