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Title: Anderson acceleration of the Jacobi iterative method: An efficient alternative to Krylov methods for large, sparse linear systems

Journal Article · · Journal of Computational Physics
 [1];  [1];  [2]
  1. Georgia Inst. of Technology, Atlanta, GA (United States)
  2. Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
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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 Anderson–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 speed-ups 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 large-scale parallel solutions of linear systems of equations.

Research Organization:
Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States)
Sponsoring Organization:
USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)
Grant/Contract Number:
AC52-07NA27344; AC52-07-NA27344; 1333500
OSTI ID:
1325883
Alternate ID(s):
OSTI ID: 1359285
Report Number(s):
LLNL-JRNL-695290
Journal Information:
Journal of Computational Physics, Vol. 306, Issue C; ISSN 0021-9991
Publisher:
ElsevierCopyright Statement
Country of Publication:
United States
Language:
English
Citation Metrics:
Cited by: 44 works
Citation information provided by
Web of Science

References (21)

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Cited By (3)

Convergence analysis of Anderson‐type acceleration of Richardson's iteration journal April 2019
Symmetry-adapted real-space density functional theory for cylindrical geometries: Application to large group-IV nanotubes journal September 2019
Accelerating ADMM for Efficient Simulation and Optimization preprint January 2019

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97 MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE
linear systems of equations
fixed-point iteration
Jacobi method
Anderson extrapolation
nonsymmetric matrix
Poisson equation
Helmholtz equation
parallel computing