Nested Krylov methods based on GCR
The author presents an extension of the GMRESR method for the solution of systems of linear equations where the orthogonality relations of the outer method GCR are preserved in the inner method GMRES\((m)\). At the end of the inner iteration the error is minimized over the space spanned by the search vectors both in the outer method and the inner method. This leads to optimal corrections to the solution in the outer method. Besides a detailed discussion of optimality, the possibility of breakdown and the continuation after breakdown implementation issues (including truncation in the outer method and the usage of BiCGStab instead of GMRES\((m)\) as inner method) are considered. Furthermore, several numerical examples are discussed.
scientific article; zbMATH DE number 908353