Twelve Ways to Fool the Masses When Giving Parallel-in-Time Results

Götschel, Sebastian; Minion, Michael L.; Ruprecht, Daniel; Speck, Robert

doi:10.1007/978-3-030-75933-9_4

Cited by 6 publications

(4 citation statements)

References 11 publications

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“…Finally, we would like to point out that the baseline used for comparison, i.e., the non-adaptive SDC operator splitting method, can but need not be the fastest algorithm for a given problem, such that the observed speedup has to be interpreted with due care [10].…”

Section: Discussionmentioning

confidence: 99%

Efficient numerical methods for simulating cardiac electrophysiology with cellular resolution

Chegini,

Froehly,

Huynh

et al. 2023

10th Edition of the International Conference on Computational Methods for Coupled Problems in Science and Engineering

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The locality of solution features in cardiac electrophysiology simulations calls for adaptive methods. Due to the overhead incurred by established mesh refinement and coarsening, however, such approaches failed in accelerating the computations. Here we investigate a different route to spatial adaptivity that is based on nested subset selection for algebraic degrees of freedom in spectral deferred correction methods. This combination of algebraic adaptivity and iterative solvers for higher order collocation time stepping realizes a multirate integration with minimal overhead. This leads to moderate but significant speedups in both monodomain and cell-by-cell models of cardiac excitation, as demonstrated at four numerical examples.Keywords cell-by-celll discretization • high-order time integration • spectral deferred correction • algebraic adaptivity • multirate integration scheme

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Section: Discussionmentioning

confidence: 99%

Efficient numerical methods for simulating cardiac electrophysiology with cellular resolution

Chegini,

Froehly,

Huynh

et al. 2023

10th Edition of the International Conference on Computational Methods for Coupled Problems in Science and Engineering

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show abstract

“…The iteration error can contract more or less at the same rate as the amplitude of the solution due to diffusion (and not at all if there is no diffusion). This means that it is easy to run into situations where reported speedups may be largely because of an unfair comparison [16]. Either the truncated modes on the coarse grid do not contribute to the accuracy of the numerical solution and could have been omitted on the fine mesh as well or the Parareal iteration was stopped too early and the parallel solution is less accurate than the fine reference.…”

Section: Discussionmentioning

confidence: 99%

“…, λ n are represented on the fine mesh, they may not actually contribute to the solution provided by Parareal. This makes it critically important to carefully investigate whether (i) Parareal and the fine serial propagator really deliver results of comparable accuracy and (ii) the spatial resolution on the fine mesh is really needed -otherwise, reported speedups might be largely meaningless [16]. This problem is illustrated for a toy example in Subsection 3.5.…”

Section: Norm Of E K Versus Norm Of Ementioning

confidence: 99%

Impact of spatial coarsening on Parareal convergence

Angel¹,

Götschel²,

Ruprecht³

2021

Preprint

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We study the impact of spatial coarsening on the convergence of the Parareal algorithm, both theoretically and numerically. For initial value problems with a normal system matrix, we prove a lower bound for the Euclidean norm of the iteration matrix. When there is no physical or numerical diffusion, an immediate consequence is that the norm of the iteration matrix cannot be smaller than unoty as soon as the coarse problem has fewer degrees-of-freedom than the fine. This prevents a theoretical guarantee for monotonic convergence, which is necessary to obtain meaningful speedups. For diffusive problems, in the worst-case where the iteration error contracts only as fast as the powers of the iteration matrix norm, making Parareal as accurate as the fine method will take about as many iterations as there are processors, making meaningful speedup impossible. Numerical examples with a non-normal system matrix show that for diffusive problems good speedup is possible, but that for non-diffusive problems the negative impact of spatial coarsening on convergence is big.

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“…However, no systematic comparison of convergence behaviour let alone efficiencies between these methods exists. There are at least three obstacles to comparing these four methods: first, there is no common formalism or notation to describe them; second, the existing analyses use very different techniques to obtain convergence bounds; third, the algorithms can be applied to many different problems in different ways with many tunable parameters, all of which affect performance [20]. Our main contribution is to address, at least for the Dahlquist test problem, the first two problems by proposing a common formalism to rigorously describe Parareal, PFASST, MGRIT and STMG using the same notation.…”

mentioning

confidence: 99%

A unified analysis framework for iterative parallel-in-time algorithms

Gander¹,

Lunet²,

Ruprecht³

et al. 2022

Preprint

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Parallel-in-time integration has been the focus of intensive research efforts over the past two decades due to the advent of massively parallel computer architectures and the scaling limits of purely spatial parallelization. Various iterative parallel-in-time (PinT) algorithms have been proposed, like Parareal, PFASST, MGRIT, and Space-Time Multi-Grid (STMG). These methods have been described using different notations and the convergence estimates that are available for some of them are difficult to compare. We describe Parareal, PFASST, MGRIT and STMG for the Dahlquist model problem using a common notation and give precise convergence estimates using generating functions. This allows us, for the first time, to directly compare their convergence. We prove that all four methods eventually converge super-linearly and compare them directly numerically. Our framework also allows us to find new methods.

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Twelve Ways to Fool the Masses When Giving Parallel-in-Time Results

Cited by 6 publications

References 11 publications

Efficient numerical methods for simulating cardiac electrophysiology with cellular resolution

Efficient numerical methods for simulating cardiac electrophysiology with cellular resolution

Impact of spatial coarsening on Parareal convergence

A unified analysis framework for iterative parallel-in-time algorithms

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