Accelerating the induced dimension reduction method using spectral information
The induced dimension reduction method (IDR(s)) [\textit{P. Sonneveld} and \textit{M. B. van Gijzen}, SIAM J. Sci. Comput. 31, No. 2, 1035--1062 (2008; Zbl 1190.65053)] is a short-recurrences Krylov method to solve non-symmetric systems of linear equations. In this work, the method is accelerated using spectral information. A Hessenberg relation is constructed from the IDR(s) residual recurrences formulas, and the eigenpairs ot the involved Hessenberg matrix are used to approximate the eigenvalues and eigenvectors. The inverses of the Ritz values are used as choices of the method's free scalars, thus leading to a self-contained variant of the Ritz-IDR(s) method [\textit{V. Simoncini} and \textit{D. B. Szyld}, SIAM J. Sci. Comput. 32, No. 4, 1898--1912 (2010; Zbl 1219.65039)] without call to an external eigensolver. In a number of numerical experiments this requires less computational time than a non-self-contained variant of the Ritz-IDR(s) method. In addition, Ritz vectors corresponding to small magnitude Ritz values are used to augment the search Krylov subspaces. The corresponding augmented methods are used to speed-up IDR(s) for the solution of sequences of systems of linear equations with identical matrices but different right-hand sides that are not available simultaneously. Spectral information gained when solving the first system is recycled in subsequent systems through augmentation with the corresponding approximate invariant subspaces. On both diffusion- and convection-dominated time-dependent convection-diffusion-reaction problems, the recycling shows a clear benefit in both the needed amount of matrix-vector products and CPU time.
scientific article; zbMATH DE number 6950466
scientific article