Convergence of Parareal with spatial coarsening

Ruprecht, Daniel

doi:10.1002/pamm.201410490

Cited by 21 publications

(23 citation statements)

References 10 publications

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“…Theorem 5.2. Suppose that the fine and corrected coarse operators satisfy Hypothesis (29) and (30). Then,…”

Section: Hypothesis 51 (I) the Phase Corrected Coarse Solution Is Lmentioning

confidence: 99%

“…Indeed, this coarsening technique provides additional speed up in some applications [23,3,21] because the coarse propagator has less grid points to compute, provided an appropriate grid restriction and interpolation operator. However as shown in [29], considerable coarse grid resolution and accurate interpolation are required in order to make the parareal iteration (2) converged.…”

Section: Introductionmentioning

confidence: 99%

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A stable parareal-like method for the second order wave equation

Nguyen

Tsai

2020

Journal of Computational Physics

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A new parallel-in-time iterative method is proposed for solving the homogeneous second-order wave equation. The new method involves a coarse scale propagator, allowing for larger time steps, and a fine scale propagator which fully resolves the medium using finer spatial grid and shorter time steps. The fine scale propagator is run in parallel for short time intervals. The two propagators are coupled in an iterative way that resembles the standard parareal method [22]. We present a data-driven strategy in which the computed data gathered from each iteration are re-used to stabilize the coupling by minimizing the energy residual of the fine and coarse propagated solutions. An example of Marmousi model is provided to demonstrate the performance of the proposed method.

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“…Theorem 5.2. Suppose that the fine and corrected coarse operators satisfy Hypothesis (29) and (30). Then,…”

Section: Hypothesis 51 (I) the Phase Corrected Coarse Solution Is Lmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A stable parareal-like method for the second order wave equation

Nguyen

Tsai

2020

Journal of Computational Physics

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“…On the other hand, building a coarse solver with explicit time integration cannot be based on an increase of the time step, because of their intrinsic CFL limitation. One can use then coarsening in space, but previous analyses in the lit-erature [29,33] show that one needs necessarily high order interpolation between the coarse and fine grids, and the coarse solver could then also suffer major efficiency loss in the context of massive space parallelization, not even talking about further high frequency error components introduced due to this process, which the iteration can not eliminate. This makes it very tricky for actual PinT methods based on multilevel techniques, and it seems unavoidable that one has to develop new techniques for the coarser levels when applying Parareal like methods to the advection problem, or more generally, hyperbolic problems.…”

Section: Application To Pint Methodsmentioning

confidence: 99%

Toward error estimates for general space-time discretizations of the advection equation

Gander

Lunet

2020

Comput. Visual Sci.

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We develop new error estimates for the one-dimensional advection equation, considering general space-time discretization schemes based on Runge–Kutta methods and finite difference discretizations. We then derive conditions on the number of points per wavelength for a given error tolerance from these new estimates. Our analysis also shows the existence of synergistic space-time discretization methods that permit to gain one order of accuracy at a given CFL number. Our new error estimates can be used to analyze the choice of space-time discretizations considered when testing Parallel-in-Time methods.

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“…As noted in [34], the convergence of Parareal with spatial coarsening will not only depend on the fine and coarse time propagators but also on the restriction and interpolation operators. Hence, we expect (5) to obtain a different convergence rate than (3).…”

Section: Variant Of Parareal Based On Explicit Time Integrators and Omentioning

confidence: 96%

Time-parallel simulation of the decay of homogeneous turbulence using Parareal with spatial coarsening

Lunet

Bodart

Gratton³

et al. 2018

Comput. Visual Sci.

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Direct Numerical Simulation of turbulent flows is a computationally demanding problem that requires efficient parallel algorithms. We investigate the applicability of the time-parallel Parareal algorithm to an instructional case study related to the simulation of the decay of homogeneous isotropic turbulence in three dimensions. We combine a Parareal variant based on explicit time integrators and spatial coarsening with the space-parallel Hybrid Navier-Stokes solver. We analyse the performance of this space-time parallel solver with respect to speedup and quality of the solution. The results are compared with reference data obtained with a classical explicit integration, using an error analysis which relies on the energetic content of the solution. We show that a single Parareal iteration is able to reproduce with high fidelity the main statistical quantities characterizing the turbulent flow field. Keywords Direct Numerical Simulation • Explicit time integrator • High-order method in space • Navier-Stokes equations • Parareal • Space-time parallelism This research is partly funded by "Région Occitanie/ Pyrénées-Méditerranée" as part of the "Développement de nouvelles stratégies pour le calcul massivement parallèleà l'échelle exa en mécanique des fluides numérique (DENSE)" project.

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Convergence of Parareal with spatial coarsening

Cited by 21 publications

References 10 publications

A stable parareal-like method for the second order wave equation

A stable parareal-like method for the second order wave equation

Toward error estimates for general space-time discretizations of the advection equation

Time-parallel simulation of the decay of homogeneous turbulence using Parareal with spatial coarsening

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