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TOWARD PARALLEL COARSE GRID CORRECTION FOR THE parareal algorithm

SIAM JOURNAL ON SCIENTIFIC COMPUTING 2018年第3期40卷 A1446-A1472页

作者： Wu, Shu-Lin Sichuan Univ Sci & Engn Sch Sci Zigong 643000 Sichuan Peoples R China

In this paper, we present an idea toward parallel coarse grid correction (CGC) for the parareal algorithm. It is well known that such a CGC procedure is often the bottleneck of speedup of the parareal algorithm. For an ODE system with initial-value condition u(0) = u(0) the idea can be explained as follows. First, we apply the G-propagator to the same ODE system but with a special condition u (0) = alpha u(T), where alpha is an element of R is a crux parameter. Second, in each iteration of the parareal algorithm the CGC procedure will be carried out by the so-called diagonalization technique established recently. The parameter alpha controls both the roundoff error arising from such a diagonalization technique and the convergence rate of the resulting parareal algorithm. We show that there exists some threshold alpha* such that the parareal algorithm with diagonalization-based CGC possesses the same convergence rate as that of the parareal algorithm with classical CGC if vertical bar alpha vertical bar <= alpha*. With vertical bar alpha vertical bar = alpha*, we show that the condition number associated with the diagonalization technique is a moderate quantity of order O (1) (and therefore the roundoff error is small) and is independent of the length of the time interval. Numerical results are given to support our fi ndings.

关键词： parareal algorithm coarse grid correction diagonalization technique convergence analysis

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A parareal algorithm for a highly oscillating Vlasov-Poisson system with reduced models for the coarse solving

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COMPUTERS & MATHEMATICS WITH APPLICATIONS 2023年 130卷 137-148页

作者： Grigori, Laura Hirstoaga, Sever A. Salomon, Julien Sorbonne Univ Project Team ALPINES Inria F-75589 Paris 12 France Univ Paris CNRS Lab Jacques Louis LJLL L F-75589 Paris 12 France Sorbonne Univ Project Team ANGE Inria F-75589 Paris 12 France

In this paper, we introduce a new strategy for solving highly oscillatory two-dimensional Vlasov-Poisson systems by means of a specific version of the parareal algorithm. The novelty consists in using reduced models, obtained from the two-scale convergence theory, for the coarse solving. The reduced models are useful to approximate the original Vlasov-Poisson model at a low computational cost since they are free of high oscillations. Both models are numerically solved in a particle-in-cell framework. We illustrate this strategy with numerical experiments based on long time simulations of a charged beam in a focusing channel and under the influence of a rapidly oscillating external electric field. On the basis of computing times, we provide an analysis of the efficiency of the parareal algorithm in terms of speedup.

关键词： Vlasov-Poisson system Two-scale convergence Multi-scale models Particle-in-cell method parareal algorithm parareal speedup

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An efficient parareal algorithm for a class of time-dependent problems with fractional Laplacian

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APPLIED MATHEMATICS AND COMPUTATION 2017年 307卷 329-341页

作者： Wu, Shulin Sichuan Univ Sci & Engn Sch Sci Zigong Sichuan Provinc Peoples R China

Time-dependent diffusion equations with fractional Laplacian have received considerable attention in recent years, for which numerical methods play an important role because a simple and analytic solution is often unavailable. We analyze in this paper a parareal algorithm for this kind of problem, which realizes parallel-in-time computation. The algorithm is iterative and uses the 3rd-order SDIRK (singly diagonally implicit Runge-Kutta) method with a small step-size Delta t as the F-propagator and the implicit-explicit Euler method with a large step-size Delta T as the G-propagator. The two step-sizes satisfy Delta T/Delta t = J with J >= 2 being an integer. Using the implicit-explicit Euler method as the G-propagator potentially improves the parallel efficiency, but complicates the convergence analysis. By employing some technical analysis, we provide a sharp estimate of the convergence rate, which is independent of the mesh ratio J and the distribution of the eigenvalues of the coefficient matrix. An extension of the results to problems with time-periodic conditions is also given. Several numerical experiments are carried out to verify the theoretical results. (C) 2017 Elsevier Inc. All rights reserved.

