Convergence analysis for parallel‐in‐time solution of hyperbolic systems

Sterck, Hans De; Friedhoff, Stephanie; Howse, Alexander J. M.; MacLachlan, Scott

doi:10.1002/nla.2271

Cited by 22 publications

(29 citation statements)

References 48 publications

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“…Instead, one should choose Ψ to be some approximation of Φ m -or equivalently, it should approximate taking m steps with Φ on the fine grid-under the constraint that its action is significantly cheaper to compute so that speed-up can be achieved. Typically Ψ is chosen through the process of rediscretizing Φ on the coarse grid, whereby it is employed with the enlarged coarse-grid time step, mΔt, 4,12,23 but other techniques have also been considered. 8,17 For completeness, we now describe all of the settings used in our numerical tests.…”

Section: Mgrit Parareal and Numerical Set-upmentioning

confidence: 99%

“…In all of these works, speed-ups over sequential time-stepping for hyperbolic PDEs are typically quite small (on the order of two to six), with slow convergence of the iteration ultimately inhibiting faster runtimes due to increased parallelism. For comparison, a speed-up on the order of 20 times was achieved for a diffusion-dominated parabolic problem in Falgout et al 9 A number of theoretical convergence analyses have been developed for Parareal and MGRIT, 5,12,13,19,[23][24][25][26][27] which have helped to explain numerical convergence results, and will likely play an important role in the design of new solvers. Furthermore, some theoretical studies have identified potential roadblocks for fast parallel-in-time convergence of hyperbolic PDEs.…”

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confidence: 99%

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Optimizing multigrid reduction‐in‐time and Parareal coarse‐grid operators for linear advection

Sterck

Falgout

Friedhoff

et al. 2021

Numerical Linear Algebra App

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Parallel-in-time methods, such as multigrid reduction-in-time (MGRIT) and Parareal, provide an attractive option for increasing concurrency when simulating time-dependent partial differential equations (PDEs) in modern high-performance computing environments. While these techniques have been very successful for parabolic equations, it has often been observed that their performance suffers dramatically when applied to advection-dominated problems or purely hyperbolic PDEs using standard rediscretization approaches on coarse grids. In this paper, we apply MGRIT or Parareal to the constant-coefficient linear advection equation, appealing to existing convergence theory to provide insight into the typically nonscalable or even divergent behavior of these solvers for this problem. To overcome these failings, we replace rediscretization on coarse grids with improved coarse-grid operators that are computed by applying optimization techniques to approximately minimize error estimates from the convergence theory. One of our main findings is that, in order to obtain fast convergence as for parabolic problems, coarse-grid operators should take into account the behavior of the hyperbolic problem by tracking the characteristic curves. Our approach is tested for schemes of various orders using explicit or implicit Runge-Kutta methods combined with upwind-finite-difference spatial discretizations. In all cases, we obtain scalable convergence in just a handful of iterations, with parallel tests also showing significant speed-ups over sequential time-stepping. K E Y W O R D Shigh-order, hyperbolic, MGRIT, multigrid, parallel-in-time, Parareal INTRODUCTIONAs compute clock speeds stagnate and core counts of parallel machines increase dramatically, parallel-in-time methods provide an attractive option for increasing concurrency when simulating time-dependent partial differential equations (PDEs). 1,2 Two well-known parallel time integration methods are Parareal 3 and multigrid reduction-in-time (MGRIT), 4

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Section: Mgrit Parareal and Numerical Set-upmentioning

confidence: 99%

mentioning

confidence: 99%

Optimizing multigrid reduction‐in‐time and Parareal coarse‐grid operators for linear advection

Sterck

Falgout

Friedhoff

et al. 2021

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“…For Runge-Kutta schemes, the number of such operations can be reduced by only computing the eigenvalues of the spatial operator L and evaluating the stability function, as opposed to computing the eigenvalues for the family of Φ for all considered Runge-Kutta schemes. While this might not be possible in general, the derivation of Fourier symbols [3] can provide another viable path to reduce the time complexity of computing such bounds. Furthermore, with prior knowledge of L (e.g., L is symmetric positive definite or skew symmetric), the need for solving an eigenvalue problem can be avoided.…”

Section: Number Of Levels Convergence Factormentioning

confidence: 99%

“…Here, N x refers to the number of degrees of freedom at one point in time, and N t refers to the number of time points. 2 For the theoretical analysis, we consider equidistant time points, δ tn = t n − t n−1 = δ t , and a time-independent one-step integrator, Φ n = Φ, for all n. 3 In matrix form, (2.1) can be written as,…”

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Multilevel Convergence Analysis of Multigrid-Reduction-in-Time

Hessenthaler¹,

Southworth²,

Nordsletten³

et al. 2020

SIAM J. Sci. Comput.

