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Explicit parallel-in-time integration of a linear acoustic-advection system

COMPUTERS & FLUIDS 2012年 59卷 72-83页

作者： Ruprecht, D. Krause, R. Univ Svizzera Italiana Inst Computat Sci CH-6906 Lugano Switzerland

The applicability of the Parareal parallel-in-time integration scheme for the solution of a linear, two-dimensional hyperbolic acoustic-advection system, which is often used as a test case for integration schemes for numerical weather prediction (NWP), is addressed. parallel-in-time schemes are a possible way to increase, on the algorithmic level, the amount of parallelism, a requirement arising from the rapidly growing number of CPUs in high performance computer systems. A recently introduced modification of the "parallel implicit time-integration algorithm" could successfully solve hyperbolic problems arising in structural dynamics. It has later been cast into the framework of Parareal. The present paper adapts this modified Parareal and employs it for the solution of a hyperbolic flow problem, where the initial value problem solved in parallel arises from the spatial discretization of a partial differential equation by a finite difference method. It is demonstrated that the modified Parareal is stable and can produce reasonably accurate solutions while allowing for a noticeable reduction of the time-to-solution. The implementation relies on integration schemes already widely used in NWP (RK-3, partially split forward Euler, forward-backward). It is demonstrated that using an explicit partially split scheme for the coarse integrator allows to avoid the use of an implicit scheme while still achieving speedup. (C) 2012 Elsevier Ltd. All rights reserved.

关键词： Numerical weather prediction parallel-in-time integration Parareal Krylov-subspace-enhancement Acoustic-advection system

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Toward fault-tolerant parallel-in-time integration with PFASST

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parallel COMPUTING 2017年 62卷 20-37页

作者： Speck, Robert Ruprecht, Daniel Forschungszentrum Julich Julich Supercomp Ctr D-52425 Julich Germany Univ Leeds Sch Mech Engn Woodhouse Lane Leeds LS2 9JT W Yorkshire England

We introduce and analyze different strategies for the parallel-in-time integration method PFASST to recover from hard faults and subsequent data loss. Since PFASST stores solutions at multiple time steps on different processors, information from adjacent steps can be used to recover after a processor has failed. PFASST's multi-level hierarchy allows to use the coarse level for correcting the reconstructed solution, which can help to minimize overhead. A theoretical model is devised linking overhead to the number of additional PFASST iterations required for convergence after a fault. The potential efficiency of different strategies is assessed in terms of required additional iterations for examples of diffusive and advective type. (C) 2016 Elsevier B.V. All rights reserved.

关键词： Algorithm-based fault tolerance Resilience parallel-in-time integration Gray-Scott model Boussinesq equations

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parallel-in-time multi-level integration of the shallow-water equations on the rotating sphere

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JOURNAL OF COMPUTATIONAL PHYSICS 2020年第0期407卷 109210-000页

作者： Hamon, Francois P. Schreiber, Martin Minion, Michael L. Lawrence Berkeley Natl Lab Ctr Computat Sci & Engn Berkeley CA 94720 USA Tech Univ Munich Chair Comp Architecture & Parallel Syst Munich Germany Lawrence Berkeley Natl Lab Dept Appl Math Berkeley CA USA TOTAL Explorat & Prod Houston TX 77002 USA

The modeling of atmospheric processes in the context of weather and climate simulations is an important and computationally expensive challenge. The temporal integration of the underlying PDEs requires a very large number of time steps, even when the terms accounting for the propagation of fast atmospheric waves are treated implicitly. Therefore, the use of parallel-in-time integration schemes to reduce the time-to-solution is of increasing interest, particularly in the numerical weather forecasting field. We present a multi-level parallel-in-time integration method combining the parallel Full Approximation Scheme in Space and time (PFASST) with a spatial discretization based on Spherical Harmonics (SH). The iterative algorithm computes multiple time steps concurrently by interweaving parallel high-order fine corrections and serial corrections performed on a coarsened problem. To do that, we design a methodology relying on the spectral basis of the SH to coarsen and interpolate the problem in space. The methods are evaluated on the shallow-water equations on the sphere using a set of tests commonly used in the atmospheric flow community. We assess the convergence of PFASST-SH upon refinement in time. We also investigate the impact of the coarsening strategy on the accuracy of the scheme, and specifically on its ability to capture the high-frequency modes accumulating in the solution. Finally, we study the computational cost of PFASST-SH to demonstrate that our scheme resolves the main features of the solution multiple times faster than the serial schemes. (C) 2019 Elsevier Inc. All rights reserved.

