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The study of parareal algorithm for the linear switched systems

APPLIED MATHEMATICS LETTERS 2022年第0期125卷 107763-107763页

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

We study in this paper the parareal (parallel-in-time) algorithm for the linear switched systems (LSS) on the basis of two coarse time subinterval divisions: the parareal algorithm based on the original switching time subinterval of the LSS (PLSS algorithm), and the parareal algorithm based on the new time subinterval division (NPLSS algorithm) for making the computation time of each processor well-balanced. These two proposed algorithms are able to compute the LSS with high-frequency oscillatory and discontinuous input efficiently. Besides, the convergence analysis of the two algorithms is provided. (C) 2021 Elsevier Ltd. All rights reserved.

关键词： Switched system parareal algorithm High-frequency oscillatory Convergence analysis

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Convergence Analysis of the parareal-Euler algorithm for Systems of ODEs with Complex Eigenvalues

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JOURNAL OF SCIENTIFIC COMPUTING 2016年第2期67卷 644-668页

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

parareal is an iterative algorithm and is characterized by two propagators and , which are respectively associated with large step size and small step size , where and is an integer. The choice Backward-Euler denotes the simplest implicit parareal solver, which we call parareal-Euler, and has been studied widely in recent years. For linear problem with being a symmetric positive definite matrix, this algorithm converges very fast and the convergence rate is insensitive to the change of J and . However, for the case that the spectrum of contains complex values, no provable results are available in the literature so far. Previous studies based on numerical plotting show that we can not expect convergence for the parareal-Euler algorithm on the whole right-hand side of the complex plane. Here, we consider a representative situation: with , i.e., the spectrum is contained in a sectorial region. Spectrum distribution of this type arises naturally for semi-discretizing a wide rang of time-dependent PDEs, e.g., the Fokker-Planck equations. We derive condition, which is independent of J and depends on only, to ensure convergence of the parareal-Euler algorithm. Numerical results for initial value and time-periodic problems are provided to support our theoretical conclusions.

关键词： parareal algorithm Backward-Euler Convergence analysis Complex eigenvalues

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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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第十届电子商务、工程及科学领域的分布式计算和应用（DCABES）国际学术研讨会

作者： Thomas Cadeau Frédéric Magoulès Applied Mathematics and Systems Laboratory Ecole Centrale Paris Chtenay-Malabry France

In this paper,a coupling strategy of the parareal algorithm with the Waveform Relaxation method is presented for the parallel solution of differential algebraic *** 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 ***,this coupled method is extended to the solution of differential algebraic *** 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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parareal-Richardson algorithm for Solving Nonlinear ODEs and PDEs

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Communications in Computational Physics 2009年第9期6卷 883-902页

作者： Shulin Wu Baochang Shi Chengming Huang School of Mathematics and Statistics Huazhong University of Science and TechnologyWuhan 430074China

The parareal algorithm,proposed firstly by Lions et al.[***,***,and ***,A”parareal”in time discretization of PDE’s,*** Ser.I Math.,332(2001),pp.661-668],is an effective algorithm to solve the timedependent problems parallel in *** algorithm has received much interest from many researchers in the past *** present in this paper a new variant of the parareal algorithm,which is derived by combining the original parareal algorithm and the Richardson extrapolation,for the numerical solution of the nonlinear ODEs and *** nonlinear problems are tested to show the advantage of the new *** accuracy of the obtained numerical solution is compared with that of its original version(i.e.,the parareal algorithm based on the same numerical method).

关键词： Parallel computation parareal algorithm Richardson extrapolation accuracy nonlinear problems

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OPTIMIZING COARSE PROPAGATORS IN parareal algorithmS

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2025年第2期47卷 A735-A761页

作者： Jin, Bangti Lin, Qingle Zhou, Zhi Chinese Univ Hong Kong Dept Math Shatin Hong Kong Peoples R China Jilin Univ Sch Math Changchun 130012 Peoples R China Hong Kong Polytech Univ Dept Appl Math Kowloon Hong Kong Peoples R China

The parareal algorithm represents an important class of parallel-in-time algorithms for solving evolution equations and has been widely applied in practice. To achieve effective speed-up, the choice of the coarse propagator in the algorithm is vital. In this work, we investigate the use of optimized coarse propagators. Building upon the error estimation framework, we present a systematic procedure for constructing coarse propagators that enjoy desirable stability and consistent order. Additionally, we provide preliminary mathematical guarantees for the resulting parareal algorithm. Numerical experiments on a variety of settings, e.g., the linear diffusion model, the Allen--Cahn model, and the viscous Burgers model, show that the optimizing procedure can significantly improve parallel efficiency when compared with the more ad hoc choice of some conventional and widely used coarse propagators.