关键词： parareal algorithm Fractional Laplacian Implicit-explicit Euler method 3rd-order SDIRK method Convergence analysis

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PARAOPT: A parareal algorithm FOR OPTIMALITY SYSTEMS

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2020年第5期42卷 A2773-A2802页

作者： Gander, Martin J. Kwok, Felix Salomon, Julien Univ Geneva Sect Math CH-1211 Geneva 4 Switzerland Univ Laval Dept Math & Stat Quebec City PQ G1V 0A6 Canada Hong Kong Baptist Univ Dept Math Kowloon Tong Hong Kong Peoples R China INRIA Paris ANGE Project Team F-75589 Paris 12 France Sorbonne Univ Lab Jacques Louis Lions CNRS F-75005 Paris France

The time parallel solution of optimality systems arising in PDE constrained optimization could be achieved by simply applying any time parallel algorithm, such as parareal, to solve the forward and backward evolution problems arising in the optimization loop. We propose here a different strategy by devising directly a new time parallel algorithm, which we call ParaOpt, for the coupled forward and backward nonlinear partial differential equations. ParaOpt is inspired by the parareal algorithm for evolution equations and thus is automatically a two-level method. We provide a detailed convergence analysis for the case of linear parabolic PDE constraints. We illustrate the performance of ParaOpt with numerical experiments for both linear and nonlinear optimality systems.

关键词： parareal algorithm optimal control preconditioning

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An analytic model for the convergence of turbulent simulations time-parallelized via the parareal algorithm

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JOURNAL OF COMPUTATIONAL PHYSICS 2013年 255卷 293-315页

作者： Reynolds-Barredo, J. M. Newman, D. E. Sanchez, R. Univ Alaska Fairbanks Dept Phys Fairbanks AK 99775 USA Univ Carlos III Madrid Dept Fis Leganes 28911 Spain

parareal is a recent time parallelization algorithm based on a predictor-corrector mechanism. Recently, it has been applied for the first time to a fully-developed plasma turbulent simulation, and a qualitative understanding of how parareal converges exists for this case. In this paper, we construct an analytical framework of the process of convergence that should be applicable to parareal simulations of general turbulent systems. This framework allows one to gain a quantitative understanding of the dependence of the convergence on the physics of the problem and the choices that must be made to implement parareal. The analytical knowledge provided by this new framework can be used to optimize the implementation of parareal. We illustrate the inner workings of the framework and demonstrate its predictive capabilities by applying it to the modeling of the parareal convergence of drift-wave plasma turbulent simulations. (c) 2013 Elsevier Inc. All rights reserved.

关键词： Time parallelization parareal algorithm Turbulent transport Plasma turbulence Magnetically confined plasmas

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CONVERGENCE ANALYSIS OF THE parareal algorithm WITH NONUNIFORM FINE TIME GRID

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SIAM JOURNAL ON NUMERICAL ANALYSIS 2024年第5期62卷 2308-2330页

作者： Wu, Shu-lin Zhou, Tao Northeast Normal Univ Sch Sch Math & Stat Changchun 130024 Peoples R China Chinese Acad Sci Acad Math & Syst Sci Inst Computat Math & Sci Engn Comp LSEC Beijing 100190 Peoples R China

In this paper, we study the convergence properties of the parareal algorithm with uniform coarse time grid and arbitrarily distributed (nonuniform) fine time grid, which may be changed at each iteration. We employ the backward-Euler method as the coarse propagator and a general single-step time-integrator as the fine propagator. Specifically, we consider two implementations of the coarse grid correction: the standard time-stepping mode and the parallel mode via the so-called diagonalization technique. For both cases, we prove that under certain conditions of the stability function of the fine propagator, the convergence factor of the parareal algorithm is not larger than that of the associated algorithm with a uniform fine time grid. Furthermore, we show that when such conditions are not satisfied, one can indeed observe degenerations of the convergence rate. The model that is used for performing the analysis is the Dahlquist test equation with nonnegative parameter, and the numerical results indicate that the theoretical results hold for nonlinear ODEs and linear ODEs where the coefficient matrix has complex eigenvalues.

关键词： parareal algorithm adaptivity diagonalization technique convergence analysis

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A POSTERIORI ERROR ANALYSIS OF TWO-STAGE COMPUTATION METHODS WITH APPLICATION TO EFFICIENT DISCRETIZATION AND THE parareal algorithm

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SIAM JOURNAL ON NUMERICAL ANALYSIS 2016年第5期54卷 2974-3002页

作者： Chaudhry, Jehanzeb Hameed Estep, Don Tavener, Simon Carey, Varis Sandelin, Jeff Univ New Mexico Dept Math & Stat Albuquerque NM 87131 USA Colorado State Univ Dept Stat Ft Collins CO 80523 USA Colorado State Univ Dept Math Ft Collins CO 80523 USA Univ Colorado Denver Dept Math & Stat Sci Denver CO 80204 USA DigitalGlobe Inc PD&L Longmont CO 80503 USA

