Multi-way Monte Carlo Method for Linear Systems

TL;DR

This paper introduces a multi-way Monte Carlo method for solving linear systems that relaxes the convergence condition from a norm constraint to a spectral radius condition, enabling faster solutions on more problems.

Contribution

It proposes a novel multi-way Markov random walk framework that broadens the applicability and improves the efficiency of Monte Carlo methods for linear systems.

Findings

The new method works under the weaker spectral radius condition.

Numerical experiments show improved speed and applicability.

The approach outperforms standard Monte Carlo methods on tested matrices.

Abstract

We study the Monte Carlo method for solving a linear system of the form $x = H x + b$. A sufficient condition for the method to work is $\| H \| < 1$, which greatly limits the usability of this method. We improve this condition by proposing a new multi-way Markov random walk, which is a generalization of the standard Markov random walk. Under our new framework we prove that the necessary and sufficient condition for our method to work is the spectral radius $\rho(H^{+}) < 1$, which is a weaker requirement than $\| H \| < 1$. In addition to solving more problems, our new method can work faster than the standard algorithm. In numerical experiments on both synthetic and real world matrices, we demonstrate the effectiveness of our new method.