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. 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 Robbe and Sadkane [M. Robbe 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 truncatio...

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