Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-23T00:09:20.788Z Has data issue: false hasContentIssue false
  • English
  • Français

Iterative solution of systems of linear differential equations

Published online by Cambridge University Press:  07 November 2008

Ulla Miekkala  and
Olavi Nevanlinna
Ulla Miekkala
Affiliation:
Helsinki University of Technology Institute of Mathematics Otakaari 1, 02150 Espoo, Finland E-mail: [email protected], [email protected]
Olavi Nevanlinna
Affiliation:
Helsinki University of Technology Institute of Mathematics Otakaari 1, 02150 Espoo, Finland E-mail: [email protected], [email protected]
Get access Rights & Permissions [Opens in a new window]

Extract

Parallel processing has made iterative methods an attractive alternative for solving large systems of initial value problems. Iterative methods for initial value problems have a history of more than a century, and in the works of Picard (1893) and Lindelöf (1894) they were given a firm theoretical basis. In particular, the superlinear convergence on finite intervals is included in Lindelöf (1894).

Type
Research Article
Information
Acta Numerica , Volume 5 , January 1996 , pp. 259 - 307
Copyright
Copyright © Cambridge University Press 1996

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Aupetit, B. (1991), A Primer on Spectral Theory, Springer, Berlin.Google Scholar
Chen, W. K. (1976), Applied Graph Theory, graphs and electrical networks, North-Holland, Amsterdam.Google Scholar
Eirola, T. (1992), ‘Sobolev characterization of solutions of dilation equations’, SIAM J. Math. Anal. 23(4), 1015–1030.Google Scholar
Gantmacher, F. R. (1959), Matrizenrechnung II, VEB Deutscher Verlag der Wissenschaften, Berlin.Google Scholar
Hadamard, J. (1893), ‘Etude sur les propriétés des fonctions entières et en particulier d'une fonction considérée par Riemann’, J. de Math. Pures et Appl. 9, 171–215.Google Scholar
Jeltsch, R. and Pohl, B. (1995), ‘Waveform relaxation with overlapping splittings’, SIAM J. Sci. Comput. 16(1), 40–49.CrossRefGoogle Scholar
Juang, F. (1990), Waveform Methods for Ordinary Differential Equations, PhD thesis, University of Illinois at Urbana-Champaign, Dept of Computer Science. Report No. UIUCDCS-R-90–1563.Google Scholar
Leimkuhler, B., Miekkala, U. and Nevanlinna, O. (1991), ‘Waveform relaxation for linear RC circuits’, Impact of Computing in Science and Engineering 3, 123–145.CrossRefGoogle Scholar
Lelarasmee, E., Ruehli, A. and Sangiovanni-Vincentelli, A. (1982), ‘The waveform relaxation method for time-domain analysis of large scale integrated circuits’, IEEE Trans. CAD 1(3), 131–145.Google Scholar
Lindelöf, E. (1894), ‘Sur l'application des méthodes d'approximations successives à l'Etude des intégrales réelles des Equations différentielles ordinaires’, J. de Math. Pures et Appl., 4e Série 10, 117–128.Google Scholar
Lubich, C. (1992), ‘Chebyshev acceleration of Picard–Lindelöf iteration’, BIT 32, 535–538.Google Scholar
Lubich, C. and Ostermann, A. (1987), ‘Multigrid dynamic iteration for parabolic equations’, BIT 27, 216–234.Google Scholar
Lumsdaine, A. and White, J. (1995), ‘Accelerating waveform relaxation methods with application to parallel semiconductor device simulation’, Numerical functional analysis and optimization 16(3,4), 395–414.Google Scholar
Lumsdaine, A. and Wu, D. (1995), Spectra and pseudospectra of waveform relaxation operators, Technical Report CSE-TR-95–14, Department of Computer Science and Engineering, University of Notre Dame, IN.Google Scholar
Manke, J. W., Dembart, B., Epton, M. A., Erisman, A. M., Lu, P., Sincovec, R. F. and Yip, E. L. (1979), Solvability of Large Scale Descriptor Systems, Boeing Computer Services Company, Seattle, WA.Google Scholar
Miekkala, U. (1989), ‘Dynamic iteration methods applied to linear DAE systems’, J. Comp. Appl. Math. 25, 133–151.Google Scholar
