Nested Krylov Methods for Shifted Linear Systems
Manuel Baumann, Martin B. van Gijzen · SIAM Journal on Scientific Computing · 2015
We consider Krylov subspace methods that are designed for sequences of shifted linear systems. For the efficient numerical solution of shifted problems, the shift-invariance property of the corresponding Krylov subspaces is used such that a Krylov basis is computed only once for all shifted systems. Preconditioners in general destroy this shift-invariance property. Known preconditioners that preserve the shift-invariance are the shift-and-invert preconditioner as well as polynomial preconditioners. In this work, we introduce a new approach to the preconditioning of multi-shift Krylov methods. In our new nested framework, we use an inner multi-shift Krylov method as a flexible preconditioner within an outer multi-shift Krylov method. In order to preserve the shift-invariance of the underlying Krylov subspaces, we require collinear residuals for the inner Krylov method. This new approach has been implemented for two possible combinations, namely, FOM-FGMRES and IDR-FQMRIDR, and has been tested for various numerical examples arising from geophysical applications.