H2-MG: A multigrid method for hierarchical rank structured matrices

TL;DR

H2-MG is a novel multigrid iterative solver that combines hierarchical H2 matrix approximations with multilevel techniques, achieving efficient and rapid solutions for large kernel matrix systems with complex geometries.

Contribution

This paper introduces H2-MG, the first multigrid method tailored for hierarchical H2 matrices, enhancing efficiency and convergence in solving large dense systems.

Findings

Linear complexity demonstrated on test problems

Effective on standard kernels and boundary element discretizations

Offers rapid convergence with reduced memory usage

Abstract

This paper presents a new fast iterative solver for large systems involving kernel matrices. Advantageous aspects of H2 matrix approximations and the multigrid method are hybridized to create the H2-MG algorithm. This combination provides the time and memory efficiency of H2 operator representation along with the rapid convergence of a multilevel method. We describe how H2-MG works, show its linear complexity, and demonstrate its effectiveness on two standard kernels and on a single-layer potential boundary element discretization with complex geometry. The current zoo of H2 solvers, which includes a wide variety of iterative and direct solvers, so far lacks a method that exploits multiple levels of resolution, commonly referred to in the iterative methods literature as ``multigrid'' from its origins in a hierarchy of grids used to discretize differential equations. This makes H2-MG a valuable addition to the collection of H2 solvers. The algorithm has potential for advancing various fields that require the solution of large, dense, symmetric positive definite matrices.