Anderson acceleration of the Jacobi iterative method: An efficient alternative to Krylov methods for large, sparse linear systems
We employ Anderson extrapolation to accelerate the classical Jacobi iterative method for large, sparse linear systems. Specifically, we utilize extrapolation at periodic intervals within the Jacobi iteration to develop the Alternating Andersonâ€“Jacobi (AAJ) method. We verify the accuracy and efficacy of AAJ in a range of test cases, including nonsymmetric systems of equations. We demonstrate that AAJ possesses a favorable scaling with system size that is accompanied by a small prefactor, even in the absence of a preconditioner. In particular, we show that AAJ is able to accelerate the classical Jacobi iteration by over four orders of magnitude, with speedups that increase as the system gets larger. Moreover, we find that AAJ significantly outperforms the Generalized Minimal Residual (GMRES) method in the range of problems considered here, with the relative performance again improving with size of the system. As a result, the proposed method represents a simple yet efficient technique that is particularly attractive for largescale parallel solutions of linear systems of equations.
 Authors:

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 Georgia Inst. of Technology, Atlanta, GA (United States)
 Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
 Publication Date:
 Report Number(s):
 LLNLJRNL695290
Journal ID: ISSN 00219991
 Grant/Contract Number:
 AC5207NA27344; AC5207NA27344
 Type:
 Accepted Manuscript
 Journal Name:
 Journal of Computational Physics
 Additional Journal Information:
 Journal Volume: 306; Journal Issue: C; Journal ID: ISSN 00219991
 Publisher:
 Elsevier
 Research Org:
 Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
 Sponsoring Org:
 USDOE
 Country of Publication:
 United States
 Language:
 English
 Subject:
 97 MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; linear systems of equations; fixedpoint iteration; Jacobi method; Anderson extrapolation; nonsymmetric matrix; Poisson equation; Helmholtz equation; parallel computing
 OSTI Identifier:
 1325883
 Alternate Identifier(s):
 OSTI ID: 1359285
Pratapa, Phanisri P., Suryanarayana, Phanish, and Pask, John E.. Anderson acceleration of the Jacobi iterative method: An efficient alternative to Krylov methods for large, sparse linear systems. United States: N. p.,
Web. doi:10.1016/j.jcp.2015.11.018.
Pratapa, Phanisri P., Suryanarayana, Phanish, & Pask, John E.. Anderson acceleration of the Jacobi iterative method: An efficient alternative to Krylov methods for large, sparse linear systems. United States. doi:10.1016/j.jcp.2015.11.018.
Pratapa, Phanisri P., Suryanarayana, Phanish, and Pask, John E.. 2015.
"Anderson acceleration of the Jacobi iterative method: An efficient alternative to Krylov methods for large, sparse linear systems". United States.
doi:10.1016/j.jcp.2015.11.018. https://www.osti.gov/servlets/purl/1325883.
@article{osti_1325883,
title = {Anderson acceleration of the Jacobi iterative method: An efficient alternative to Krylov methods for large, sparse linear systems},
author = {Pratapa, Phanisri P. and Suryanarayana, Phanish and Pask, John E.},
abstractNote = {We employ Anderson extrapolation to accelerate the classical Jacobi iterative method for large, sparse linear systems. Specifically, we utilize extrapolation at periodic intervals within the Jacobi iteration to develop the Alternating Andersonâ€“Jacobi (AAJ) method. We verify the accuracy and efficacy of AAJ in a range of test cases, including nonsymmetric systems of equations. We demonstrate that AAJ possesses a favorable scaling with system size that is accompanied by a small prefactor, even in the absence of a preconditioner. In particular, we show that AAJ is able to accelerate the classical Jacobi iteration by over four orders of magnitude, with speedups that increase as the system gets larger. Moreover, we find that AAJ significantly outperforms the Generalized Minimal Residual (GMRES) method in the range of problems considered here, with the relative performance again improving with size of the system. As a result, the proposed method represents a simple yet efficient technique that is particularly attractive for largescale parallel solutions of linear systems of equations.},
doi = {10.1016/j.jcp.2015.11.018},
journal = {Journal of Computational Physics},
number = C,
volume = 306,
place = {United States},
year = {2015},
month = {12}
}