Randomly sparsified Richardson iteration: A dimension-independent sparse linear solver

TL;DR

This paper introduces a novel randomized sparsification algorithm for solving high-dimensional sparse linear systems, providing a rigorous mathematical analysis and demonstrating faster convergence than traditional Monte Carlo methods.

Contribution

It extends random sparsification techniques to linear system solving and offers a complete theoretical analysis of the method's convergence properties.

Findings

Faster-than-Monte Carlo convergence rate established.

Algorithm effective even when solution vectors are too large to store.

Applicable to arbitrarily high-dimensional sparse systems.

Abstract

Recently, a class of algorithms combining classical fixed point iterations with repeated random sparsification of approximate solution vectors has been successfully applied to eigenproblems with matrices as large as $10^{108} \times 10^{108}$. So far, a complete mathematical explanation for their success has proven elusive. The family of methods has not yet been extended to the important case of linear system solves. In this paper we propose a new scheme based on repeated random sparsification that is capable of solving sparse linear systems in arbitrarily high dimensions. We provide a complete mathematical analysis of this new algorithm. Our analysis establishes a faster-than-Monte Carlo convergence rate and justifies use of the scheme even when the solution vector itself is too large to store.