New Randomized Global Generalized Minimum Residual (RGl-GMRES) method

TL;DR

This paper introduces a novel randomized algorithm, RGl-GMRES, for efficiently solving large-scale linear systems with multiple right-hand sides, leveraging matrix sketching and providing new convergence insights.

Contribution

It develops a new randomized global GMRES algorithm using matrix sketching and offers convergence analysis and practical validation for large-scale problems.

Findings

RGl-GMRES is competitive with existing methods.

The method reduces computational dimension via matrix sketching.

Convergence bounds are established for diagonalizable matrices.

Abstract

In this paper, we develop a new Randomized Global Generalized Minimum Residual (RGlGMRES) algorithm for efficiently computing solutions to large scale linear systems with multiple right hand sides.The proposed method builds on a recently developed randomized global Gram Schmidt process, in which sketched Frobenius inner products are employed to approximate the exact Frobenius inner products of high-dimensional matrices. We give some new convergence results of the randomized global GMRES method for multiple linear systems. In the case where the coefficient matrix A is diagonalizable, we derive new upper bounds for the randomized Frobenius norm of the residual. In this paper, we study how to introduce matrix sketching in this algorithm. It allows us to reduce the dimension of the problem in one of the main steps of the algorithm. To validate the effectiveness and practicality of this approach, we conduct several numerical experiments, which demonstrate that our RGl-GMRES method is competitive with the GlGMRES method for solving large scale problems with multiple right-hand sides.