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Parallel integration of ODEs based on convolution algorithms

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Abstract

We consider parallel computing methods for ODEs based on the potential parallelism of convolution algorithms. Detailed presentations of fast convolution algorithms are provided. Implementations of our methods on signal processors and special processors are described. The main emphasis is concentrated on applications of powerful parallel signal processing facilities in ODE computations. By using convolution algorithms we show some treatments of parallel computing in finite fields. As an example of our approach we discuss integrating Lorenz's model.

Zusammenfassung

Wir behandeln parallele numerische Methoden für gewöhnliche Differentiagleichungen, die auf Parallelismus von konvolutiven Algorithmen basieren. Ausführliche Darstellungen der schnellen konvolutiven Berechnungen mit spezieller Hardware werden präsentiert. Der Schwerpunkt dieser Arbeit ist die Anwendung von leistungsfähigen parallelen Signal-Prozessoren in der Berechnung gewöhnlicher Differentialgleichungen. Unter Verwendung von Konvolutiven Algorithmen erläutern wir einige Anwendungen von parallelem Rechnen in endlichen Körpern. Als Beispiel unseres Zugangs diskutieren wir die Integration des Lorenz-Modells.

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References

  1. Ashworth, M., Lyne, A.: A segmented FFT algorithm for vector computers. Parallel Comput.6, 217–224 (1988).

    Google Scholar 

  2. Bellen, A., Zennaro, M.: Parallel algorithms for initial value problems for difference and differential equations. J. Comput. Appl. Math.25, 341–350 (1989).

    Google Scholar 

  3. Blahut, R.: Fast algorithms for digital signal processing. New York: Addison-Wesley 1985.

    Google Scholar 

  4. Boglaev, Yu.: An iterative method for the approximate solution of singularly perturbed problems. Soviet Math. Dokl., AMS17, 543–547 (1976).

    Google Scholar 

  5. Boglaev, Yu.: Construction by iterates of fundamental solutions of systems of linear ODE in general position. Soviet Math. Dokl., AMS37, 760–764 (1988).

    Google Scholar 

  6. Boglaev, Yu.: Parallel integration of ODEs by vector field decomposition. Int. J. Comp. Math.47, 43–58 (1993).

    Google Scholar 

  7. Evans, D., Megson, G.: Construction of extrapolation tables by systolic arrays for solving ordinary differential equations. Parallel Comput.4, 33–48 (1987).

    Google Scholar 

  8. Gear, C.: Parallel methods for ordinary differential equations. Calcolo25, 1–20 (1988).

    Google Scholar 

  9. van der Houwen, P. J., Sommeijer, B. P.: Iterated Runge-Kutta methods on parallel computers. SIAM J. Sci. Statist. Comput.12, 1000–1028 (1991).

    Google Scholar 

  10. Hutson, V. C. L., Pym, J. S.: Applications of functional analysis and operator theory. London, New York: Academic Press 1980.

    Google Scholar 

  11. Jackson, K. R., Norsett, S. P.: The potential for parallelism in Runge-Kutta methods. Part 1: RK formulas in standard form. Univ. of Toronto, Rep. 239/90, Dept. of Computer Science (1990).

  12. Krasnoselski, M.: Shift operators on trajectories of differential equations. (in Russian). Moscow: Nauka 1966.

    Google Scholar 

  13. Kung, S. Y., Whitehouse, H. J., Kailath, T. (eds.): VLSI and modern signal processing. Englewood Cliffs, N.J.: Prentice-Hall 1985.

    Google Scholar 

  14. Lee, E. A.: Programmable DSPs: A Brief Overview. IEEE MICRO10, 14–16 (1990).

    Google Scholar 

  15. Lichtenberg, A., Liberman, M.: Regular and stochastic motion. Berlin, Heidelberg, New York: Springer 1983.

    Google Scholar 

  16. Lorenz, E.: Deterministic nonperiodic flow. J. Atmos. Sci.20, 130–141 (1963).

    Google Scholar 

  17. Miller, R. K.: Optical computers: the next frontier in computing1. Englewood/Fort Lee-N.J.: Technical Insights 1986.

    Google Scholar 

  18. Miranker, W. L.: Numerical methods for stiff equations and singular perturbation problems. Dordrecht, Boston, London: D. Reidel 1981.

    Google Scholar 

  19. Pollard, J.: The fast Fourier transform in finite fields. Math. Comp.25, 356–374 (1976).

    Google Scholar 

  20. Preston, K., Jr.: Coherent optical computers. New York: McGraw-Hill 1972.

    Google Scholar 

  21. Swarztrauber, O.: Multiprocessor FFT's.Parallel Comput.5, 197–210 (1987).

    Google Scholar 

  22. Tam, H. W.: Parallel methods for the numerical solution of ordinary differential equations. Univ. of Illinois, Rep UIUCDCS-R-89-1516 (1989).

  23. Temperton, C.: Implementation of a prime factor FFT algorithm on CRAY-1. Parallel Comput.6, 99–108 (1988).

    Google Scholar 

  24. Umeo, H.: A design of time-optimum and register-number-minimum systolic convolvers. Parallel Comput.12, 285–299 (1989).

    Google Scholar 

  25. Wasow, W.: Asymptotic expansions for ordinary differential equations. New York: John Wiley & Sons 1965.

    Google Scholar 

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  1. Computer Science Division Institute for High Temperatures, Russian Academy of Sciences, Izhorskaya 13/19, 127412, Moscow, Russia

    Y. P. Boglaev

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  1. Y. P. Boglaev
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Boglaev, Y.P. Parallel integration of ODEs based on convolution algorithms. Computing 51, 185–207 (1993). https://doi.org/10.1007/BF02238533

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  • DOI: https://doi.org/10.1007/BF02238533

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