On Advanced Monte Carlo Methods for Linear Algebra on Advanced Accelerator Architectures

TL;DR

This paper explores the performance and scalability of advanced Monte Carlo methods for linear algebra problems on modern accelerator architectures, using numerical experiments to evaluate their effectiveness as preconditioners and solvers.

Contribution

It introduces computational experiments with the Markov Chain Monte Carlo Matrix Inversion method on various accelerators, assessing its performance and scalability in linear algebra applications.

Findings

Benefits of the Monte Carlo method as a preconditioner identified

Scalability issues and deficiencies highlighted through experiments

Performance comparison with traditional iterative methods like GMRES and BICGstab

Abstract

In this paper we present computational experiments with the Markov Chain Monte Carlo Matrix Inversion ($(\text{MC})^2\text{MI}$) on several accelerator architectures and investigate their impact on performance and scalability of the method. The method is used as a preconditioner and for solving the corresponding system of linear equations iterative methods, such as generalized minimal residuals (GMRES) or bi-conjugate gradient (stabilized) (BICGstab), are used. Numerical experiments are carried out to highlight the benefits and deficiencies of both approaches and to assess their overall usefulness in light of scalability of the method.