Variants of the block GMRES method for solving multi-shifted linear systems with multiple right-hand sides simultaneously

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 Krylov subspace methods for this problem class. Shifted block Krylov subspace methods are computationally attractive to use as they can preserve the shift-invariance property of the block Krylov subspace and can use much larger search spaces for solving sequences of multi-shifted and multiple right-hand sides linear systems simultaneously. In Chapter 1 we introduce the background about Krylov subspace methods and state the main research problems of this class of methods. In Chapter 2, we develop a new variant of the restarted shifted block Krylov method augmented with eigenvectors. In Chapter 3, we introduce a new flexible and deflated variant of the shifted block Krylov method solving the whole sequence of multi-shifted linear systems simultaneously based on an initial deflation strategy. In Chapter 4, we exploit the inexact breakdown strategy to develop a new shifted, augmented and deflated block Krylov method. In Chapter 5, we present spectrally preconditioned and initially deflated variants of block iterative Krylov solvers. Finally, in Chapter 6 we summarize the main characteristics of the algorithms proposed in the thesis, compare their numerical performance and draw some plans for future research.