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‎references/index.html‎

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<h2 id="year2024" class="year">2024 <span class="count-stat"></span></h2>
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BetckeEtAl2024
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<span id="BetckeEtAl2024">M. M. Betcke, L. M. Kreusser, and D. Murari, “Parallel-in-Time Solutions with Random Projection Neural Networks,” arXiv:2408.09756v1 [math.NA], 2024 [Online]. Available at: <a href="http://arxiv.org/abs/2408.09756v1" target="_blank">http://arxiv.org/abs/2408.09756v1</a></span>
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BibTeX entry <code>BetckeEtAl2024</code>
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<pre class="abstract">@unpublished{BetckeEtAl2024,
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author = {Betcke, Marta M. and Kreusser, Lisa Maria and Murari, Davide},
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howpublished = {arXiv:2408.09756v1 [math.NA]},
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title = {Parallel-in-Time Solutions with Random Projection Neural Networks},
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url = {http://arxiv.org/abs/2408.09756v1},
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year = {2024}
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}
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This paper considers one of the fundamental parallel-in-time methods for the solution of ordinary differential equations, Parareal, and extends it by adopting a neural network as a coarse propagator. We provide a theoretical analysis of the convergence properties of the proposed algorithm and show its effectiveness for several examples, including Lorenz and Burgers’ equations. In our numerical simulations, we further specialize the underpinning neural architecture to Random Projection Neural Networks (RPNNs), a 2-layer neural network where the first layer weights are drawn at random rather than optimized. This restriction substantially increases the efficiency of fitting RPNN’s weights in comparison to a standard feedforward network without negatively impacting the accuracy, as demonstrated in the SIR system example.
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GanderEtAl2024
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<span id="GanderEtAl2024">M. J. Gander, M. Ohlberger, and S. Rave, “A Parareal algorithm without Coarse Propagator?,” arXiv:2409.02673v1 [math.NA], 2024 [Online]. Available at: <a href="http://arxiv.org/abs/2409.02673v1" target="_blank">http://arxiv.org/abs/2409.02673v1</a></span>
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author = {Gander, Martin J. and Ohlberger, Mario and Rave, Stephan},
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howpublished = {arXiv:2409.02673v1 [math.NA]},
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title = {A Parareal algorithm without Coarse Propagator?},
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url = {http://arxiv.org/abs/2409.02673v1},
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The Parareal algorithm was invented in 2001 in order to parallelize the solution of evolution problems in the time direction. It is based on parallel fine time propagators called F and sequential coarse time propagators called G, which alternatingly solve the evolution problem and iteratively converge to the fine solution. The coarse propagator G is a very important component of Parareal, as one sees in the convergence analyses. We present here for the first time a Parareal algorithm without coarse propagator, and explain why this can work very well for parabolic problems. We give a new convergence proof for coarse propagators approximating in space, in contrast to the more classical coarse propagators which are approximations in time, and our proof also applies in the absence of the coarse propagator. We illustrate our theoretical results with numerical experiments, and also explain why this approach can not work for hyperbolic problems.
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