Applications of time parallelization
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PINT 2019
Applications of time parallelization Benjamin W. Ong1
· Jacob B. Schroder2
Received: 6 December 2019 / Accepted: 15 July 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract This review article serves to summarize the many advances in time-parallel computations since the excellent review article by Gander, “50 years of Time Parallel Integration” (Gander, in: 50 years of time parallel time integration. Multiple shooting and time domain decomposition, Springer, Berlin, 2015). We focus, when possible, on applications of time parallelism and the observed speedup and efficiency, highlighting the challenges and benefits of parallel time computations. The applications covered range from numerous PDE-based simulations (both hyperbolic and parabolic), to PDE-constrained optimization, powergrid simulations, and machine learning. The time-parallel methods covered range from various iterative schemes (multigrid, waveform, multiple shooting, domain decomposition) to direct time-parallel methods. Keywords Time-parallelization · Multigrid-in-time · Waveform relaxation · Parallel-in-time Mathematics Subject Classification 34-02 · 65-02 · 65-Y05
1 Introduction Numerical simulations are increasingly important in the study of complex systems in engineering, life sciences, medicine, chemistry, physics, and even non-traditional fields such as social sciences. Computer models and simulations, often referred to as the “third pillar of science” [110], allow us to leverage modern supercomputers as virtual laboratories and experimental facilities. However, a brief glance at technological advancements in microprocessors (Fig. 1) shows that future speedup for computational simulations will come through using increasing numbers of cores and not through faster clock speeds. Thus as spatial parallelism techniques saturate, parallelization in the time direction offers a promising avenue for leveraging modern supercomputers as they can work in tandem with existing spatial parallelism to pro-
vide a multiplicative increase in concurrency [41]. The need for time-parallel integration is being driven by this massively parallel nature of modern computer architectures. Parallelization in the time direction is special because of the causality principle: solutions later in time are determined by solutions earlier in time. Algorithms trying to use the time direction for parallelization must account for this causality principle. Research on parallel-in-time integration started at least 50 years ago with the work of Nievergelt [102]. Since then, various approaches have been explored, with the review articles [41,54] and book [14] providing excellent introductions. In this manuscript, we focus on applications of time parallelism, with an emphasis on recent articles published after Gander’s review article [41]. The manuscript and discussion is organized loosely using the following four categories introduced by Gander [41]:
Communicated by Daniel Ruprecht.
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Jacob B. Schroder [email protected] Benjamin W. Ong ongbw@mtu.
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