Absolute value preconditioning for symmetric indefinite linear systems

Details Absolute value preconditioning for symmetric indefinite linear systems Absolute value preconditioning for symmetric indefinite linear systems

2011-04-23

arxiv.org/abs/1104.4530v1

Mathematics

Numerical Analysis

Scientific paper

We introduce a novel strategy for constructing symmetric positive definite (SPD) preconditioners for linear systems with symmetric indefinite coefficient matrices. The strategy is motivated by the observation that the preconditioned minimal residual method with the inverse of the absolute value of the coefficient matrix as a preconditioner converges to the exact solution of the system in at most two steps. Neither the exact absolute value of the coefficient matrix, nor its exact inverse are computationally feasible to construct in general. However, as the proof of concept, we provide two practical examples of SPD preconditioners, which are based on the suggested approach, called absolute value preconditioning. The first example is for strictly (block) diagonally dominant coefficient matrices, where we propose using the inverse to the absolute value of the (block) diagonal as the preconditioner. Our second example is less intuitive. We consider a model problem with a shifted discrete negative Laplacian, and suggest a geometric multigrid preconditioner, where the inverse to the absolute value of the coefficient matrix appears only on the coarse grid, while operations on finer grids are based on the Laplacian. Our numerical tests demonstrate practical effectiveness of the new multigrid preconditioner for moderately small shifts.

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