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 bo...
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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.
We construct a space-time parallel method for solving parabolic partial differential equations by coupling the parareal algorithm in time with overlapping domain decomposition in space. The goal is to obtain a discret...
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We construct a space-time parallel method for solving parabolic partial differential equations by coupling the parareal algorithm in time with overlapping domain decomposition in space. The goal is to obtain a discretization consisting of "local" problems that can be solved on parallel computers efficiently. However, this introduces significant sources of error that must be evaluated. Reformulating the original parareal algorithm as a variational method and implementing a finite element discretization in space enables an adjoint-based a posteriori error analysis to be performed. Through an appropriate choice of adjoint problems and residuals the error analysis distinguishes between errors arising due to the temporal and spatial discretizations, as well as between the errors arising due to incomplete parareal iterations and incomplete iterations of the domain decomposition solver. We first develop an error analysis for the parareal method applied to parabolic partial differential equations, and then refine this analysis to the case where the associated spatial problems are solved using overlapping domain decomposition. These constitute our time parallel algorithm and space-time parallel algorithm respectively. Numerical experiments demonstrate the accuracy of the estimator for both algorithms and the iterations between distinct components of the error.
The numerical simulation of atherosclerotic plaque growth is computationally prohibitive, since it involves a complex cardiovascular fluid-structure interaction (FSI) problem with a characteristic time scale of millis...
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The numerical simulation of atherosclerotic plaque growth is computationally prohibitive, since it involves a complex cardiovascular fluid-structure interaction (FSI) problem with a characteristic time scale of milliseconds to seconds, as well as a plaque growth process governed by reaction-diffusion equations, which takes place over several months. In this work we combine a temporal homogenization approach, which separates the problem in computationally expensive FSI problems on a micro scale and a reaction-diffusion problem on the macro scale, with parallel time-stepping algorithms. It has been found in the literature that parallel time-stepping algorithms do not perform well when applied directly to the FSI problem. To circumvent this problem, a parareal algorithm is applied on the macro-scale reaction-diffusion problem instead of the micro-scale FSI problem. We investigate modifications in the coarse propagator of the parareal algorithm, in order to further reduce the number of costly micro problems to be solved. The approaches are tested in detailed numerical investigations based on serial simulations.& COPY;2023 Elsevier Inc. All rights reserved.
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 unders...
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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.
We propose a new strategy for solving by the parareal algorithm highly oscillatory ordinary differential equations which are characteristics of a six-dimensional Vlasov equation. For the coarse solvers we use reduced ...
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We propose a new strategy for solving by the parareal algorithm highly oscillatory ordinary differential equations which are characteristics of a six-dimensional Vlasov equation. For the coarse solvers we use reduced models, obtained from the two-scale asymptotic expansions in [4]. Such reduced models have a low computational cost since they are free of high oscillations. The parareal method allows to improve their accuracy in a few iterations through corrections by fine solvers of the full model. We demonstrate the accuracy and the efficiency of the strategy in numerical experiments of short time and long time simulations of charged particles submitted to a large magnetic field. In addition, the convergence of the parareal method is obtained uniformly with respect to the vanishing stiff parameter. (c) 2021 Elsevier Inc. All rights reserved. Superscript/Subscript Available
This paper presents a highly parallelizable parallel-in-time algorithm for efficient solution of nonlinear time-periodic problems. It is based on the time-periodic extension of the parareal method, known to accelerate...
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This paper presents a highly parallelizable parallel-in-time algorithm for efficient solution of nonlinear time-periodic problems. It is based on the time-periodic extension of the parareal method, known to accelerate sequential computations via parallelization on the fine grid. The proposed approach reduces the complexity of the periodic parareal solution by introducing a simplified Newton algorithm, which allows an additional parallelization on the coarse grid. In particular, at each Newton iteration a multiharmonic correction is performed, which converts the block-cyclic periodic system in the time domain into a block-diagonal system in the frequency domain, thereby solving for each frequency component in parallel. The convergence analysis of the method is discussed for a one-dimensional model problem. The introduced algorithm and several existing solution approaches are compared via their application to the eddy current problem for both linear and nonlinear models of a coaxial cable. Performance of the considered methods is also illustrated for a three-dimensional transformer model.
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 t...
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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.
A parareal algorithm based on an exponential theta-scheme is proposed for the stochastic Schrodinger equation with weak damping and additive noise. It proceeds as a two-level temporal parallelizable integrator with th...
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A parareal algorithm based on an exponential theta-scheme is proposed for the stochastic Schrodinger equation with weak damping and additive noise. It proceeds as a two-level temporal parallelizable integrator with the exponential theta-scheme as the integrator on the coarse grid. The proposed algorithm in the linear case increases the convergence order from one to k for theta is an element of [0, 1] \ {1/2}. In particular, the convergence order increases to 2k when theta = 1/2 due to the symmetry of the algorithm. The convergence condition for longtime simulation is also established for the proposed algorithm in the nonlinear case, which indicates the superiority of implicit schemes. Numerical experiments are dedicated to illustrating the convergence order of the algorithm for different choices of theta, as well as the error for different iterated number k.
We report a new parallel iterative algorithm for time-dependent differential equations by combining the known waveform relaxation (WR) technique with the classical parareal algorithm. The parallelism can be simultaneo...
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We report a new parallel iterative algorithm for time-dependent differential equations by combining the known waveform relaxation (WR) technique with the classical parareal algorithm. The parallelism can be simultaneously exploited in both sub-systems by WR and time by parareal. We also provide a sharp estimation on errors for the new algorithm. The iterations of parareal and WR are balanced to optimize the performance of the algorithm. Furthermore, the parallel speedup and efficiency of the new approach are analyzed by comparing with the classical parareal algorithm and the WR technique, respectively. Numerical experiments are carried out to verify the effectiveness of the theoretic work. (C) 2012 IMACS. Published by Elsevier B.V. All rights reserved.
This paper investigates a novel parallel technique based on the spectral deferred correction (SDC) method and a compensation step for solving first-order evolution problems, and we call it para-SDC method for convenie...
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This paper investigates a novel parallel technique based on the spectral deferred correction (SDC) method and a compensation step for solving first-order evolution problems, and we call it para-SDC method for convenience. The standard SDC method is used in parallel with a rough initial guess and a Picard integral equation with high precision initial condition is acted as a compensator. The goal of this paper is to show how these processes can be parallelized and how to improve the efficiency. During the SDC step an implicit or semi-implicit method can be used for stiff problems which is always time-consuming, therefore that's why we do this procedure in parallel. Due to a better initial condition of parallel intervals after the SDC step, the goal of compensation step is to get a better approximation and also avoid of solving an implicit problem again. During the compensation step an explicit Picard scheme is taken based on the numerical integration with polynomial interpolation on Gauss Radau II nodes, which is almost no time consumption, obviously, that's why we do this procedure in serial. The convergency analysis and the parallel efficiency of our method are also discussed. Several numerical experiments and an application for simulation Allen-Cahn equation are presented to show the accuracy, stability, convergence order and efficiency features of para-SDC method.
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