Least Angle Regression Coarsening in Bootstrap Algebraic Multigrid

TL;DR

This paper introduces an adaptive coarsening method for bootstrap algebraic multigrid that models interpolation as a local regression problem, improving coarsening and stability in algebraic multigrid methods.

Contribution

It develops a novel adaptive coarsening scheme based on local regression, integrating machine learning concepts into algebraic multigrid to enhance coarsening and stability.

Findings

Effective coarsening scheme demonstrated through numerical experiments

Improved stability of the interpolation operator achieved

Sparse responses enable better variable coupling analysis

Abstract

The bootstrap algebraic multigrid framework allows for the adaptive construction of algebraic multigrid methods in situations where geometric multigrid methods are not known or not available at all. While there has been some work on adaptive coarsening in this framework in terms of algebraic distances, coarsening is the part of the adaptive bootstrap setup that is least developed. In this paper we try to close this gap by introducing an adaptive coarsening scheme that views interpolation as a local regression problem. In fact the bootstrap algebraic multigrid setup can be understood as a machine learning ansatz that learns the nature of smooth error by local regression. In order to turn this idea into a practical method we modify least squares interpolation to both avoid overfitting of the data and to recover a sparse response that can be used to extract information about the coupling strength amongst variables like in classical algebraic multigrid. In order to improve the so-found coarse grid we propose a post-processing to ensure stability of the resulting least squares interpolation operator. We conclude with numerical experiments that show the viability of the chosen approach.