Topics in matrix analysis
SIAM Journal on Matrix Analysis and Applications
A flexible inner-outer preconditioned GMRES algorithm
SIAM Journal on Scientific Computing
Krylov subspace methods for solving large Lyapunov equations
SIAM Journal on Numerical Analysis
An iterative method for nonsymmetric systems with multiple right-hand sides
SIAM Journal on Scientific Computing
SIAM Journal on Matrix Analysis and Applications
A hybrid block GMRES method for nonsymmetric systems with multiple right-hand sides
Proceedings of the 6th international congress on Computational and applied mathematics
Restarted GMRES preconditioned by deflation
Journal of Computational and Applied Mathematics
Three-dimensional perfectly matched layer for the absorption of electromagnetic waves
Journal of Computational Physics
Matrix computations (3rd ed.)
A block variant of the GMRES method on massively parallel processors
Parallel Computing
Adaptively Preconditioned GMRES Algorithms
SIAM Journal on Scientific Computing
ABLE: An Adaptive Block Lanczos Method for Non-Hermitian Eigenvalue Problems
SIAM Journal on Matrix Analysis and Applications
Global FOM and GMRES algorithms for matrix equations
Applied Numerical Mathematics
A Lanczos-type method for multiple starting vectors
Mathematics of Computation
Multigrid
A block GMRES method augmented with eigenvectors
Applied Mathematics and Computation
FQMR: A Flexible Quasi-Minimal Residual Method with Inexact Preconditioning
SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Two-Sided Arnoldi and Nonsymmetric Lanczos Algorithms
SIAM Journal on Matrix Analysis and Applications
Flexible Inner-Outer Krylov Subspace Methods
SIAM Journal on Numerical Analysis
On Improving Linear Solver Performance: A Block Variant of GMRES
SIAM Journal on Scientific Computing
Functions of Matrices: Theory and Computation (Other Titles in Applied Mathematics)
Functions of Matrices: Theory and Computation (Other Titles in Applied Mathematics)
Restarted block-GMRES with deflation of eigenvalues
Applied Numerical Mathematics - 6th IMACS International symposium on iterative methods in scientific computing
A perfectly matched layer for the absorption of electromagnetic waves
Journal of Computational Physics
Multigrid Smoothers for Ultraparallel Computing
SIAM Journal on Scientific Computing
A block GCROT(m,k) method for linear systems with multiple right-hand sides
Journal of Computational and Applied Mathematics
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In a wide number of applications in computational science and engineering the solution of large linear systems of equations with several right-hand sides given at once is required. Direct methods based on Gaussian elimination are known to be especially appealing in that setting. Nevertheless, when the dimension of the problem is very large, preconditioned block Krylov space solvers are often considered as the method of choice. The purpose of this paper is thus to present iterative methods based on block restarted GMRES that allow variable preconditioning for the solution of linear systems with multiple right-hand sides. The use of flexible methods is especially of interest when approximate possibly iterative solvers are considered in the preconditioning phase. First we introduce a new variant of block flexible restarted GMRES that includes a strategy for detecting when a linear combination of the systems has approximately converged. This explicit block size reduction is often called deflation. We analyze the main properties of this flexible method based on deflation and notably prove that the Frobenius norm of the block residual is always nonincreasing. We also present a flexible variant based on both deflation and truncation to especially be used in case of limited memory. Finally we illustrate the numerical behavior of these flexible block methods for large industrial simulations arising in geophysics, where indefinite linear systems of size up to 1 billion unknowns with multiple right-hand sides have been successfully solved in a parallel distributed memory environment.