A generic multiresolution preconditioner for sparse symmetric systems

TL;DR

This paper presents a versatile multiresolution preconditioner for symmetric linear systems that adapts to the specific system without geometric assumptions, improving efficiency and applicability.

Contribution

It introduces a new preconditioner based on Multiresolution Matrix Factorization that automatically discovers an effective wavelet basis tailored to the system.

Findings

Effective preconditioning for various symmetric systems

Fast preconditioner-vector multiplication

Invariant to row/column ordering

Abstract

We introduce a new general purpose multiresolution preconditioner for symmetric linear systems. Most existing multiresolution preconditioners use some standard wavelet basis that relies on knowledge of the geometry of the underlying domain. In constrast, based on the recently proposed Multiresolution Matrix Factorization (MMF) algorithm, we construct a preconditioner that discovers a custom wavelet basis adapted to the given linear system without making any geometric assumptions. Some advantages of the new approach are fast preconditioner-vector products, invariance to the ordering of the rows/columns, and the ability to handle systems of any size. Numerical experiments on finite difference discretizations of model PDEs and off-the-shelf matrices illustrate the effectiveness of the MMF preconditioner.