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50 Years of Time Parallel Time Integration

  • Conference paper
Multiple Shooting and Time Domain Decomposition Methods

Part of the book series: Contributions in Mathematical and Computational Sciences ((CMCS,volume 9))

Abstract

Time parallel time integration methods have received renewed interest over the last decade because of the advent of massively parallel computers, which is mainly due to the clock speed limit reached on today’s processors. When solving time dependent partial differential equations, the time direction is usually not used for parallelization. But when parallelization in space saturates, the time direction offers itself as a further direction for parallelization. The time direction is however special, and for evolution problems there is a causality principle: the solution later in time is affected (it is even determined) by the solution earlier in time, but not the other way round. Algorithms trying to use the time direction for parallelization must therefore be special, and take this very different property of the time dimension into account.We show in this chapter how time domain decomposition methods were invented, and give an overview of the existing techniques. Time parallel methods can be classified into four different groups: methods based on multiple shooting, methods based on domain decomposition and waveform relaxation, space-time multigrid methods and direct time parallel methods. We show for each of these techniques the main inventions over time by choosing specific publications and explaining the core ideas of the authors. This chapter is for people who want to quickly gain an overview of the exciting and rapidly developing area of research of time parallel methods.

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Notes

  1. 1.

    “In diesem Artikel studieren wir verschiedene Versionen einer Klasse paralleler Algorithmen, die ursprünglich von A. Bellen und M. Zennaro für Differenzengleichungen konzipiert und von ihnen ‘across the steps’ Methode genannt worden ist.”

  2. 2.

    “Pour commencer, on expose l’idée sur l’exemple simple.”

  3. 3.

    “C’est alors un exercice que de montrer la:”

  4. 4.

    “Actually this method of continuing the computation is highly inefficient and is not recommended”, see [59].

  5. 5.

    “The spectacular growth in the scale of integrated circuits being designed in the VLSI era has generated the need for new methods of circuit simulation. “Standard” circuit simulators, such as SPICE and ASTAP, simply take too much CPU time and too much storage to analyze a VLSI circuit”, see [52].

  6. 6.

    “Note that since the oscillator is highly non unidirectional due to the feedback from v 3 to the NOR gate, the convergence of the iterated solutions is achieved with the number of iterations being proportional to the number of oscillating cycles of interest”, see [52].

  7. 7.

    This example had already been proposed by Fox in 1954.

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Acknowledgements

The author is very thankful for the comments of Stefan Vandewalle, which greatly improved this manuscript and also made the content more complete. We thank the Bibliotheque de Geneve for granting permission to reproduce pictures from the original sources.

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  1. Section of Mathematics, University of Geneva, Rue du Lievre 2-4, CP 64, 1211, Geneva 4, Switzerland

    Martin J. Gander

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  1. Heidelberg University, Institute for Applied Mathematics, Heidelberg, Germany

    Thomas Carraro

  2. Heidelberg University, Institute for Applied Mathematics, Heidelberg, Germany

    Michael Geiger

  3. Heidelberg University, Interdisciplinary Center for Scientific Computing, Heidelberg, Germany

    Stefan Körkel

  4. Heidelberg University, Institute for Applied Mathematics, Heidelberg, Germany

    Rolf Rannacher

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Gander, M.J. (2015). 50 Years of Time Parallel Time Integration. In: Carraro, T., Geiger, M., Körkel, S., Rannacher, R. (eds) Multiple Shooting and Time Domain Decomposition Methods. Contributions in Mathematical and Computational Sciences, vol 9. Springer, Cham. https://doi.org/10.1007/978-3-319-23321-5_3

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