A theoretical comparison of the Arnoldi and GMRES algorithms
SIAM Journal on Scientific and Statistical Computing
SIAM Journal on Scientific and Statistical Computing
A flexible inner-outer preconditioned GMRES algorithm
SIAM Journal on Scientific Computing
New insights in GMRES-like methods with variable preconditioners
Journal of Computational and Applied Mathematics
Matrix market: a web resource for test matrix collections
Proceedings of the IFIP TC2/WG2.5 working conference on Quality of numerical software: assessment and enhancement
The finite element discretization for stream-function problems on multiply connected domains
Journal of Computational Physics
Accuracy and Stability of Numerical Algorithms
Accuracy and Stability of Numerical Algorithms
Theory of Inexact Krylov Subspace Methods and Applications to Scientific Computing
SIAM Journal on Scientific Computing
Inexact Krylov Subspace Methods for Linear Systems
SIAM Journal on Matrix Analysis and Applications
Restarted GMRES with inexact matrix–vector products
NAA'04 Proceedings of the Third international conference on Numerical Analysis and its Applications
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This paper studies computational aspects of Krylov methods for solving linear systems where the matrix-vector products dominate the cost of the solution process because they have to be computed via an expensive approximation procedure. In recent years, so-called relaxation strategies for tuning the precision of the matrix-vector multiplications in Krylov methods have proved to be effective for a range of problems. In this paper, we will argue that the gain obtained from such strategies is often limited. Another important strategy for reducing the work in the matrix-vector products is preconditioning the Krylov method by another iterative Krylov method. Flexible Krylov methods are Krylov methods designed for this situation. We combine these two approaches for reducing the work in the matrix-vector products. Specifically, we present strategies for choosing the precision of the matrix-vector products in several flexible Krylov methods as well as for choosing the accuracy of the variable preconditioner such that the overall method is as efficient as possible. We will illustrate this computational scheme with a Schur-complement system that arises in the modeling of global ocean circulation.