关键词： parareal algorithm parabolic problems single step integrator convergence factor

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parareal Neural Networks Emulating a Parallel-in-Time algorithm

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IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS 2024年第5期35卷 6353-6364页

作者： Lee, Youngkyu Park, Jongho Lee, Chang-Ock Korea Adv Inst Sci & Technol Dept Math Sci Daejeon 34141 South Korea Korea Adv Inst Sci & Technol Nat Sci Res Inst Daejeon 34141 South Korea

As deep neural networks (DNNs) become deeper, the training time increases. In this perspective, multi-CPU parallel computing has become a key tool in accelerating the training of DNNs. In this article, we introduce a novel methodology to construct a parallel neural network that can utilize multiple GPUs simultaneously from a given DNN. We observe that layers of DNN can be interpreted as the time steps of a time-dependent problem and can be parallelized by emulating a parallel-in-time algorithm called parareal. The parareal algorithm consists of fine structures which can be implemented in parallel and a coarse structure that gives suitable approximations to the fine structures. By emulating it, the layers of DNN are torn to form a parallel structure, which is connected using a suitable coarse network. We report accelerated and accuracy-preserved results of the proposed methodology applied to VGG-16 and ResNet-1001 on several datasets.

关键词： Deep neural network (DNN) parallel computing parareal algorithm time-dependent problem

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A new parareal waveform relaxation algorithm for time-periodic problems

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INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS 2015年第2期92卷 377-393页

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

We present a new parareal waveform relaxation algorithm for time-periodic problems which performs the parallelism both in sub-systems and in time. The new parareal waveform relaxation algorithm only needs to solve an initial-value coarse problem in each iteration instead of the periodic coarse problem in the classical parareal waveform relaxation algorithm. The convergence result of the new parareal waveform relaxation algorithm is then proved with a linear bound at most on the convergence factor. Numerical experiments including some parallel experiments illustrate our analysis and the effectiveness of the new parareal waveform relaxation algorithm finally.

关键词： parareal algorithm time-periodic problems waveform relaxation convergence analysis parallel computing 65L20 65L70

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A parareal waveform relaxation algorithm for semi-linear parabolic partial differential equations

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2012年第17期236卷 4245-4263页

作者： Liu, Jun Jiang, Yao-Lin Xi An Jiao Tong Univ Dept Math Sci Xian 710049 Shaanxi Peoples R China China Univ Petr Coll Sci Qingdao 266580 Shandong Peoples R China

We report a new parallel iterative algorithm for semi-linear parabolic partial differential equations (PDEs) by combining a kind of waveform relaxation (WR) techniques into the classical parareal algorithm. The parallelism can be simultaneously exploited by WR and parareal in different directions. We provide sharp error estimations for the new algorithm on bounded time domain and on unbounded time domain, respectively. The iterations of the parareal and the WR are balanced to optimize the performance of the algorithm. Furthermore, the speedup and the parallel efficiency of the new approach are analyzed. Numerical experiments are carried out to verify the effectiveness of the theoretic work. (C) 2012 Elsevier B.V. All rights reserved.

关键词： parareal algorithm Waveform relaxation Hybrid parallelism Convergence

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A SUPERLINEAR CONVERGENCE ESTIMATE FOR THE parareal SCHWARZ WAVEFORM RELAXATION algorithm

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2019年第2期41卷 A1148-A1169页

作者： Gander, Martin J. Jiang, Yao-Lin Song, Bo Univ Geneva Sect Math CH-1211 Geneva 4 Switzerland Xi An Jiao Tong Univ Sch Math & Stat Xian 710049 Shaanxi Peoples R China Northwestern Polytech Univ Dept Appl Math Xian 710072 Shaanxi Peoples R China

The parareal Schwarz waveform relaxation algorithm is a new space-time parallel algorithm for the solution of evolution partial differential equations. It is based on a decomposition of the entire space-time domain both in space and in time into smaller space-time subdomains, and then computes by an iteration in parallel on all these small space-time subdomains a better and better approximation of the overall solution in space-time. The initial conditions in the space-time subdomains are updated using a parareal mechanism, while the boundary conditions are updated using Schwarz waveform relaxation techniques. A first precursor of this algorithm was presented 15 years ago, and while the method works well in practice, the convergence of the algorithm is not yet understood, and to analyze it is technically difficult. We present in this paper for the first time an accurate superlinear convergence estimate when the algorithm is applied to the heat equation. We illustrate our analysis with numerical experiments including cases not covered by the analysis, which opens up many further research directions.

关键词： Schwarz waveform relaxation parareal algorithm parareal Schwarz waveform relaxation

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ON parareal algorithmS FOR SEMILINEAR PARABOLIC STOCHASTIC PDEs

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SIAM JOURNAL ON NUMERICAL ANALYSIS 2020年第1期58卷 254-278页

作者： Brehier, Charles-Edouard Wang, Xu Univ Claude Bernard Lyon 1 Univ Lyon CNRS UMR5208Inst Camille Jordan F-69622 Villeurbanne France Purdue Univ Dept Math W Lafayette IN 47907 USA

parareal algorithms are studied for semilinear parabolic stochastic partial differential equations to achieve a "parallel-in-real-time" implementation. These algorithms proceed as two-level integrators, with the fine integrator being given by the exponential Euler scheme in this work. Two choices for the coarse integrator are considered: the linear implicit Euler scheme and the exponential Euler scheme. It is proved that as the number of iterations increases, the order of convergence is limited by the regularity of the noise, whereas for the exponential Euler case, the order of convergence always increases. The influences on the performance of the parareal algorithms, of the choice of the coarse integrator, of the regularity of the noise, and of the number of parareal iterations are also illustrated by extensive numerical experiments.

关键词： parareal algorithm parabolic stochastic partial differential equations strong convergence implicit Euler scheme exponential Euler scheme

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