Random Sampling and Efficient Algorithms for Multiscale PDEs

TL;DR

This paper introduces a novel numerical framework utilizing random sampling to efficiently solve multiscale PDEs, achieving asymptotic preservation and homogenization without relying on detailed analytical models.

Contribution

The proposed method is a new, general approach that does not depend on specific PDE asymptotics, enabling efficient multiscale PDE solutions through random sampling.

Findings

Achieves asymptotic preserving property for kinetic equations

Enables numerical homogenization for elliptic equations with rough media

Demonstrates efficiency and generality across different multiscale PDEs

Abstract

We describe a numerical framework that uses random sampling to efficiently capture low-rank local solution spaces of multiscale PDE problems arising in domain decomposition. In contrast to existing techniques, our method does not rely on detailed analytical understanding of specific multiscale PDEs, in particular, their asymptotic limits. We present the application of the framework on two examples --- a linear kinetic equation and an elliptic equation with rough media. On these two examples, this framework achieves the asymptotic preserving property for the kinetic equations and numerical homogenization for the elliptic equations.