Low-rank approximation to heterogeneous elliptic problems

Abstract

In this work, we investigate the low-rank approximation of elliptic problems in heterogeneous media by means of Kolmogrov <em>n</em>-width and asymptotic expansion. This class of problems arises in practical applications involving high-contrast media, and their efficient approximation often relies crucially on certain low-rank structure of the solutions. We provide conditions on the permeability coefficient <em>κ</em> that ensure a favorable low-rank approximation. These conditions are expressed in terms of the distribution of the inclusions in the coefficient <em>κ</em>, e.g., the values, locations and diameters of the heterogeneous regions. Further, we provide a new asymptotic analysis for high-contrast elliptic problems based on the perfect conductivity problems and layer potential techniques. These results provide theoretical underpinnings for several multiscale model reduction algorithms.