Miekkala, U. (1991), Theory for iterative solution of large dynamical systems using parallel computations, PhD thesis, Helsinki University of Technology.Google Scholar
Miekkala, U. (1996), Remarks on WR method with overlapping splittings. In preparation.Google Scholar
Miekkala, U. and Nevanlinna, O. (1987 a), ‘Convergence of dynamic iteration methods for initial value problems’, SIAM J. Sci. Stat. Comput. 8(4), 459–482.Google Scholar
Miekkala, U. and Nevanlinna, O. (1987 b), ‘Sets of convergence and stability regions’, BIT 27, 554–584.Google Scholar
Miekkala, U. and Nevanlinna, O. (1992), ‘Quasinilpotency of the operators in Picard–Lindelöf iteration’, Numer. Funct. Anal, and Optimiz. 13(1,2), 203–221.Google Scholar
Miekkala, U., Nevanlinna, O. and Ruehli, A. (1990), Convergence and circuit partitioning aspects for waveform relaxation, in Proceedings of the Fifth Distributed Memory Computing Conference, Charleston, South Carolina (Walker, D. and Stout, Q., eds), IEEE Computer Society Press, Los Alamitos, CA, pp. 605–611.Google Scholar
Nevanlinna, O. (1989 a), A note on Picard-Lindelöf iteration, in Numerical Methods for Ordinary Differential Equations, Proceedings of the Workshop held in L'Aquila (Italy), Sept. 16–18, 1987. Vol. 1386 of Lecture Notes in Mathematics (Bellen, A., Gear, C. W. and Russo, E., eds), Springer.Google Scholar
Nevanlinna, O. (1989 b), ‘Remarks on Picard-Lindelöf iteration, PART I’ BIT 29, 328–346.Google Scholar
Nevanlinna, O. (1989 c), ‘Remarks on Picard Lindelöf iteration, PART II’, BIT 29, 535–562.Google Scholar
Nevanlinna, O. (1990 a), ‘Linear acceleration of Picard-Lindelöf iteration’, Numer. Math. 57, 147–156.Google Scholar
Nevanlinna, O. (1990 b), ‘Power bounded prolongations and Picard-Lindelöf iteration’, Numer. Math. 58, 479–501.Google Scholar
Nevanlinna, O. (1991), Waveform relaxation always converges for RC-circuits, in Proc. of NASECOD VII, held in April 8–12, 1991, Colorado, Front Range Press, Colorado.Google Scholar
Nevanlinna, O. (1993), Convergence of iterations for linear equations, Lectures in Mathematics, ETH Zürich, Birkhäuser, Basel.Google Scholar
Nevanlinna, O. and Odeh, F. (1987), ‘Remarks on the convergence of waveform relaxation method’, Numer. Fund. Anal, and Optimiz. 9(3,4), 435–445.CrossRefGoogle Scholar
Picard, E. (1893), ‘Sur l'application des méthodes d'approximations successives à l'étude de certaines équations différentielles ordinaires’, J. Math. Pures. Appl., 4e série 9, 217–271.Google Scholar
Piirila, O.ä(1993), ‘Questions and notions related to quasialgebraicity in Banach algebras’, Annales Academia Scientiarum Fennica, Series A, Mathematica Dissertationes. Helsinki.Google Scholar
Reichelt, M., White, J. and Allen, J. (1995), ‘Optimal convolution SOR acceleration of waveform relaxation with application to parallel simulation of semiconductor devices’, SIAM J. Sci. Comput. 16(5), 1137–1158.Google Scholar
Skeel, R. (1989), ‘Waveform iteration and the shifted Picard splitting’, SIAM J. Sci. Stat. Comput. 10(4), 756–776.CrossRefGoogle Scholar
Spijker, M. N. (1991), ‘On a conjecture by LeVeque and Trefethen related to the Kreiss matrix theorem’, BIT 31, 551–555.Google Scholar
Trefethen, L. N. (1992), Pseudospectra and matrices, in Numerical Analysis (Griffiths, D. F. and Watson, G. A., eds), Longman, Harlow, UK, pp. 234–266.Google Scholar
Vainberg, M. M. and Trenogin, V. A. (1974), Theory of branching of solutions of non-linear equations, Noordhoff International Publishing, Leyden.Google Scholar
Vandewalle, S. (1992), The Parallel Solution of Parabolic Partial Differential Equations by Multigrid Waveform Relaxation Methods, PhD thesis, Katholieke Universiteit Leuven, Belgium.Google Scholar
Vandewalle, S. (1993), Parallel Multigrid Waveform Relaxation for Parabolic Problems, B.G. Teubner, Stuttgart.Google Scholar
White, J. and Sangiovanni-Vincentelli, A. (1987), Relaxation Techniques for the Simulation of VLSI Circuits, Kluwer, Boston.Google Scholar