Despite the growing interest in parallel-in-time methods as an approach to accelerate numerical simulations in atmospheric modeling, improving their stability and convergence remains substantial challenge for their ap...
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Despite the growing interest in parallel-in-time methods as an approach to accelerate numerical simulations in atmospheric modeling, improving their stability and convergence remains substantial challenge for their application to operational models. In this work, we study temporal parallelization of the shallow water equations on the rotating sphere combined with time-stepping schemes commonly used in atmospheric modeling due to their stability properties, namely an Eulerian implicit-explicit (IMEX) method and a semi-Lagrangian semi-implicit method (SL-SI-SETTLS). The main goal is to investigate the performance of parallel-in-time methods, namely Parareal and Multigrid Reduction in time (MGRIT) when these well-established schemes are used on the coarse discretization levels and provide insights on how they can be improved better performance. We begin by performing an analytical stability study of Parareal and MGRIT applied to a linearized ordinary differential equation depending on some temporal parallelization parameters, including the choice of a coarse scheme. Next, we perform numerical simulations two standard tests in atmospheric modeling to evaluate the stability, convergence, and speedup provided by the parallel-in-time methods compared to a fine reference solution computed serially. We also conduct a detailed investigation on the influence of artificial viscosity and hyperviscosity approaches, applied on the coarse discretization levels, on the performance of the temporal parallelization. Both the analytical stability study and the numerical simulations indicate a poorer stability behavior when SL-SI-SETTLS is used on the coarse levels, compared to the IMEX scheme. With the IMEX scheme, a better trade-off between convergence, stability, and speedup compared to serial simulations can be obtained under proper parameters and artificial viscosity choices, opening the perspective of the potential competitiveness for realistic models.
A long-standing issue in the parallel-in-time community is the poor convergence of standard iterative parallel-in-time methods for hyperbolic partial differential equations (PDEs), and for advection-dominated PDEs mor...
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A long-standing issue in the parallel-in-time community is the poor convergence of standard iterative parallel-in-time methods for hyperbolic partial differential equations (PDEs), and for advection-dominated PDEs more broadly. Here, a local Fourier analysis (LFA) convergence theory is derived for the two-level variant of the iterative parallel-in-time method of multigrid reduction-in-time (MGRIT). This closed-form theory allows for new insights into the poor convergence of MGRIT for advection-dominated PDEs when using the standard approach of rediscretizing the fine-grid problem on the coarse grid. Specifically, we show that this poor convergence arises, at least in part, from inadequate coarse-grid correction of certain smooth Fourier modes known as characteristic components, which was previously identified as causing poor convergence of classical spatial multigrid on steady-state advection-dominated PDEs. We apply this convergence theory to show that, for certain semi-Lagrangian discretizations of advection problems, MGRIT convergence using rediscretized coarse-grid operators cannot be robust with respect to CFL number or coarsening factor. A consequence of this analysis is that techniques developed for improving convergence in the spatial multigrid context can be re-purposed in the MGRIT context to develop more robust parallel-in-time solvers. This strategy has been used in recent work to great effect;here, we provide further theoretical evidence supporting the effectiveness of this approach.
parallel-in-time integration has been the focus of intensive research efforts over the past two decades due to the advent of massively parallel computer architectures and the scaling limits of purely spatial paralleli...
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parallel-in-time integration has been the focus of intensive research efforts over the past two decades due to the advent of massively parallel computer architectures and the scaling limits of purely spatial parallelization. Various iterative parallel-in-time algorithms have been proposed, like PARAREAL, PFASST, MGRIT, and Space-time Multi-Grid (STMG). These methods have been described using different notation, and the convergence estimates that are available are difficult to compare. We describe PARAREAL, PFASST, MGRIT, and STMG for the Dahlquist model problem using a common notation and give precise convergence estimates using generating functions. This allows us, for the first time, to directly compare their convergence. We prove that all four methods eventually converge superlinearly, and we also compare them numerically. The generating function framework provides further opportunities to explore and analyze existing and new methods.
Probabilistic numerical solvers for ordinary differential equations (ODEs) treat the numerical simulation of dynamical systems as problems of Bayesian state estimation. Aside from producing posterior distributions ove...
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Probabilistic numerical solvers for ordinary differential equations (ODEs) treat the numerical simulation of dynamical systems as problems of Bayesian state estimation. Aside from producing posterior distributions over ODE solutions and thereby quantifying the numerical approximation error of the method itself, one less-often noted advantage of this formalism is the algorithmic exibility gained by formulating numerical simulation in the framework of Bayesian filtering and smoothing. In this paper, we leverage this exibility and build on the time-parallel formulation of iterated extended Kalman smoothers to formulate a parallel-in-time probabilistic numerical ODE solver. Instead of simulating the dynamical system sequentially in time, as done by current probabilistic solvers, the proposed method processes all time steps in parallel and thereby reduces the computational complexity from linear to logarithmic in the number of time steps. We demonstrate the effectiveness of our approach on a variety of ODEs and compare it to a range of both classic and probabilistic numerical ODE solvers.
To solve optimization problems with parabolic PDE constraints, often methods working on the reduced objective functional are used. They are computationally expensive due to the necessity of solving both the state equa...
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To solve optimization problems with parabolic PDE constraints, often methods working on the reduced objective functional are used. They are computationally expensive due to the necessity of solving both the state equation and a backward-in-time adjoint equation to evaluate the reduced gradient in each iteration of the optimization method. In this study, we investigate the use of the parallel-in-time method PFASST in the setting of PDE-constrained optimization. In order to develop an efficient fully time-parallel algorithm, we discuss different options for applying PFASST to adjoint gradient computation, including the possibility of doing PFASST iterations on both the state and the adjoint equations simultaneously. We also explore the additional gains in efficiency from reusing information from previous optimization iterations when solving each equation. Numerical results for both a linear and a nonlinear reaction-diffusion optimal control problem demonstrate the parallel speedup and efficiency of different approaches.
In this paper, we investigate the influence of time integration methods on the performance of the Parareal method for the computation of eddy current problems. Parareal is a method that allows parallelization in time ...
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ISBN:
(纸本)9781509010325
In this paper, we investigate the influence of time integration methods on the performance of the Parareal method for the computation of eddy current problems. Parareal is a method that allows parallelization in time domain. The time interval is split into many smaller time intervals (as many as the number of available CPUs) with a fine time grid and solved in parallel allowing to capture the finescale details of the solution. An approximation of initial conditions for these fine problems is obtained by solving a cheap, time-dependent problem defined on a coarse grid for the entire time interval. The method has been successfully implemented for the problem of eddy currents using the implicit backward Euler method. In this paper we will investigate the influence of the time stepping methods on the convergence and the complexity of the Parareal algorithm.
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