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Convergence analysis of some second-order parareal algorithms

IMA JOURNAL OF NUMERICAL ANALYSIS 2015年第3期35卷 1315-1341页

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

In the past 10 years, the 'parareal' (parallel-in-time) algorithm has attracted lots of attention thanks to its excellent performance in scientific computing. The parareal algorithm is iterative and is characterized by two propagators G and F which are associated with a coarse step size Delta T and a fine step size Delta t, respectively, where Delta T = J Delta t and J >= 2 is an integer. When we apply this algorithm to large-scale systems of ordinary differential equations obtained by semidiscretizing partial differential equations, two questions arise naturally. (I) Is the error between the iterate and the target solution contractive at each iteration for any choice of the discretization parameters Delta T, J and Delta x? (II) How small can the contraction factor be and can such a contraction factor be independent of the discretization parameters? For linear problems u' = Au + g with symmetric negative-definite matrix A, when the implicit Euler method is used as both the G-and F-propagators, positive answers to these two questions were given by Mathew et al. (2010, SIAM J. Sci. Comput., 32, 1180-1200) and the contraction factor can be bounded by 0.298 for any choice of the discretization parameters. In this paper, for the case that the implicit Euler method is used as the G-propagator, we provide a positive answer to (I) for three second-order F-propagators: the trapezoidal method, the TR/BDF2 method and the two-stage diagonally implicit Runge-Kutta (2s-DIRK) method. For (II), we prove that the contraction factors can be bounded by 0.316 and 1/3 for the 2s-DIRK method and the TR/BDF2 method (provided the parameter gamma involved in TR/BDF2 satisfies gamma is an element of[0.043, 0.977]), respectively, and both bounds are independent of the discretization parameters. For the trapezoidal method, we show that a uniform bound (less than 1) of the contraction factor does not exist. Numerical results are presented to validate the theoretical prediction.

关键词： parareal algorithm implicit Euler method TR/BDF2 method trapezoidal method 2s-DIRK method

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Analysis for parareal algorithms applied to Hamiltonian differential equations

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2014年第parta期259卷 2-13页

作者： Gander, Martin J. Hairer, Ernst Univ Geneva Sect Math CH-1211 Geneva 4 Switzerland

Long-time integrations are an important issue in the numerical solution of Hamiltonian systems. They are time consuming and it is natural to consider the use of parallel architectures for reasons of efficiency. In this context the parareal algorithm has been proposed by several authors. The present work is a theoretical study of the parareal algorithm when it is applied to Hamiltonian differential equations. The idea of backward error analysis is employed to get insight into the long-time behavior of numerical approximations. One of the main results is that convergence of the parareal iterations restricts the length of the time window. For nearly integrable systems its length is bounded by the square root of the inverse of the accuracy of the coarse integrator. The theoretical bounds are confirmed by numerical experiments. (C) 2013 Elsevier B.V. All rights reserved.

关键词： Hamiltonian differential equations parareal algorithm Parallel architectures Symplectic methods Backward error analysis Long-time integration

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ANALYSIS OF TWO parareal algorithmS FOR TIME-PERIODIC PROBLEMS

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2013年第5期35卷 A2393-A2415页

作者： Gander, Martin J. Jiang, Yao-Lin Song, Bo Zhang, Hui Univ Geneva Sect Math CH-1211 Geneva 4 Switzerland Xinjiang Univ Coll Math & Syst Sci Urumqi 830046 Peoples R China Xi An Jiao Tong Univ Sch Math & Stat Xian 710049 Peoples R China

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.

关键词： parareal algorithm time-periodic problems linear convergence multiple shooting

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A fast second-order parareal solver for fractional optimal control problems

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JOURNAL OF VIBRATION AND CONTROL 2018年第15期24卷 3418-3433页

作者： Wu, Shu-Lin Huang, Ting-Zhu Sichuan Univ Sci & Engn Sch Math & Stat Zigong 643000 Sichuan Peoples R China Univ Elect Sci & Technol China Sch Math Sci Chengdu Sichuan Peoples R China

The gradient projection technique has recently been used to solve the optimal control problems governed by a fractional diffusion equation. It lies in repeatedly solving the state and co-state equations derived from the optimality conditions, and the Crank-Nicolson (CN) scheme, which gives a second-order numerical solution, is a widely used method to solve these two equations. The goal of this paper is to implement the CN scheme in a parallel-in-time manner in the framework of the parareal algorithm. Because of the stiffness of the approximation matrix of the fractional operator, direct use of the CN scheme results in a convergence factor satisfying 1 as x0 for the parareal algorithm, where x denotes the space step-size. Here, we provide a new idea to let the parareal algorithm use the CN scheme as the basic component possessing a constant convergence factor approximate to 1/5, which is independent of x. Numerical results are provided to show the efficiency of the proposed algorithm.

