ILAS2016 — 11–15 July 2016 — KU Leuven, Belgium

20th Conference of the International Linear Algebra Society (ILAS)

20th ILAS Conference

Contributed talks on Linear Systems

Thu 14:00–14:30, Room AV 03.12, Chair: Kirk Soodhalter
A block GMRES-DR method for solving linear systems with multiple shifts and multiple right-hand sides
Dong-Lin Sun (University of Groningen)
Joint work with Ting-Zhu Huang (University of Electronic Science and Technology of China); Yan-Fei Jing (University of Electronic Science and Technology of China); Bruno Carpentieri (Nottingham Trent University)

We propose a new shifted block Krylov subspace algorithm (named as BGMRES-DR-Sh) that solves all of the linear systems with multiple shifts and multiple right-hand sides simultaneously, and it has the ability to dump the negative effect of small eigenvalues from the convergence. Our method can be seen as an extension of the BGMRES-DR method proposed by Morgan in [1] to the case of shifted linear systems. This approach can be computationally more efficient than applying Krylov subspace methods to the sequence of linear systems with multiple shifts, or block Krylov subspace methods to each single shifted linear system with multiple right-hand sides separately.

In our talk we present the main lines of development of the BGMRES-DR-Sh method, describing its theoretical properties, and we report on numerical experiments on a set of representative matrix problems having size up to 1.5M unknowns arising from different fields. We analyse various aspects such as performance comparison against other Krylov methods, the effect of the seed selection strategy, the effect of the preconditioning technique, and the sensitivity of our method to the accuracy of the eigen-computation. Finally, in the last part of the presentation we show how to improve the performance of the proposed BGMRES-DR-Sh by combining it with initial deflation.

  1. R. B. Morgan. Restarted block-GMRES with deflation of eigenvalues. Applied Numerical Mathematics, 54(2):222–236, 2005.