PARAEXP: A Parallel Integrator for Linear Initial-Value Problems

Gander, Martin J.; Güttel, Stefan

doi:10.1137/110856137

Cited by 89 publications

(113 citation statements)

References 45 publications

(46 reference statements)

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“…It would be interesting to investigate if then acceleration methods could be used, such as Krylov subspace methods, for which interior variants were investigated in [5,12]. It would also be interesting to develop the more recent PARAEXP [7,8] algorithm for the time parallel solution of linear time-periodic problems.…”

Section: Discussionmentioning

confidence: 99%

Analysis of Two Parareal Algorithms for Time-Periodic Problems

Gander¹,

Jiang²,

Song³

et al. 2013

SIAM J. Sci. Comput.

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Abstract. The parareal algorithm, which permits us to solve evolution problems in a time parallel fashion, has created a lot of attention over the past decade. The algorithm has its roots in the multiple shooting method for boundary value problems, which in the parareal algorithm is applied to initial value problems, with a particular coarse approximation of the Jacobian matrix. It is therefore of interest to formulate parareal-type algorithms for time-periodic problems, which also couple the end of the time interval with the beginning, and to analyze their performance in this context. We present and analyze two parareal algorithms for time-periodic problems: one with a periodic coarse problem and one with a nonperiodic coarse problem. An interesting advantage of the algorithm with the nonperiodic coarse problem is that no time-periodic problems need to be solved during the iteration, since on the time subdomains, the problems are not time-periodic either. We prove for both linear and nonlinear problems convergence of the new algorithms, with linear bounds on the convergence. We also extend these results to evolution partial differential equations using Fourier techniques. We illustrate our analysis with numerical experiments, both for model problems and the realistic application of a nonlinear cooled reverse-flow reactor system of partial differential equations.

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Section: Discussionmentioning

confidence: 99%

Analysis of Two Parareal Algorithms for Time-Periodic Problems

Gander¹,

Jiang²,

Song³

et al. 2013

SIAM J. Sci. Comput.

Self Cite

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“…Observe that the source is only nonzero on the subinterval [T j−1 , T j ). We follow the ideas of the Paraexp method [21] and note that the nonhomogeneous part of the ODE requires most of the computational work in the EBK method. An accurate SVD approximation of the source term (2.4) generally requires more singular values to be retained, increasing the dimensions of the block Krylov subspace (2.7).…”

Section: Parallelization Of Linear Problemsmentioning

confidence: 99%

“…2.2. In principle, this algorithm is identical to the Paraexp method [21]. The only practical difference is that in our implementation both the nonhomogeneous and the homogeneous part of the subproblems are solved by the EBK method.…”

Section: Parallelization Of Linear Problemsmentioning

confidence: 99%

“…We have verified that the SVD approximation is sufficiently accurate, i.e., the approximation error, measured in the L 2 -norm, is less than the tolerance of the EBK method. We compare the parallel efficiency of the Paraexp-EBK implementation with a convential implementation of the Paraexp algorithm [21], where the nonhomogeneous problem is integrated with the Crank-Nicolson (CN) scheme. The linear system is solved directly using MATLAB.…”

Section: Parallel Efficiencymentioning

confidence: 99%

“…Also, we mention parallel methods that facilitate parallelism by replacing the exact solves in implicit schemes by approximate ones [7], and parallel methods based on operator splitting [4]. Finally, there is the Paraexp method [21] for parallel integration of linear initial-value problems. This algorithm is based on the fact that linear initial-value problems can be decomposed into homogeneous and nonhomogeneous subproblems.…”

mentioning

confidence: 99%

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A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations

Kooij

Botchev

Geurts

2017

Journal of Computational and Applied Mathematics

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Abstract. A parallel time integration method for nonlinear partial differential equations is proposed. It is based on a new implementation of the Paraexp method for linear partial differential equations (PDEs) employing a block Krylov subspace method. For nonlinear PDEs the algorithm is based on our Paraexp implementation within a waveform relaxation. The initial value problem is solved iteratively on a complete time interval. Nonlinear terms are treated as a source term, provided by the solution from the previous iteration. At each iteration, the problem is decoupled into independent subproblems by the principle of superposition. The decoupled subproblems are solved fast by exponential integration, based on a block Krylov method. The new time integration is demonstrated for the one-dimensional advection-diffusion equation and the viscous Burgers equation. Numerical experiments confirm excellent parallel scaling for the linear advection-diffusion problem, and good scaling in case the nonlinear Burgers equation is simulated.Key words. parallel computing, exponential integrators, partial differential equations, parallel in time, block Krylov subspace.AMS subject classifications. 65F60, 65L05, 65Y051. Introduction. Recent developments in hardware architecture, bringing to practice computers with hundreds of thousands of cores, urge the creation of new, as well as a major revision of existing numerical algorithms [14]. To be efficient on such massively parallel platforms, the algorithms need to employ all possible means to parallelize the computations. When solving partial differential equations (PDEs) with a time-dependent solution, an important way to parallelize the computations is, next to the parallelization across space, parallelization across time. This adds a new dimension of parallelism with which the simulations can be implemented. In this paper we present a new time-parallel integration method extending the Paraexp method [21] to nonlinear partial differential equations using Krylov methods and waveform relaxation.Several approaches to parallelize the simulation of time-dependent solutions in time can be distinguished. The first important class of the methods are the waveform relaxation methods [6,38,44,56], including the space-time multigrid methods for parabolic PDEs [8,24,30,37]. The key idea is to start with an approximation to the numerical solution for the whole time interval of interest and update the solution, solving an easier-to-solve approximate system in time. The Parareal method [36], which attracted significant attention recently, is a prime example related to the class of waveform relaxation methods [19].Parallel Runge-Kutta methods and general linear methods, where the parallelism is determined and restricted by the number of stages or steps, form another class of the time-parallel methods [8,12,52]. Time-parallel schemes can also be obtained

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ParaExp Using Leapfrog as Integrator for High‐Frequency Electromagnetic Simulations

2017

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Recently, ParaExp was proposed for the time integration of linear hyperbolic problems. It splits the time interval of interest into subintervals and computes the solution on each subinterval in parallel. The overall solution is decomposed into a particular solution defined on each subinterval with zero initial conditions and a homogeneous solution propagated by the matrix exponential applied to the initial conditions. The efficiency of the method depends on fast approximations of this matrix exponential based on recent results from numerical linear algebra. This paper deals with the application of ParaExp in combination with Leapfrog to electromagnetic wave problems in time domain. Numerical tests are carried out for a simple toy problem and a realistic spiral inductor model discretized by the Finite Integration Technique.

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PARAEXP: A Parallel Integrator for Linear Initial-Value Problems

Cited by 89 publications

References 45 publications

Analysis of Two Parareal Algorithms for Time-Periodic Problems

Analysis of Two Parareal Algorithms for Time-Periodic Problems

A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations

ParaExp Using Leapfrog as Integrator for High‐Frequency Electromagnetic Simulations

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