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This paper presents a multilevel convergence framework for multigrid-reduction-intime (MGRIT) as a generalization of previous two-grid estimates. The framework provides a priori upper bounds on the convergence of MGRIT V-and F-cycles, with different relaxation schemes, by deriving the respective residual and error propagation operators. The residual and error operators are functions of the time stepping operator, analyzed directly and bounded in norm, both numerically and analytically. We present various upper bounds of different computational cost and varying sharpness. These upper bounds are complemented by proposing analytic formulae for the approximate convergence factor of V-cycle algorithms that take the number of fine grid time points, the temporal coarsening factors, and the eigenvalues of the time stepping operator as parameters.The paper concludes with supporting numerical investigations of parabolic (anisotropic diffusion) and hyperbolic (wave equation) model problems. We assess the sharpness of the bounds and the quality of the approximate convergence factors. Observations from these numerical investigations demonstrate the value of the proposed multilevel convergence framework for estimating MGRIT convergence a priori and for the design of a convergent algorithm. We further highlight that observations in the literature are captured by the theory, including that two-level Parareal and multilevel MGRIT with F-relaxation do not yield scalable algorithms and the benefit of a stronger relaxation scheme. An important observation is that with increasing numbers of levels MGRIT convergence deteriorates for the hyperbolic model problem, while constant convergence factors can be achieved for the diffusion equation. The theory also indicates that L-stable Runge-Kutta schemes are more amendable to multilevel parallel-in-time integration with MGRIT than A-stable Runge-Kutta schemes.Key words. multilevel convergence theory, multigrid-reduction-in-time (MGRIT), parallel-intime, multigrid, analytic upper bounds, a priori estimates 1. Introduction. Modern computer architectures enable massively parallel computations for systems under numerical investigation. While clock rates of recent highperformance computing architectures have largely become stagnant, increased concurrency continues to reduce the time-to-solution, allowing for increased complexity of computational models and accuracy of computed quantities.Spatial domain decomposition (DD) methods are a wide-spread class of parallelization techniques to exploit parallelism in numerical simulations. Many DD methods are straightforward to implement and scalable in parallel up to the point that communication tasks become dominant over computation tasks. Thus, spatial parallelism may saturate without exploiting the full potential of the available hardware.Parallel-in-time methods [37,15] increase the amount of parallelism that can be exploited by introducing parallelism in the temporal domain. Many such methods exist, including waveform relaxation [35,47], space-time ...

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Independence of placement for local Fourier analysis

2021

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Local Fourier analysis (LFA) serves a prominent role in the prediction of the convergence factor of multigrid methods for discretizations of PDEs. However, few discussions about the implementation of LFA for complicated discretizations of PDEs exist, such as higher order finite elements, as staggered meshes could lead to complex LFA representations of the grid‐transfer operator. In this work, we prove that the LFA representation for d‐dimensional PDEs is independent of the placement of degrees of freedom (DoFs). Intuitively speaking, the seeding of the unknowns has no direct effect on the LFA presentation, instead, it serves as a unitary transformation between different representations. Thus, different LFA representations for a given PDE have the same spectrum and norm. Furthermore, we provide a uniform representation in terms of the location of the unknowns, named simple representation, where we allocate all of the DoFs at nodes, resulting in a simple and unified way to compute the symbols of the discrete operators, especially for the grid‐transfer operators. This simple representation can contribute to the generalization of the implementation of LFA for different types of discretizations and different problems, especially for higher order discretization methods.

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Convergence analysis for parallel‐in‐time solution of hyperbolic systems

Cited by 22 publications

References 48 publications

Optimizing multigrid reduction‐in‐time and Parareal coarse‐grid operators for linear advection

Optimizing multigrid reduction‐in‐time and Parareal coarse‐grid operators for linear advection

Multilevel Convergence Analysis of Multigrid-Reduction-in-Time

Independence of placement for local Fourier analysis

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