Advertisement

An Accelerated Block-Parallel Newton Method via Overlapped Partitioning

  • Yurong Chen
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 40)

Summary

This paper presents an overlapped block-parallel Newton method for solving large nonlinear systems. The graph partitioning algorithms are first used to partition the Jacobian into weakly coupled overlapping blocks. Then the simplified Newton iteration is directly performed, with the diagonal blocks and the overlapping solutions assembled in a weighted average way at each iteration. In the algorithmic implementation, an accelerated technique has been proposed to reduce the number of iterations. The conditions under which the algorithm is locally and semi-locally convergent are studied. Numerical results from solving power flow equations are presented to support our study.

Keywords

Newton Method Krylov Subspace Sandia National Laboratory Inexact Newton Method Good Initial Guess 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. P. N. Brown and Y. Saad. Hybrid Krylov methods for nonlinear system of equations. SIAM J. Sci. Stat. Comput., 11(3):450–481, 1990.MathSciNetCrossRefGoogle Scholar
  2. X. C. Cai and D. E. Keyes. Nonlinear proconditioned inexact Newton algorithms. SIAM J. Sci. Comput., 24(1):183–200, 2002.MathSciNetCrossRefGoogle Scholar
  3. Y. Chen and D. Cai. Inexact overlapped block Broyden methods for solving nonlinear equations. Appl. Math. Comput., 136(2/3):215–228, 2003.MathSciNetCrossRefGoogle Scholar
  4. R. Dembo, S. Eisenstat, and T. Steihaug. Inexact Newton methods. Siam J. Numer. Anal., 19(2):400–408, 1982.MathSciNetCrossRefGoogle Scholar
  5. A. Frommer. Parallel nonlinear multisplitting methods. Numer. Math., 56:269–282, 1989.MATHMathSciNetCrossRefGoogle Scholar
  6. W. D. Gropp, D. E. Keyes, L. C. McInnes, and M. D. Tidriri. Globalized Newton-Krylov-Schwarz algorithms and software for parallel implicit CFD. Int. J. High Performance Computing Applications, 14:102–136, 2000.CrossRefGoogle Scholar
  7. B. Hendrickson and R. Leland. The Chaco user's guide, version 2.0. Technical report, Sandia National Laboratories, Albuquerque, NM, July 1995. Tech. Rep. SAND 95-2344.Google Scholar
  8. D. A. Knoll and D. E. Keyes. Jacobian-free Newton-Krylov methods: A survey of approaches and applications. Int. J. High Performance Computing Applications, 2003. to appear.Google Scholar
  9. J. M. Ortega and W. C. Rheinboldt. Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, 1970.Google Scholar
  10. Y. Saad. Krylov subspace methods for solving unsymmetric linear systems. Math. Comp., 37:105–126, 1981.MATHMathSciNetCrossRefGoogle Scholar
  11. Y. Saad and M. Schultz. GMRES: A generalized mininum residual algorithm for solving non-symmetric linear systems. SIAM J. Sci. Stat. Comput., 7:856–869, 1986.MathSciNetCrossRefGoogle Scholar
  12. G. Yang, L. C. Dutto, and M. Fortin. Inexact block Jacobi-Broyden methods for solving nonlinear systems of equations. SIAM J. Sci. Comput., 18(5):1367–1392, 1997.MathSciNetCrossRefGoogle Scholar
  13. A. I. Zecevic and D. D. Siljak. A block-parallel Newton method via overlapping epsilon decompositions. SIAM J. Matrix Analysis and Applications, 15(3):824–844, 1994.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Yurong Chen
    • 1
  1. 1.Lab. of Parallel ComputingInstitute of Software, CASChina

Personalised recommendations