关键词： parareal algorithm fractional optimal control convergence analysis parameter optimization numerical computation

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CONVERGENCE ANALYSIS FOR THREE parareal SOLVERS

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2015年第2期37卷 A970-A992页

作者： Wu, Shu-Lin Zhou, Tao Sichuan Univ Sci & Engn Sch Sci Zigong 643000 Sichuan Peoples R China Chinese Acad Sci Inst Computat Math & Sci Engn Comp AMSS Beijing Peoples R China

We analyze in this paper the convergence properties of the parareal algorithm for the symmetric positive definite problem u' + Au = g. The coarse propagator G is fixed to the backward-Euler method and three time integrators are chosen for the F-propagator: the trapezoidal rule, the third-order diagonal implicit Runge-Kutta (RK) (DIRK) method, and the fourth-order Gauss RK method. It is well known that the parareal-Euler algorithm using the backward-Euler method for F and G converges rapidly, but less is known when one uses for F the trapezoidal rule, or the fourth-order Gauss RK method, especially when the mesh ratio J (= Delta T/Delta t) is small. We show that for a specified lambda(max)(the maximal eigenvalue of A or its upper bound), there exists some critical J(cri) such that the parareal solvers derived from these three choices of F converge as fast as parareal-Euler, provided J >= Jcri. Precisely, for F the trapezoidal rule and the fourth-order Gauss RK method, Jcri depends on Delta T, Delta t, and lambda(max) and we present concise formulas to calculate Jcri, while for F the third-order DIRK method, J(cri) = 4, independently of these parameters. Numerical examples with applications in fractional PDEs and uncertainty quantification are presented to support the theoretical predictions.

关键词： parareal algorithm SPD problems time-dependent PDEs trapezoidal rule third-order DIRK method fourth-order Gauss Runge-Kutta method

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Fast parareal iterations for fractional diffusion equations

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JOURNAL OF COMPUTATIONAL PHYSICS 2017年 329卷 210-226页

作者： Wu, Shu-Lin Zhou, Tao Sichuan Univ Sci & Engn Sch Sci Zigong Sichuan Peoples R China Chinese Acad Sci AMSS Inst Computat Math & Sci Engn Comp LSEC Beijing Peoples R China

Numerical methods for fractional PDEs is a hot topic recently. This work is concerned with the parareal algorithm for system of ODEs u'(t) + Au(t) = f that arising from semi-discretizations of time-dependent fractional diffusion equations with nonsymmetric Riemann-Liouville fractional derivatives. The spatial semi-discretization of this kind of fractional derivatives often results in a coefficient matrix A with spectrum sigma(A) satisfying sigma(A)subset of S(eta) := {lambda is an element of C : R(lambda) >=eta, F(lambda) is an element of R}, where eta > 0 is a measure of dissipativity of the differential equations. To accelerate the parareal algorithm, we propose a scaled model u'(t) + 1/alpha Au(t) = f (with alpha > 0) to serve the coarse grid correction, which is an important component of our parareal algorithm. Given eta and alpha, we derive a sharp bound of the convergence factor of the parareal iterations. Moreover, by minimizing such a bound we get optimized scaling factor alpha(opt). It is shown that, compared to alpha =1 (i.e., the classical implementation pattern of the coarse grid correction), the optimized scaling factor significantly improves the convergence rate. Numerical examples are presented to support the theoretical finding. (C) 2016 Elsevier Inc. All rights reserved.

关键词： parareal algorithm Fractional PDEs Convergence analysis Parameter optimization

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Mechanisms for the convergence of time-parallelized, parareal turbulent plasma simulations

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JOURNAL OF COMPUTATIONAL PHYSICS 2012年第23期231卷 7851-7867页

作者： Reynolds-Barredo, J. M. Newman, D. E. Sanchez, R. Samaddar, D. Berry, L. A. Elwasif, W. R. Univ Alaska Dept Phys Fairbanks AK 99775 USA Univ Carlos III Madrid Dept Fis Leganes 28911 Spain ITER Org F-13115 St Paul Les Durance France Oak Ridge Natl Lab Div Fus Energy Oak Ridge TN 37831 USA Oak Ridge Natl Lab Comp Sci & Math Div Oak Ridge TN 37831 USA

parareal is a recent algorithm able to parallelize the time dimension in spite of its sequential nature. It has been applied to several linear and nonlinear problems and, very recently, to a simulation of fully-developed, two-dimensional drift wave turbulence. The mere fact that parareal works in such a turbulent regime is in itself somewhat unexpected, due to the characteristic sensitivity of turbulence to any change in initial conditions. This fundamental property of any turbulent system should render the iterative correction procedure characteristic of the parareal method inoperative, but this seems not to be the case. In addition, the choices that must be made to implement parareal (division of the temporal domain, election of the coarse solver and so on) are currently made using trial-and-error approaches. Here, we identify the mechanisms responsible for the convergence of parareal of these simulations of drift wave turbulence. We also investigate which conditions these mechanisms impose on any successful parareal implementation. The results reported here should be useful to guide future implementations of parareal within the much wider context of fully-developed fluid and plasma turbulent simulations. (C) 2012 Elsevier Inc. All rights reserved.

