A deflated minimal block residual method for the solution of non-hermitian linear systems with multiple right-hand sides

In this paper we address the solution of linear systems of equations with multiple right-hand sides given at once. When the dimension of the problem is known to be large, preconditioned block Krylov subspace methods are usually considered as the method of choice. Nevertheless to be effective in terms of computational operations it is known that these methods must incorporate a strategy for detecting when a linear combination of the systems has approximately converged. This strategy usually leads to an explicit block size reduction often called deflation. While initial deflation or deflation at the beginning of a cycle are nowadays popular, block Krylov subspace methods incorporating deflation at each iteration are quite rare. The purpose of this paper is thus to extend the block flexible restarted GMRES method to variants that allow the use of deflation at each iteration when solving multiple right-hand side problems given at once. The main goal of deflation will be then to reduce the cost of each iteration of the block Krylov subspace method by judiciously choosing which information to use and which information to be postponed. For the purpose of analysis we introduce a new minimal block residual method that incorporates block size reduction at each iteration named Deflated Minimal Block Residual method. First we study its main mathematical properties. We notably show that the Frobenius norm of the block residual is always nonincreasing. Second we justify the choice of the deflation strategy based on a nonincreasing behaviour of the singular values of the scaled block true residual. Third we propose a variant of the deflated block residual method that includes truncation at each iteration. Finally we discuss the computational cost of the algorithm and the possibility of breakdown in exact arithmetic. Numerical experiments on two different problems ∗TOTAL, Centre Scientifique et Technique Jean Féger, avenue de Larribau F-64000 Pau, France †INPT-IRIT, University of Toulouse and ENSEEIHT, 2 Rue Camichel, BP 7122, F-31071 Toulouse Cedex 7, France ‡CERFACS, 42 Avenue Gaspard Coriolis, F-31057 Toulouse Cedex 1, France §CERFACS and HiePACS project joint INRIA-CERFACS Laboratory, 42 Avenue Gaspard Coriolis, F-31057 Toulouse Cedex 1, France ¶CNPq fellowship, Brazil. Applied Math. Dep., IME-UERJ, R. S. F. Xavier, 524, 629D, 20559-900, Rio de Janeiro, RJ, Brazil.