skip to main content


Title: 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 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.
 [1] ;  [1] ;  [2]
  1. Georgia Inst. of Technology, Atlanta, GA (United States)
  2. Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
Publication Date:
Report Number(s):
Journal ID: ISSN 0021-9991
Grant/Contract Number:
AC52-07NA27344; AC52-07-NA27344
Accepted Manuscript
Journal Name:
Journal of Computational Physics
Additional Journal Information:
Journal Volume: 306; Journal Issue: C; Journal ID: ISSN 0021-9991
Research Org:
Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
Sponsoring Org:
Country of Publication:
United States
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
OSTI Identifier:
Alternate Identifier(s):
OSTI ID: 1359285