Parareal for Diffusion Problems with Space- and Time-Dependent Coefficients

Ruprecht, Daniel; Speck, Robert; Krause, Rolf

doi:10.1007/978-3-319-18827-0_37

Cited by 4 publications

(5 citation statements)

References 12 publications

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“…This matches the analysis for constant-coefficient diffusive problems in [23]. The marginal affect of a time-dependent diffusion coefficient is in line with the study of Parareal's convergence for space-and time-dependent diffusion coefficients in [45]. See also the comments in 2.2.1.…”

Section: Resultssupporting

confidence: 85%

A stencil-based implementation of Parareal in the C++ domain specific embedded language STELLA

Arteaga

Ruprecht

Krause

2015

Applied Mathematics and Computation

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In view of the rapid rise of the number of cores in modern supercomputers, time-parallel methods that introduce concurrency along the temporal axis are becoming increasingly popular. For the solution of time-dependent partial differential equations, these methods can add another direction for concurrency on top of spatial parallelization. The paper presents an implementation of the time-parallel Parareal method in a C++ domain specific language for stencil computations (STELLA). STELLA provides both an OpenMP and a CUDA backend for a shared memory parallelization, using the CPU or GPU inside a node for the spatial stencils. Here, we intertwine this node-wise spatial parallelism with the time-parallel Parareal. This is done by adding an MPI-based implementation of Parareal, which allows us to parallelize in time across nodes. The performance of Parareal with both backends is analyzed in terms of speedup, parallel efficiency and energy-to-solution for an advection-diffusion problem with a time-dependent diffusion coefficient.

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Section: Resultssupporting

confidence: 85%

A stencil-based implementation of Parareal in the C++ domain specific embedded language STELLA

Arteaga

Ruprecht

Krause

2015

Applied Mathematics and Computation

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“…To assess the impact these jumps have on the convergence of Parareal, Figure 5 gives a comparison of the defect for the brick and mortar problem (red) and a reference configuration with D lip = D cor = 10 −3 throughout the whole domain. For the setup studied here, in line with the findings for 2D problems in [31], the jump in coefficients has almost no effect on how Parareal convergence. Experiments not documented here suggest that a larger T (that is, a final configuration closer to the steady state) can lead to a larger detrimental effect of coefficient jumps: However, even there this only resulted in a small number of additional iterations required for convergence.…”

Section: Effect Of Spatially Varying Coefficientssupporting

confidence: 87%

“…problem, it already has jumps in the diffusion coefficients of several orders of magnitude on a highly anisotropic domain. The article is an extension of a previous study of a 2D problem on a domain with a much simpler structure [31].…”

Section: Discussionmentioning

confidence: 99%

“…Theory for diffusive problems with constant coefficients can also be found in [6]. For 2D diffusion with space-time dependent coefficients, numerical experiments showed only a marginal reduction in convergence speed [31]. In [4], the small impact of a time dependent viscosity on Parareal's convergence for an advection-diffusion problem is demonstrated.…”

Section: Introductionmentioning

confidence: 98%

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Numerical simulation of skin transport using Parareal

Kreienbuehl

Naegel

Ruprecht

et al. 2015

Comput. Visual Sci.

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In-silico investigation of skin permeation is an important but also computationally demanding problem. To resolve all scales involved in full detail will not only require exascale computing capacities but also suitable parallel algorithms. This article investigates the applicability of the time-parallel Parareal algorithm to a brick and mortar setup, a precursory problem to skin permeation. The C++ library Lib4PrM implementing Parareal is combined with the UG4 simulation framework, which provides the spatial discretization and parallelization. The combination's performance is studied with respect to convergence and speedup. It is confirmed that anisotropies in the domain and jumps in diffusion coefficients only have a minor impact on Parareal's convergence. The influence of load imbalances in time due to differences in number of iterations required by

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“…The solution computed by provides a correction for ℱ and therefore the overall algorithm converges. The parareal algorithm has received a lot of attention over the past few years (Bal 2005;Dai and Maday 2013;Gander and Vandewalle 2007;Wu and Zhou 2015), and it has been successfully applied to a variety of problems, such as parabolic problems (Gander and Vandewalle 2007;Ruprecht et al 2016), hyperbolic problems (Dai and Maday 2013), optimal control problems (Du et al 2013;Maday et al 2007;Mathew et al 2010), Navier-Stokes equations (Trindade and Pereira 2006) and non-differential equations (Bal and Maday 2002). The parallel efficiency of the parareal algorithm is far from ideal, it is proportional to 1 / k, here k denotes the number of iterations (Bal 2005;Farhat and Chandesris 2003).…”

Section: Introductionmentioning

confidence: 99%

Highly parallel space-time domain decomposition methods for parabolic problems

Shao

Cai

2019

CCF Trans. HPC

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In the past few years, the number of processor cores of top ranked supercomputers has increased drastically. It is challenging to design efficient parallel algorithms that offer such a high degree of parallelism, especially for certain time-dependent problems because of the sequential nature of "time". To increase the degree of parallelization, some parallel-in-time algorithms have been developed. In this paper, we give an overview of some recently introduced parallel-in-time methods, and present in detail the class of space-time Schwarz methods, including the standard and the restricted versions, for solving parabolic partial differentialequations. Some numerical experiments carried out on a parallel computer with a large number of processor cores for three-dimensional problems are given to show the parallel scalability of the methods. In the end of the paper, we provide a comparison of the parallel-in-time algorithms with a traditional algorithm that is parallelized only in space.

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Parareal for Diffusion Problems with Space- and Time-Dependent Coefficients

Cited by 4 publications

References 12 publications

A stencil-based implementation of Parareal in the C++ domain specific embedded language STELLA

A stencil-based implementation of Parareal in the C++ domain specific embedded language STELLA

Numerical simulation of skin transport using Parareal

Highly parallel space-time domain decomposition methods for parabolic problems

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