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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P R O P O S I T I O N S belonging to the thesis
Variants of the block GMRES method for solving multi-shifted linear
systems with multiple right-hand sides simultaneously
by
Donglin Sun
1. Shifted block Krylov subspace methods are effective iterative methods for solving linear systems with multiple shifts and multiple right-hand sides that arise frequently in large-scale scientific and engineering simulations.
– Chapters 2, 3, and 4
2. The shifted block GMRES method augmented with eigenvectors at each restart has the ability to mitigate the negative effects on the convergence due to eigen-values near zero.
– Chapter 2
3. One difficulty with the implementation and use of shifted block Krylov subspace methods is their lack of robustness due to the presence of approximately linearly dependent right-hand sides or block residuals. The initial deflation strategy can handle the approximate rank deficiency of the block residuals effectively restoring fast convergence.
– Chapter 3
4. Detecting inexact breakdowns in the inner block Arnoldi procedure can improve significantly the robustness of shifted block Krylov subspace methods. Recycling spectral information at restart can help to speed up the iterative procedure.
– Chapter 4
5. Spectral preconditioning combined with an adaptive eigenvalue recycling strat-egy can restore the superlinear convergence of the restarted block GMRES method for the simultaneous solution of sequences of linear systems with mul-tiple right-hand sides that often arise from the discretization of either partial differential equations or integral equations.
– Chapter 5
6. Different deflation strategies can be developed and applied in combination with shifted block Krylov subspace method. Identifying the best deflation technique in terms of robustness and implementation efficiency is an open research question that deserves further analysis.
– Chapter 6
7. Learning needs to be combined with thinking. Without thinking, one cannot go deep; without learning, one cannot go far.
