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Title: Krylov-Subspace Recycling via the POD-Augmented Conjugate Gradient Method.


Abstract not provided.

; ;
Publication Date:
Research Org.:
Sandia National Lab. (SNL-CA), Livermore, CA (United States)
Sponsoring Org.:
USDOE National Nuclear Security Administration (NNSA)
OSTI Identifier:
Report Number(s):
Journal ID: ISSN 0895--4798; 645693
DOE Contract Number:
Resource Type:
Resource Relation:
Journal Volume: 37; Journal Issue: 3; Conference: Proposed for presentation at the 2016 SIAM Annual Meeting held July 11-15, 2016 in Boston, MA.
Country of Publication:
United States

Citation Formats

Carlberg, Kevin Thomas, Forstall, Virginia, and Tuminaro, Raymond S. Krylov-Subspace Recycling via the POD-Augmented Conjugate Gradient Method.. United States: N. p., 2016. Web. doi:10.1137/16M1057693.
Carlberg, Kevin Thomas, Forstall, Virginia, & Tuminaro, Raymond S. Krylov-Subspace Recycling via the POD-Augmented Conjugate Gradient Method.. United States. doi:10.1137/16M1057693.
Carlberg, Kevin Thomas, Forstall, Virginia, and Tuminaro, Raymond S. 2016. "Krylov-Subspace Recycling via the POD-Augmented Conjugate Gradient Method.". United States. doi:10.1137/16M1057693.
title = {Krylov-Subspace Recycling via the POD-Augmented Conjugate Gradient Method.},
author = {Carlberg, Kevin Thomas and Forstall, Virginia and Tuminaro, Raymond S.},
abstractNote = {Abstract not provided.},
doi = {10.1137/16M1057693},
journal = {},
number = 3,
volume = 37,
place = {United States},
year = 2016,
month = 7

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  • This paper presents a new Krylov-subspace-recycling method for efficiently solving sequences of linear systems of equations characterized by varying right-hand sides and symmetric-positive-definite matrices. As opposed to typical truncation strategies used in recycling such as deflation, we propose a truncation method inspired by goal-oriented proper orthogonal decomposition (POD) from model reduction. This idea is based on the observation that model reduction aims to compute a low-dimensional subspace that contains an accurate solution; as such, we expect the proposed method to generate a low-dimensional subspace that is well suited for computing solutions that can satisfy inexact tolerances. In particular, we proposemore » specific goal-oriented POD `ingredients' that align the optimality properties of POD with the objective of Krylov-subspace recycling. To compute solutions in the resulting 'augmented' POD subspace, we propose a hybrid direct/iterative three-stage method that leverages 1) the optimal ordering of POD basis vectors, and 2) well-conditioned reduced matrices. Numerical experiments performed on solid-mechanics problems highlight the benefits of the proposed method over existing approaches for Krylov-subspace recycling.« less
  • Many problems from engineering and the sciences require the solution of sequences of linear systems where the matrix and right-hand side change from one system to the next, and the linear systems are not available simultaneously. We review a class of Krylov subspace methods for sequences of linear systems, which can significantly reduce the cost of solving the next system in the sequence by 'recycling' subspace information from previous systems. These methods have been successfully applied to sequences of linear systems arising from several different application areas. We analyze a particular method, GCRO-DR, that recycles approximate invariant subspaces, and establishmore » residual bounds that suggest a convergence rate similar to one obtained by removing select eigenvector components from the initial residual. We review implications of this analysis, which suggests problem classes where we expect this technique to be particularly effective. From this analysis and related numerical experiments we also demonstrate that recycling the invariant subspace corresponding to the eigenvalues of smallest absolute magnitude is often not the best choice, especially for nonsymmetric problems, and that GCRO-DR will, in practice, select better subspaces. These results suggest possibilities for improvement in the subspace selection process.« less
  • Abstract not provided.
  • A subspace adaptation of the Coleman-Li trust region and interior method is proposed for solving large-scale bound-constrained minimization problems. This method can be implemented with either sparse Cholesky factorization or conjugate gradient computation. Under reasonable conditions the convergence properties of this subspace trust region method are as strong as those of its full-space version. Computational performance on various large test problems is reported; advantages of the approach are demonstrated. The experience indicates that the proposed method represents an efficient way to solve large bound-constrained minimization problems.