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

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:
 [1];  [2];  [1]
  1. Sandia National Lab. (SNL-CA), Livermore, CA (United States)
  2. Univ. of Maryland, College Park, MD (United States)
Publication Date:
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:
Journal Article: 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.
Carlberg, Kevin, Forstall, Virginia, & Tuminaro, Ray. Krylov-Subspace Recycling via the POD-Augmented Conjugate-Gradient Method. United States. doi:10.1137/16M1057693.
Carlberg, Kevin, Forstall, Virginia, and Tuminaro, Ray. Fri . "Krylov-Subspace Recycling via the POD-Augmented Conjugate-Gradient Method". United States. doi:10.1137/16M1057693. https://www.osti.gov/servlets/purl/1251146.
@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},
issn = {0895-4798},
number = 3,
volume = 37,
place = {United States},
year = {2016},
month = {1}
}

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