Manifold Learning and Nonlinear Homogenization

TL;DR

This paper introduces a manifold learning-inspired domain decomposition framework for efficiently solving nonlinear multiscale PDEs, leveraging local tangent space approximations to improve computational efficacy without requiring detailed analytical multiscale understanding.

Contribution

The paper presents a novel, versatile method for nonlinear multiscale PDEs that uses manifold learning techniques to compress local solution spaces, enhancing efficiency and applicability.

Findings

Significant efficiency improvements in solving multiscale PDEs

Effective application to oscillatory media and nonlinear radiative transfer equations

Method does not depend on detailed asymptotic analysis

Abstract

We describe an efficient domain decomposition-based framework for nonlinear multiscale PDE problems. The framework is inspired by manifold learning techniques and exploits the tangent spaces spanned by the nearest neighbors to compress local solution manifolds. Our framework is applied to a semilinear elliptic equation with oscillatory media and a nonlinear radiative transfer equation; in both cases, significant improvements in efficacy are observed. This new method does not rely on detailed analytical understanding of the multiscale PDEs, such as their asymptotic limits, and thus is more versatile for general multiscale problems.