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

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parareal OPERATOR SPLITTING TECHNIQUES FOR MULTI-SCALE REACTION WAVES: NUMERICAL ANALYSIS AND STRATEGIES

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ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS-MODELISATION MATHEMATIQUE ET ANALYSE NUMERIQUE 2011年第5期45卷 825-852页

作者： Duarte, Max Massot, Marc Descombes, Stephane Ecole Cent Paris CNRS Lab EM2C UPR 288 F-92295 Chatenay Malabry France Univ Nice Sophia Antipolis CNRS Lab JA Dieudonne UMR 6621 F-06108 Nice 02 France

In this paper, we investigate the coupling between operator splitting techniques and a time parallelization scheme, the parareal algorithm, as a numerical strategy for the simulation of reaction-diffusion equations modelling multi-scale reaction waves. This type of problems induces peculiar difficulties and potentially large stiffness which stem from the broad spectrum of temporal scales in the nonlinear chemical source term as well as from the presence of large spatial gradients in the reactive fronts, spatially very localized. In a series of previous studies, the numerical analysis of the operator splitting as well as the parareal algorithm has been conducted and such approaches have shown a great potential in the framework of reaction-diffusion and convection-diffusion-reaction systems. However, complementary studies are needed for a more complete characterization of such techniques for these stiff configurations. Therefore, we conduct in this work a precise numerical analysis that considers the combination of time operator splitting and the parareal algorithm in the context of stiff reaction fronts. The impact of the stiffness featured by these fronts on the convergence of the method is thus quantified, and allows to conclude on an optimal strategy for the resolution of such problems. We finally perform some numerical simulations in the field of nonlinear chemical dynamics that validate the theoretical estimates and examine the performance of such strategies in the context of academical one-dimensional test cases as well as multi-dimensional configurations simulated on parallel architecture.

关键词： parareal algorithm operator splitting convergence analysis reaction-diffusion multi-scale waves

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Parallel in time approximation of the lattice Boltzmann method for laminar flows

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JOURNAL OF COMPUTATIONAL PHYSICS 2014年 270卷 577-586页

作者： Randles, Amanda Kaxiras, Efthimios Harvard Univ Dept Phys Cambridge MA 02138 USA Harvard Univ Sch Engn & Appl Sci Cambridge MA 02138 USA

Fluid dynamics simulations using grid-based methods, such as the lattice Boltzmann equation, can benefit from parallel-in-space computation. However, for a fixed-size simulation of this type, the efficiency of larger processor counts will saturate when the number of grid points per core becomes too small. To overcome this fundamental strong scaling limit in space-parallel approaches, we present a novel time-parallel version of the lattice Boltzmann method using the parareal algorithm. This method is based on a predictor-corrector scheme combined with mesh refinement to enable the simulation of larger number of time steps. We present results of up to a 32x increase in speed for a model system consisting of a cylinder with conditions for laminar flow. The parallel gain obtainable is predicted with strong accuracy, providing a quantitative understanding of the potential impact of this method. (C) 2014 Elsevier Inc. All rights reserved.

关键词： Lattice Boltzmann Fluid dynamics parareal algorithm Parallel in time methods High performance computing

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A fully parallel in time and space algorithm for simulating the electrical activity of a neural tissue

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JOURNAL OF NEUROSCIENCE METHODS 2016年第0期257卷 17-25页

作者： Bedez, Mathieu Belhachmi, Zakaria Haeberle, Olivier Greget, Renaud Moussaoui, Saliha Bouteiller, Jean-Marie Bischoff, Serge Rhenovia Pharma SA Mulhouse France Univ Haute Alsace Lab LMIA EA3993 Mulhouse France Univ Haute Alsace Lab MIPS EA2332 Mulhouse France Firalis SAS Huningue France Univ So Calif Dept Biomed Engn Los Angeles CA 90089 USA

Background: The resolution of a model describing the electrical activity of neural tissue and its propagation within this tissue is highly consuming in term of computing time and requires strong computing power to achieve good results. New method: In this study, we present a method to solve a model describing the electrical propagation in neuronal tissue, using parareal algorithm, coupling with parallelization space using CUDA in graphical processing unit (GPU). Results: We applied the method of resolution to different dimensions of the geometry of our model (1-D, 2-D and 3-D). The GPU results are compared with simulations from a multi-core processor cluster, using message-passing interface (MPI), where the spatial scale was parallelized in order to reach a comparable calculation time than that of the presented method using GPU. A gain of a factor 100 in term of computational time between sequential results and those obtained using the GPU has been obtained, in the case of 3-D geometry. Given the structure of the GPU, this factor increases according to the fineness of the geometry used in the computation. Comparison with existing method(s): To the best of our knowledge, it is the first time such a method is used, even in the case of neuroscience. Conclusion: Parallelization time coupled with GPU parallelization space allows for drastically reducing computational time with a fine resolution of the model describing the propagation of the electrical signal in a neuronal tissue. (C) 2015 Elsevier B.V. All rights reserved.

关键词： GPU Neural activity parareal algorithm MPI Partial differential equations

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