A Modified Block Flexible GMRES Method with Deflation at Each Iteration for the Solution of Non-Hermitian Linear Systems with Multiple Right-Hand Sides
Henri Calandra, Serge Gratton, Rafael Lago, Xavier Vasseur, Luiz Mariano Carvalho · SIAM Journal on Scientific Computing · 2013
We propose a variant of the block GMRES method for the solution of linear systems of equations with multiple right-hand sides. We investigate a deflation strategy to detect when a linear combination of approximate solutions is already known that avoids performing expensive computational operations with the system matrix. This is especially useful when the cost of the preconditioner is supposed to be larger than the cost of orthogonalization in the block Arnoldi procedure. We specifically focus on the block GMRES method incorporating deflation at the end of each iteration proposed by Robbé and Sadkane [M. Robbé and M. Sadkane, Linear Algebra Appl., 419 (2006), pp. 265--285]. We extend their contribution by proposing that deflation be performed also at the beginning of each cycle. This change leads to a modified least-squares problem to be solved at each iteration and gives rise to a different behavior especially when multiple restarts are required to reach convergence. Additionally we investigate truncation techniques, aiming at reducing the computational cost of the iteration. This is particularly useful when the number of right-hand sides is large. Finally, we address the case of variable preconditioning, an important feature when iterative methods are used as preconditioners, as investigated here. The numerical experiments performed in a parallel environment show the relevance of the proposed variant on a challenging application related to geophysics. A savings of up to 35% in terms of computational time---at the same memory cost---is obtained with respect to the original method on this application.