University of Groningen
Variants of the block GMRES method for solving multi-shifted linear systems with multiple
right-hand sides simultaneously
Sun, Donglin
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Publication date: 2019
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Sun, D. (2019). Variants of the block GMRES method for solving multi-shifted linear systems with multiple right-hand sides simultaneously. University of Groningen.
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Summary
This thesis concerns with the development of efficient Krylov subspace methods for solving sequences of large and sparse linear systems with multiple shifts and multiple right-hand sides given simultaneously. The need to solve this mathematical problem efficiently arises frequently in large-scale scientific and engineering applications. We introduce new robust variants of the shifted block Generalized Minimum Residual Method (GMRES) that can preserve the shift-invariance property of the block Krylov subspace for this problem class.
In Chapter 1 we present the background about Krylov subspace methods and multi-shifted linear systems with multiple right-hand sides, we discuss the strengths and problems of this popular class of methods, and state the main objectives of the thesis. As the convergence of Krylov methods depends to a large extent on the eigenvalue distribution of the linear system, it is observed that in many cases “removing” the smallest eigenvalues can greatly accelerate the iterative solution. In Chapter 2, we develop a new variant of the restarted shifted block GMRES method augmented with eigenvectors that has the ability to mitigate the negative effects of small eigenvalues on the iterations, resulting in faster convergence.
Another difficulty with the implementation and use of shifted block Krylov subspace methods for solving sequences of linear systems with multiple shifts and multiple right-hand sides is their lack of robustness due to the presence of approximately linearly dependent right-hand sides vectors. In Chapter 3, we introduce a new deflated variant of the shifted block GMRES method, based on an initial deflation strategy, that can detect near rank deficiency of the block residuals while solving the whole sequence of multi-shifted linear systems simultaneously. Further, we present a preconditioned variant of the new Krylov solver that maintains the block shift-invariance property.
In Chapter 4, we exploit the inexact breakdown strategy to develop a new shifted, augmented and deflated block GMRES method that can handle effectively numerical instabilities due to the occurrence of inexact breakdowns in the block Arnoldi procedure by detecting the approximately
148 Summary
linearly dependent directions of the Krylov basis, and additionally it recycles spectral information at restart to achieve faster convergence.
In Chapter 5, we consider the problem of solving simultaneously sequences of unshifted linear systems with multiple right-hand sides, a problem that often occurs in the numerical solution of partial differential and boundary integral equations. We present a new class of block iterative Krylov solvers that exploit approximate invariant subspaces recycled over the iterations to adapt an existing preconditioner and have the ability to handle the approximate linear dependence of the block of right-hand sides quite effectively. Our numerical experiments show that the new family of methods can outperform conventional block Krylov algorithms for solving general linear systems arising from the discretization of the Dirac equation and boundary integral equations in electromagnetics scattering.
In Chapter 6 we summarize the main characteristics of the algorithms proposed in the thesis, and compare their numerical performance for solving sequences of multi-shifted linear systems with multiple right-hand sides arising in computational fluid dynamics and electromagnetics applications. Finally, we draw some plans for future research.