关键词： parallel-in-time integration Multi-level spectral deferred corrections Spherical harmonics Shallow-water equations on the sphere Atmospheric flows Climate and weather simulations

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A parallel-IN-time COLLOCATION METHOD USING DIAGONALIZATION: THEORY AND IMPLEMENTATION FOR LINEAR PROBLEMS

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COMMUNICATIONS IN APPLIED MATHEMATICS AND COMPUTATIONAL SCIENCE 2023年第1期18卷 55-85页

作者： Caklovic, Gayatri Speck, Robert Frank, Martin Julich Supercomp Ctr Julich Germany Karlsruhe Inst Technol Dept Math Karlsruhe Germany

We present and analyze a parallel implementation of a parallel-in-time collocation method based on alpha-circulant preconditioned Richardson iterations. While many papers explore this family of single-level, time-parallel "all-at-once" integrators from various perspectives, performance results of actual parallel runs are still scarce. This leaves a critical gap, because the efficiency and applicability of any parallel method heavily rely on the actual parallel performance, with only limited guidance from theoretical considerations. Further, challenges like selecting good parameters, finding suitable communication strategies, and performing a fair comparison to sequential time-stepping methods can be easily missed. In this paper, we first extend the original idea of these fixed point iterative approaches based on alpha-circulant preconditioners to high-order collocation methods, adding yet another level of parallelization in time "across the method". We derive an adaptive strategy to select a new alpha-circulant preconditioner for each iteration during runtime for balancing convergence rates, round-off errors, and inexactness of inner system solves for the individual time-steps. After addressing these more theoretical challenges, we present an open-source space- and time-parallel implementation and evaluate its performance for two different test problems.

关键词： parallel-in-time integration iterative methods diagonalization collocation high-performance computing petsc4py

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Influence of the phase accuracy of the coarse solver calculation on the convergence of the parareal method iteration for hyperbolic PDEs

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COMPUTING AND VISUALIZATION IN SCIENCE 2018年第3-4期19卷 97-108页

作者： Iizuka, Mikio Ono, Kenji Kyushu Univ Res Inst Informat Technol Nishi Ku 744 Motooka Fukuoka 8190395 Japan

Gander and Petcu (ESAIM Proc 25:114-129, 2008) reported that, theoretically, the convergence of the parareal method iteration for hyperbolic PDEs is strongly influenced by the phase (frequency) accuracy of the coarse solver calculation. However, no numerical study has clearly shown this. Therefore, through numerical tests, we investigate the influence of the phase accuracy of the coarse solver calculation on the convergence of the parareal method iteration for hyperbolic PDEs. First, we consider a simple harmonic motion and a multi-DOF mass-spring system (MDMSS) as examples of hyperbolic PDEs using the modified Newmark- method (Mizuta et al. in J JSCE 268:15-21, 1977), which can provide the exact phase of the time integration of a simple harmonic motion. Based on the results of the numerical tests, we show that the convergence of the parareal method iteration for hyperbolic PDEs is approximately independent of the parameters of parallel-in-time integration (PinT) and instead is dependent primarily on the phase accuracy of the coarse solver calculation. In addition, we show that reducing the number of bases in the reduced basis method (RBM) (Chen et al., in: Rozza (ed) Reduced order methods for modeling and computational reduction, MS and a modeling, simulation and applications, vol 9, Springer, Berlin, pp 187-214, 2014) causes the saturation of a decrease in an error during the parareal iteration for the MDMSS using the mode analysis method. The RBM is expected to make available accurate phase calculation in the coarse solver by maintaining the time step width as same as that of the fine solver. Second, we investigate whether the same saturation appears for the linear advection-diffusion equation when we use the RBM. We use the time evolution basis method in the RBM for the linear advection-diffusion equation. As a result, we show that reducing the number of bases causes the saturation of the decrease in the error in the linear advection-diffusion equation. Based on

关键词： parallel-in-time integration Parareal method Hyperbolic PDE Convergence Phase error Reduced basis method

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PFASST-ER: combining the parallel full approximation scheme in space and time with parallelization across the method

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COMPUTING AND VISUALIZATION IN SCIENCE 2020年第1-4期23卷 12-12页

作者： Schoebel, Ruth Speck, Robert Tech Univ Dresden Inst Numer Math Dresden Germany Forschungszentrum Julich Julich Supercomp Ctr Julich Germany

To extend prevailing scaling limits when solving time-dependent partial differential equations, the parallel full approximation scheme in space and time (PFASST) has been shown to be a promising parallel-in-time integrator. Similar to space-time multigrid, PFASST is able to compute multiple time-steps simultaneously and is therefore in particular suitable for large-scale applications on high performance computing systems. In this work we couple PFASST with a parallel spectral deferred correction (SDC) method, forming an unprecedented doubly time-parallel integrator. While PFASST provides global, large-scale "parallelization across the step", the inner parallel SDC method allows integrating each individual time-step "parallel across the method" using a diagonalized local Quasi-Newton solver. This new method, which we call "PFASST with Enhanced concuRrency" (PFASST-ER), therefore exposes even more temporal concurrency. For two challenging nonlinear reaction-diffusion problems, we show that PFASST-ER works more efficiently than the classical variants of PFASST and can use more processors than time-steps.