We consider numerical methods for initial value problems that employ a two-stage approach consisting of solution on a relatively coarse discretization followed by solution on a relatively fine discretization. Examples include adaptive error control, parallel-in-time solution schemes, and efficient solution of adjoint problems for computing a posteriori error estimates. We describe a general formulation of two-stage computations and then perform a general a posteriori error analysis based on computable residuals and solution of an adjoint problem. The analysis accommodates various variations in the two-stage computation and in the formulation of the adjoint problems. We apply the analysis to computing "dual-weighted" a posteriori error estimates, developing novel algorithms for efficient solution that take into account cancellation of error, and to the parareal algorithm. We test the various results using several numerical examples.

关键词： a posteriori error estimation adjoint problem cancellation of error efficient time-step selection initial value problem parallel-in-time parareal algorithm two-stage computation

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Analysis of a New Krylov subspace enhanced parareal algorithm for time-periodic problems

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NUMERICAL algorithmS 2024年第1期97卷 289-310页

作者： Song, Bo Wang, Jing-Yi Jiang, Yao-Lin Northwestern Polytech Univ Sch Math & Stat 1 Dongxiang Rd Xian 710129 Shaanxi Peoples R China Xi An Jiao Tong Univ Sch Math & Stat 28 West Xianning Rd Xian 710049 Shaanxi Peoples R China

The classical parareal algorithm for time-periodic problems, solving a periodic-like coarse problem, called the periodic parareal algorithm with periodic coarse problem (PP-PC), usually converges slowly. In this paper, we present a new parallel-in-time algorithm for time-periodic problems based on the classical PP-PC algorithm and the Krylov subspace method. In this new algorithm, a new propagator derived by the Krylov subspace is chosen as the coarse propagator instead of the classical coarse propagator in the PP-PC algorithm. And because of the special characteristic of time-periodic problems, the Krylov subspace enhanced PP-PC algorithm needs to solve a periodic coarse problem on the coarse time grid on each iteration. We provide two different kinds of theoretical bounds under different assumptions for the proposed algorithm. Numerical results illustrate our analysis with two effective theoretical bounds for the heat equation, the wave equation, and the viscous Burgers equation, where we could also find that the new proposed algorithm converges faster than the classical PP-PC algorithm.

关键词： parareal algorithm Krylov subspace Time-periodic problems Convergence analysis

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Analysis of a new parareal algorithm based on waveform relaxation method for time-periodic problems

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NUMERICAL algorithmS 2014年第3期67卷 599-622页

作者： Song, Bo Jiang, Yao-Lin Xi An Jiao Tong Univ Sch Math & Stat Xian 710049 Shaanxi Peoples R China

We present a new parallel algorithm for time-periodic problems by combining the waveform relaxation method and the parareal algorithm, which performs the parallelism both in sub-systems and in time. In the new algorithm, the waveform relaxation propagator is chosen as a new fine propagator instead of the classical fine propagator. And because of the characteristic of time-periodic problems, the new parareal waveform relaxation algorithm needs to solve a periodic coarse problem at the coarse level in each iteration. The new algorithm is proved to converge linearly at most. Then the theoretic parallel efficiency of the new algorithm is also considered. Numerical experiments confirm our analysis finally.

关键词： parareal algorithm Waveform relaxation Time-periodic problems Convergence analysis

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Coupling the parareal algorithm with the Waveform Relaxation method for the solution of Differential Algebraic Equations

Coupling the Parareal algorithm with the Waveform Relaxation...

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10th International Symposium on Distributed Computing and Applications to Business, Engineering and Science (DCABES)

作者： Cadeau, Thomas Magoules, Frederic Ecole Cent Paris Appl Math & Syst Lab Chatenay Malabry France

ISBN: (纸本)9780769544151

In this paper, a coupling strategy of the parareal algorithm with the Waveform Relaxation method is presented for the parallel solution of differential algebraic equations. The classical Waveform Relaxation (in space) method and the parareal (in time) method are first recalled, followed by the introduction of a coupled parareal-Waveform Relaxation method recently introduced for the solution of partial differential equations. Here, this coupled method is extended to the solution of differential algebraic equations. Numerical experiments, performed on parallel multicores architectures, illustrate the impressive performances of this new method.

关键词： Differential Algebraic Equations Domain decomposition method Waveform Relaxation method parareal algorithm Parallel and distributed computing

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