A Modified Block Flexible GMRES Method with Deflation at Each Iteration for the Solution of Non-Hermitian Linear Systems with Multiple Right-Hand Sides

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. 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, Exact and inexact breakdowns in the block GMRES method, Linear Algebra Appl., 419 (2006), pp. 265-285]. We extend their contribution by proposing the deflation to be also performed at the beginning of each cycle. This change leads to a modified least-squares problem to be solved at each iteration and gives raise to a different behavior especially when the method is often restarted. 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 saving 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.