A Flexible Generalized Conjugate Residual Method with Inner Orthogonalization and Deflated Restarting

Abstract

This work is concerned with the development and study of a minimum residual norm subspace method based on the Generalized Conjugate Residual method with inner Orthogonalization (GCRO) method that allows flexible preconditioning and deflated restarting for the solution of non-symmetric or non-Hermitian linear systems. First we recall the main features of Flexible Generalized Minimum Residual with deflated restarting (FGMRES-DR), a recently proposed algorithm of the same family but based on the GMRES method. Next we introduce the new inner-outer subspace method named FGCRO-DR. A theoretical comparison of both algorithms is then made in the case of flexible preconditioning. It is proved that FGCRO-DR and FGMRES-DR are algebraically equivalent if a collinearity condition is satisfied. While being nearly as expensive as FGMRES-DR in terms of computational operations per cycle, FGCRO-DR offers the additional advantage to be suitable for the solution of sequences of slowly changing linear systems (where both the matrix and right-hand side can change) through subspace recycling. Numerical experiments on the solution of multidimensional elliptic partial differential equations show the efficiency of FGCRO-DR when solving sequences of linear systems.

Preconditioning the solution of the time-dependent neutron diffusion equation by recycling Krylov subspaces

“…Flexible GCRO-DR method [1] is an inner-outer method that combines GCRO as the outer method and Flexible GMRES-DR as the inner method, and it allows deflated restarting and subspace recycling. GCRO method is used to compute optimal approximation over a given set of search vectors in the sense that the residual is minimized, and the inner method …”

“…One possibility is to use the GMRES-DR method [5], which in each external iteration recycles an approximate invariant subspace to deflate eigenvalues of smallest magnitude. In particular, to obtain approximate invariant subspaces we have used a variant of GMRES-DR method known as the FGMRES-DR method [1]. The algorithm known as FGMRES(m) to solve a system Ax = b , of dimension n, relies on the Arnoldi relation…”

Block Iterative Methods and Recycling for Improved Scalability of Linear Solvers

“…A MATLAB implementation has ever since been available to try this method, with either left or right preconditioning. A flexible variant of GCRO has then been proposed [27], and eventually a flexible variant of GCRO-DR was derived [28]. In the latter reference, it is proved that under certain circumstances, FGCRO-DR is algebraically equivalent to FGMRES-DR [30], another flexible variant of well-established iterative method that uses recycling to improve the numerical efficiency of restarted GMRES [31].…”

“…As first proposed by Parks et al [22], and as implemented in Belos, GCRO-DR cannot handle variable preconditioning. A first flexible variant of GCRO-DR was proposed by Carvalho et al [28]. In the corresponding Technical Report 3 , the authors 3 http://www.cerfacs.fr/algor/reports/2010/TR PA 10 10.pdf…”

“…We also extend this study to the case of nonvariable sequence of linear systems-i.e., when only righthand sides are changing but not the linear operator itself-and to variable preconditioning, as already done theoretically [27], [28]. In section IV, we use PETSc to generate large linear systems on up to 8,196 cores.…”