Multipreconditioned GMRES for Shifted Systems
Tania Bakhos, Peter Kitanidis, Scott Ladenheim, Arvind K. Saibaba, and, Daniel Szyld
TL;DR
This paper introduces a multipreconditioned GMRES method for efficiently solving shifted linear systems, reducing computational cost by leveraging multiple preconditioners without requiring matrix-vector products with shifted matrices.
Contribution
It presents a novel implementation of GMRES with multiple preconditioners that grows linearly with the number of preconditioners, improving efficiency for shifted systems.
Findings
Effective in solving shifted systems in hydrology and matrix functions
Reduces computational cost by avoiding shifted matrix-vector products
Numerical results demonstrate improved efficiency and effectiveness
Abstract
An implementation of GMRES with multiple preconditioners (MPGMRES) is proposed for solving shifted linear systems with shift-and-invert preconditioners. With this type of preconditioner, the Krylov subspace can be built without requiring the matrix-vector product with the shifted matrix. Furthermore, the multipreconditioned search space is shown to grow only linearly with the number of preconditioners. This allows for a more efficient implementation of the algorithm. The proposed implementation is tested on shifted systems that arise in computational hydrology and the evaluation of different matrix functions. The numerical results indicate the effectiveness of the proposed approach.
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