Recycling for sequences of linear systems by proper orthogonal decomposition augmented Krylov subspaces

Recycling for sequences of linear systems by proper orthogonal decomposition augmented Krylov subspaces

Carlberg, Kevin; Tsuji, Paul; Forstall, Virginia

Abstract:

Sequences of sparse linear systems arise in many applications, including iterative methods for PDE-constrained optimization. In such cases, inexact solves are often su cient to guarantee convergence, wherein a forcing sequence defines modest solver tolerances such that the computed solutions satisfy. Krylov-subspace recycling methods accelerate convergence for such problems by reusing (truncated) information generated during the solution of previous linear systems. In particular, they search for solutions in the subspace Km + Y, where the subspace Y contains information from previous solves and `augments'' the Krylov subspace Km. Existing recycling techniques (e.g., de ation, optimal truncation) are tailored to improve convergence toward exact solutions; as a result, they do not always realize convergence acceleration when inexact solutions are sought. Instead, we propose a recycling technique inspired by model reduction that aims to e ciently compute inexact solutions. In particular, we construct Y by a goal-oriented proper orthogonal decomposition (POD) of previous search directions, where the POD inner product enables eficient computation of solutions in Y.

Sequences of sparse linear systems arise in many applications, including iterative methods for PDE-constrained optimization. In such cases, inexact solves are often su cient to guarantee convergence, wherein a forcing sequence defines modest solver tolerances such that the computed solutions satisfy. Krylov-subspace recycling methods accelerate convergence for such problems by reusing (truncated) information generated during the solution of previous linear systems. In particular, they search for solutions in the subspace Km + Y, where the subspace Y contains information from previous solves and `augments'' the Krylov subspace Km. Existing recycling techniques (e.g., de ation, optimal truncation) are tailored to improve convergence toward exact solutions; as a result, they do not always realize convergence acceleration when inexact solutions are sought. Instead, we propose a recycling technique inspired by model reduction that aims to e ciently compute inexact solutions. In particular, we construct Y by a goal-oriented proper orthogonal decomposition (POD) of previous search directions, where the POD inner product enables eficient computation of solutions in Y.

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Como citar:

Carlberg, Kevin; Tsuji, Paul; Forstall, Virginia; "Recycling for sequences of linear systems by proper orthogonal decomposition augmented Krylov subspaces", p-21-21. In: Proceedings of the 13th International Symposium on Multiscale, Multifunctional and Functionally Graded Materials [=Blucher Material Science Proceedings, v.1, n.1]. São Paulo: Blucher, 2014.
ISSN 23589337, DOI