Low-rank approximation for multiscale PDEs

TL;DR

This paper introduces a unified framework utilizing randomized SVD and manifold learning to efficiently approximate multiscale PDEs, demonstrated on radiative transfer and elliptic equations with rough media.

Contribution

It presents a novel, unified approach for multiscale PDEs using random sampling, combining randomized SVD and manifold learning for low-rank feature reconstruction.

Findings

Effective low-rank approximation of multiscale PDEs demonstrated

Framework applicable to radiative transfer and elliptic equations

Improved computational efficiency for multiscale problems

Abstract

Historically, analysis for multiscale PDEs is largely unified while numerical schemes tend to be equation-specific. In this paper, we propose a unified framework for computing multiscale problems through random sampling. This is achieved by incorporating randomized SVD solvers and manifold learning techniques to numerically reconstruct the low-rank features of multiscale PDEs. We use multiscale radiative transfer equation and elliptic equation with rough media to showcase the application of this framework.