Analysis of a Krylov subspace enhanced parareal algorithm for linear problems

Gander, Martin J.; Petcu, Mădălina

doi:10.1051/proc:082508

Cited by 46 publications

(46 citation statements)

References 7 publications

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“…Krylov acceleration methods have already been studied for serial SDC methods for both ODEs and DAEs [35; 36; 12], as well as for the traditional parareal method [28], although the effectiveness of these methods for large scale PDEs has not yet been demonstrated. Another possibility concerns the use of iterative solvers within implicit or semi-implicit temporal methods for PDEs.…”

Section: Discussionmentioning

confidence: 99%

“…The key observation in [53] is that the Ᏺ propagator in traditional parareal approaches makes no use of the previously computed solution on the same interval (a recent alternative approach to reusing information appears in [28]). It is shown how the use of an iterative method can be combined with parareal to improve the solution from the previous parareal iteration rather than computing a solution from scratch.…”

Section: Introductionmentioning

confidence: 99%

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A hybrid parareal spectral deferred corrections method

Minion

2010

CAMCoS

127

129

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The parareal algorithm introduced in 2001 by Lions, Maday, and Turinici is an iterative method for the parallelization of the numerical solution of ordinary differential equations or partial differential equations discretized in the temporal direction. The temporal interval of interest is partitioned into successive domains which are assigned to separate processor units. Each iteration of the parareal algorithm consists of a high accuracy solution procedure performed in parallel on each domain using approximate initial conditions and a serial step which propagates a correction to the initial conditions through the entire time interval. The original method is designed to use classical single-step numerical methods for both of these steps. This paper investigates a variant of the parareal algorithm first outlined by Minion and Williams in 2008 that utilizes a deferred correction strategy within the parareal iterations. Here, the connections between parareal, parallel deferred corrections, and a hybrid parareal-spectral deferred correction method are further explored. The parallel speedup and efficiency of the hybrid methods are analyzed, and numerical results for ODEs and discretized PDEs are presented to demonstrate the performance of the hybrid approach.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A hybrid parareal spectral deferred corrections method

Minion

2010

CAMCoS

127

129

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“…Again, the jumping coefficients affect Parareal's convergence only marginally. Note that interpreting variants like the "Krylov-subspace-enhanced Parareal", introduced in Gander and Petcu [2008] and studied further in , as a non-stationary fixed point iteration could be an interesting approach for a mathematical analysis.…”

Section: Error Bound From Singular Valuesmentioning

confidence: 99%

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

Ruprecht

Speck

Krause

2016

Lecture Notes in Computational Science and Engineering

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“…It is interesting to note that the new parareal algorithms for time-periodic problems do not have any superlinear convergence regime, in sharp contrast to the classical parareal algorithm for initial value problems. 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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Analysis of a Krylov subspace enhanced parareal algorithm for linear problems

Cited by 46 publications

References 7 publications

A hybrid parareal spectral deferred corrections method

A hybrid parareal spectral deferred corrections method

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

Analysis of Two Parareal Algorithms for Time-Periodic Problems

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