Efficient variants of the CMRH method for solving a sequence of multi-shifted non-Hermitian linear systems simultaneously

TL;DR

This paper develops efficient variants of the CMRH method tailored for solving multiple non-Hermitian linear systems with shifts simultaneously, improving convergence speed through flexible preconditioning and demonstrating effectiveness in PDE and FDE applications.

Contribution

The paper introduces new variants of the CMRH method, including a flexible version with variable preconditioning, optimized for multi-shifted non-Hermitian systems, with analysis and numerical validation.

Findings

Enhanced convergence with flexible preconditioning.

Superior performance compared to existing Krylov methods.

Effective in solving PDEs and FDEs numerically.

Abstract

Multi-shifted linear systems with non-Hermitian coefficient matrices arise in numerical solutions of time-dependent partial/fractional differential equations (PDEs/FDEs), in control theory, PageRank problems, and other research fields. We derive efficient variants of the restarted Changing Minimal Residual method based on the cost-effective Hessenberg procedure (CMRH) for this problem class. Then, we introduce a flexible variant of the algorithm that allows to use variable preconditioning at each iteration to further accelerate the convergence of shifted CMRH. We analyse the performance of the new class of methods in the numerical solution of PDEs and FDEs, also against other multi-shifted Krylov subspace methods.