QMR-Based Projection Techniques for the Solution of Non-Hermitian Systems with Multiple Right-Hand Sides

Authors:
Misha Kilmer;Eric Miller;Carey Rappaport
Affiliations:
-;-;-
Venue:
SIAM Journal on Scientific Computing
Year:
2001

Citing 0
Cited 5

Deflated and Restarted Symmetric Lanczos Methods for Eigenvalues and Linear Equations with Multiple Right-Hand Sides

SIAM Journal on Scientific Computing
Computing and Deflating Eigenvalues While Solving Multiple Right-Hand Side Linear Systems with an Application to Quantum Chromodynamics

SIAM Journal on Scientific Computing
A block IDR(s) method for nonsymmetric linear systems with multiple right-hand sides

Journal of Computational and Applied Mathematics
Restarting the Nonsymmetric Lanczos Algorithm for Eigenvalues and Linear Equations Including Multiple Right-Hand Sides

SIAM Journal on Scientific Computing
Accelerating data uncertainty quantification by solving linear systems with multiple right-hand sides

Numerical Algorithms

Quantified Score

Hi-index	0.00

Visualization

Abstract

In this work we consider the simultaneous solution of large linear systems of the form Ax(j) = b(j), j=1,. . . ,K, where A is sparse and non-Hermitian. We describe single-seed and block-seed projection approaches to these multiple right-hand side problems that are based on the QMR and block QMR algorithms, respectively. We use (block) QMR to solve the (block) seed system and generate the relevant biorthogonal subspaces. Approximate solutions to the nonseed systems are simultaneously generated by minimizing their appropriately projected (block) residuals. After the initial (block) seed has converged, the process is repeated by choosing a new (block) seed from among the remaining nonconverged systems and using the previously generated approximate solutions as initial guesses for the new seed and nonseed systems. We give theory for the single-seed case that helps explain the convergence behavior under certain conditions. Implementation details for both the single-seed and block-seed algorithms are discussed and advantages of the block-seed algorithm in cache-based serial and parallel environments are noted. The computational savings of our methods over using QMR to solve each system individually are illustrated in two examples.