Frequency analysis of preconditioned waveform relaxation iterations

Andrzej Augustynowicz; Zdzisław Jackiewicz

Applicationes Mathematicae (1999)

  • Volume: 26, Issue: 2, page 229-242
  • ISSN: 1233-7234

Abstract

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The error analysis of preconditioned waveform relaxation iterations for differential systems is presented. This analysis extends and refines previous results by Burrage, Jackiewicz, Nørsett and Renaut by incorporating all terms in the expansion of the error of waveform relaxation iterations in the Laplace transform domain. Lower bounds for the size of the window of rapid convergence are also obtained. The theory is illustrated for waveform relaxation methods applied to differential systems resulting from semi-discretization of the heat equation in one and two dimensions. This theory and some heuristic arguments predict that preconditioning is most effective for the first few iterations.

How to cite

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Augustynowicz, Andrzej, and Jackiewicz, Zdzisław. "Frequency analysis of preconditioned waveform relaxation iterations." Applicationes Mathematicae 26.2 (1999): 229-242. <http://eudml.org/doc/219235>.

@article{Augustynowicz1999,
abstract = {The error analysis of preconditioned waveform relaxation iterations for differential systems is presented. This analysis extends and refines previous results by Burrage, Jackiewicz, Nørsett and Renaut by incorporating all terms in the expansion of the error of waveform relaxation iterations in the Laplace transform domain. Lower bounds for the size of the window of rapid convergence are also obtained. The theory is illustrated for waveform relaxation methods applied to differential systems resulting from semi-discretization of the heat equation in one and two dimensions. This theory and some heuristic arguments predict that preconditioning is most effective for the first few iterations.},
author = {Augustynowicz, Andrzej, Jackiewicz, Zdzisław},
journal = {Applicationes Mathematicae},
keywords = {error analysis; waveform relaxation; convergence; preconditioning; splitting; Laplace transform; semi-discretization; heat equation},
language = {eng},
number = {2},
pages = {229-242},
title = {Frequency analysis of preconditioned waveform relaxation iterations},
url = {http://eudml.org/doc/219235},
volume = {26},
year = {1999},
}

TY - JOUR
AU - Augustynowicz, Andrzej
AU - Jackiewicz, Zdzisław
TI - Frequency analysis of preconditioned waveform relaxation iterations
JO - Applicationes Mathematicae
PY - 1999
VL - 26
IS - 2
SP - 229
EP - 242
AB - The error analysis of preconditioned waveform relaxation iterations for differential systems is presented. This analysis extends and refines previous results by Burrage, Jackiewicz, Nørsett and Renaut by incorporating all terms in the expansion of the error of waveform relaxation iterations in the Laplace transform domain. Lower bounds for the size of the window of rapid convergence are also obtained. The theory is illustrated for waveform relaxation methods applied to differential systems resulting from semi-discretization of the heat equation in one and two dimensions. This theory and some heuristic arguments predict that preconditioning is most effective for the first few iterations.
LA - eng
KW - error analysis; waveform relaxation; convergence; preconditioning; splitting; Laplace transform; semi-discretization; heat equation
UR - http://eudml.org/doc/219235
ER -

References

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  1. [1] K. Burrage, Parallel and Sequential Methods for Ordinary Differential Equations, Oxford Univ. Press, Oxford, 1995. Zbl0838.65073
  2. [2] K. Burrage, Z. Jackiewicz, S. P. Nørsett and R. Renaut, Preconditioning waveform relaxation iterations for differential systems, BIT 36 (1996), 54-76. Zbl0864.65049
  3. [3] Z. Jackiewicz, B. Owren and B. D. Welfert, Pseudospectra of waveform relaxation operators, Computers Math. Appl. 36 (1998), 67-85. Zbl0932.65080
  4. [4] B. Leimkuhler, Estimating waveform relaxation convergence, SIAM J. Sci. Comput. 14 (1993), 872-889. Zbl0787.65039
  5. [5] E. Lelerasmee, A. Ruehli and A. Sangiovanni-Vincentelli, The waveform relaxation method for time domain analysis of large scale integrated circuits, IEEE Trans. on CAD of IC and Systems 1 (1982), 131-145. 
  6. [6] U. Miekkala and O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems, SIAM J. Sci. Statist. Comput. 8 (1987), 459-482. Zbl0625.65063
  7. [7] O. Nevanlinna, Remarks on Picard-Lindelöf iteration, Part I, BIT 29 (1989), 328-346. Zbl0673.65037

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