关键词： parallel-in-time integration parallel full approximation scheme in space and time Spectral deferred corrections parallelization across the method parallelization across the step Quasi-Newton

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Multigrid methods with space-time concurrency

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COMPUTING AND VISUALIZATION IN SCIENCE 2017年第4-5期18卷 123-143页

作者： Falgout, R. D. Friedhoff, S. Kolev, Tz. V. MacLachlan, S. P. Schroder, J. B. Vandewalle, S. Lawrence Livermore Natl Lab Ctr Appl Sci Comp POB 808L-561 Livermore CA 94551 USA Katholieke Univ Leuven Dept Comp Sci Celestijnenlaan 200aBox 2402 B-3001 Leuven Belgium Berg Univ Wuppertal Fak Math & Nat Wissensch D-42097 Wuppertal Germany Mem Univ Newfoundland Dept Math & Stat St John NF Canada

We consider the comparison of multigrid methods for parabolic partial differential equations that allow space-time concurrency. With current trends in computer architectures leading towards systems with more, but not faster, processors, space-time concurrency is crucial for speeding up time-integration simulations. In contrast, traditional time-integration techniques impose serious limitations on parallel performance due to the sequential nature of the time-stepping approach, allowing spatial concurrency only. This paper considers the three basic options of multigrid algorithms on space-time grids that allow parallelism in space and time: coarsening in space and time, semicoarsening in the spatial dimensions, and semicoarsening in the temporal dimension. We develop parallel software and performance models to study the three methods at scales of up to 16K cores and introduce an extension of one of them for handling multistep time integration. We then discuss advantages and disadvantages of the different approaches and their benefit compared to traditional space-parallel algorithms with sequential time stepping on modern architectures.

关键词： Multigrid methods Space-time discretizations parallel-in-time integration

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Toward error estimates for general space-time discretizations of the advection equation

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COMPUTING AND VISUALIZATION IN SCIENCE 2020年第1-4期23卷 16-16页

作者： Gander, Martin J. Lunet, Thibaut Univ Geneva Sect Math 2-4 Rue LievreCase Postale 64 CH-1211 Geneva 4 Switzerland

We develop new error estimates for the one-dimensional advection equation, considering general space-time discretization schemes based on Runge-Kutta methods and finite difference discretizations. We then derive conditions on the number of points per wavelength for a given error tolerance from these new estimates. Our analysis also shows the existence of synergistic space-time discretization methods that permit to gain one order of accuracy at a given CFL number. Our new error estimates can be used to analyze the choice of space-time discretizations considered when testing parallel-in-time methods.

关键词： Advection equation Space-time discretization Error estimates parallel-in-time integration Parareal

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parallelizing spectral deferred corrections across the method

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COMPUTING AND VISUALIZATION IN SCIENCE 2018年第3-4期19卷 75-83页

作者： Speck, Robert Forschungszentrum Julich Julich Supercomp Ctr Julich Germany

In this paper we present two strategies to enable "parallelization across the method" for spectral deferred corrections (SDC). Using standard low-order time-stepping methods in an iterative fashion, SDC can be seen as preconditioned Picard iteration for the collocation problem. Typically, a serial Gau-Seidel-like preconditioner is used, computing updates for each collocation node one by one. The goal of this paper is to show how this process can be parallelized, so that all collocation nodes are updated simultaneously. The first strategy aims at finding parallel preconditioners for the Picard iteration and we test three choices using four different test problems. For the second strategy we diagonalize the quadrature matrix of the collocation problem directly. In order to integrate non-linear problems we employ simplified and inexact Newton methods. Here, we estimate the speed of convergence depending on the time-step size and verify our results using a non-linear diffusion problem.

关键词： Spectral deferred corrections parallel-in-time integration Preconditioning Simplified Newton

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Parareal with a Physics-Informed Neural Network as Coarse Propagator 29th

Parareal with a Physics-Informed Neural Network as Coarse Pr...

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29th International Conference on parallel and Distributed Computing (Euro-Par)

作者： Ibrahim, Abdul Qadir Goetschel, Sebastian Ruprecht, Daniel Hamburg Univ Technol Inst Math Chair Computat Math Hamburg Germany

ISBN: (纸本)9783031396977;9783031396984

parallel-in-time algorithms provide an additional layer of concurrency for the numerical integration of models based on time-dependent differential equations. Methods like Parareal, which parallelize across multiple time steps, rely on a computationally cheap and coarse integrator to propagate information forward in time, while a parallelizable expensive fine propagator provides accuracy. Typically, the coarse method is a numerical integrator using lower resolution, reduced order or a simplified model. Our paper proposes to use a physics-informed neural network (PINN) instead. We demonstrate for the Black-Scholes equation, a partial differential equation from computational finance, that Parareal with a PINN coarse propagator provides better speedup than a numerical coarse propagator. Training and evaluating a neural network are both tasks whose computing patterns are well suited for GPUs. By contrast, mesh-based algorithms with their low computational intensity struggle to perform well. We show that moving the coarse propagator PINN to a GPU while running the numerical fine propagator on the CPU further improves Parareal's single-node performance. This suggests that integrating machine learning techniques into parallel-in-time integration methods and exploiting their differences in computing patterns might offer a way to better utilize heterogeneous architectures.

关键词： Parareal parallel-in-time integration PINN Machine learning GPUs heterogeneous architectures

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