Krylov-Subspace Recycling via the POD-Augmented Conjugate-Gradient Method

Carlberg, Kevin; Forstall, Virginia; Tuminaro, Ray

doi:10.1137/16M1057693

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

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Abstract

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 propose 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.

Authors:

Carlberg, Kevin ^[1]; Forstall, Virginia ^[2]; Tuminaro, Ray ^[1]

Sandia National Lab. (SNL-CA), Livermore, CA (United States)
Univ. of Maryland, College Park, MD (United States)

Publication Date:: Fri Jan 01 00:00:00 EST 2016

Research Org.:: Sandia National Lab. (SNL-CA), Livermore, CA (United States)

Sponsoring Org.:: USDOE National Nuclear Security Administration (NNSA)

OSTI Identifier:: 1251146

Report Number(s):: SAND-2016-0828J
Journal ID: ISSN 0895-4798; 618975

Grant/Contract Number:: AC04-94AL85000

Resource Type:: Accepted Manuscript

Journal Name:: SIAM Journal on Matrix Analysis and Applications

Additional Journal Information:: Journal Volume: 37; Journal Issue: 3; Journal ID: ISSN 0895-4798

Publisher:: SIAM

Country of Publication:: United States

Language:: English

Subject:: 97 MATHEMATICS AND COMPUTING; Krylov-subspace recycling; proper orthogonal decomposition; augmented Krylov methods; model reduction; conjugate-gradient method

Citation Formats


                    Carlberg, Kevin, Forstall, Virginia, and Tuminaro, Ray. Krylov-Subspace Recycling via the POD-Augmented Conjugate-Gradient Method.  United States: N. p., 2016. 
Web.  doi:10.1137/16M1057693.

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                    Carlberg, Kevin, Forstall, Virginia, & Tuminaro, Ray. Krylov-Subspace Recycling via the POD-Augmented Conjugate-Gradient Method.  United States.  https://doi.org/10.1137/16M1057693

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                    Carlberg, Kevin, Forstall, Virginia, and Tuminaro, Ray. Fri .  
"Krylov-Subspace Recycling via the POD-Augmented Conjugate-Gradient Method".  United States.  https://doi.org/10.1137/16M1057693.  https://www.osti.gov/servlets/purl/1251146.

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@article{osti_1251146,

  title        = {Krylov-Subspace Recycling via the POD-Augmented Conjugate-Gradient Method},

  author       = {Carlberg, Kevin and Forstall, Virginia and Tuminaro, Ray},

  abstractNote = {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 propose 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.},

  doi          = {10.1137/16M1057693},

  journal      = {SIAM Journal on Matrix Analysis and Applications},

  number       = 3,

  volume       = 37,

  place        = {United States},

  year         = {Fri Jan 01 00:00:00 EST 2016},

  month        = {Fri Jan 01 00:00:00 EST 2016}

}

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