Many problems in science and engineering require the solution of a sequence of linear systems. Our interest is in the case of large-scale systems solved by preconditioned Krylov methods. We investigate how to obtain an overall solution procedure that is cheaper, in terms of total solution costs, than either reusing the same preconditioner without modifications or recomputing the preconditioner from scratch for each linear system.
Given a factorized preconditioner for a (reference) matrix of the sequence, update preconditioning techniques enhance the quality of such preconditioner through low-cost updates containing information on the subsequent matrices. In this talk, we propose and analyze preconditioning techniques for sequences of linear systems arising in optimization. We address the problem of preconditioning two classes of sequences: sequences of nonsymmetric matrices arising in Jacobian-free Newton-Krylov methods and sequences of shifted symmetric and positive definite matrices from trust-region and regularization procedures.
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