A multi-level spectral deferred correction method

Time parallel time integration methods have received renewed interest over the last decade because of the advent of massively parallel computers, which is mainly due to the clock speed limit reached on today's processors. When solving time dependent partial differential equations, the time direction is usually not used for parallelization. But when parallelization in space saturates, the time direction offers itself as a further direction for parallelization. The time direction is however special, and for evolution problems there is a causality principle: the solution later in time is affected (it is even determined) by the solution earlier in time, but not the other way round. Algorithms trying to use the time direction for parallelization must therefore be special, and take this very different property of the time dimension into account. We show in this chapter how time domain decomposition methods were invented, and give an overview of the existing techniques. Time parallel methods can be classified into four different groups: methods based on multiple shooting, methods based on domain decomposition and waveform relaxation, space-time multigrid methods and direct time parallel methods. We show for each of these techniques the main inventions over time by choosing specific publications and explaining the core ideas of the authors. This chapter is for people who want to quickly gain an overview of the exciting and rapidly developing area of research of time parallel methods.

show abstract

“…The method has successfully been applied to non-linear problems in [19,77,76], but there is so far no convergence analysis for PFASST.…”

Section: Letũ(t) Be An Approximation With Error E(t) := U(t) −ũ(T) Imentioning

confidence: 99%

50 Years of Time Parallel Time Integration

Gander

2015

Contributions in Mathematical and Computational Sciences

270

226

View full text Add to dashboard Cite

show abstract

“…Space-time multigrid methods were developed in [20,30,41,25,40,26,27,43], and reached good F-cycle convergence behavior when appropriate semi-coarsening and extension operators are used. For a variant for non-linear problems, see [4,36,35].We present and analyze here a new space-time parallel multigrid algorithm that has excellent strong and weak scalability properties on large scale parallel computers. As a model problem we consider the heat equation in a bounded domain Ω ⊂ R d , d = 1, 2, 3 with boundary Γ := ∂Ω on the bounded time interval [0, T ],…”

mentioning

confidence: 99%

mentioning

confidence: 99%

Analysis of a New Space-Time Parallel Multigrid Algorithm for Parabolic Problems

Gander¹,

Neumüller²

2016

SIAM J. Sci. Comput.

125

View full text Add to dashboard Cite

Abstract. We present and analyze a new space-time parallel multigrid method for parabolic equations. The method is based on arbitrarily high order discontinuous Galerkin discretizations in time, and a finite element discretization in space. The key ingredient of the new algorithm is a block Jacobi smoother. We present a detailed convergence analysis when the algorithm is applied to the heat equation, and determine asymptotically optimal smoothing parameters, a precise criterion for semi-coarsening in time or full coarsening, and give an asymptotic two grid contraction factor estimate. We then explain how to implement the new multigrid algorithm in parallel, and show with numerical experiments its excellent strong and weak scalability properties.Key words. Space-time parallel methods, multigrid in space-time, DG-discretizations, strong and weak scalability, parabolic problems AMS subject classifications. 65N55, 65F10, 65L601. Introduction. About ten years ago, clock speeds of processors have stopped increasing, and the only way to obtain more performance is by using more processing cores. This has led to new generations of supercomputers with millions of computing cores, and even today's small devices are multicore. In order to exploit these new architectures for high performance computing, algorithms must be developed that can use these large numbers of cores efficiently. When solving evolution partial differential equations, the time direction offers itself as a further direction for parallelization, in addition to the spatial directions, and the parareal algorithm [29,31,1,37,18,9] has sparked renewed interest in the area of time parallelization, a field that is now just over fifty years old, see the historical overview [8]. We are interested here in space-time parallel methods, which can be based on the two fundamental paradigms of domain decomposition or multigrid. Domain decomposition methods in spacetime lead to waveform relaxation type methods, see [17,7,19] for classical Schwarz waveform relaxation, [12,13,10,11,2] for optimal and optimized variants, and [28,33,15] for Dirichlet-Neumann and Neumann-Neumann waveform relaxation. The spatial decompositions can be combined with parareal to obtain algorithms that run on arbitrary decompositions of the space-time domain into space-time subdomains, see [32,14]. Space-time multigrid methods were developed in [20,30,41,25,40,26,27,43], and reached good F-cycle convergence behavior when appropriate semi-coarsening and extension operators are used. For a variant for non-linear problems, see [4,36,35].We present and analyze here a new space-time parallel multigrid algorithm that has excellent strong and weak scalability properties on large scale parallel computers. As a model problem we consider the heat equation in a bounded domain Ω ⊂ R d , d = 1, 2, 3 with boundary Γ := ∂Ω on the bounded time interval [0, T ],

show abstract

“…Inversion of this preconditioner is inherently serial, but the goal is to keep this serial part as small as possible by applying it on the coarse level only, just as the Parareal method does. Thus, we need coarsening strategies in place to reduce the costs on the coarser levels . To this end, we introduce blockwise restriction and interpolation

{boldT}_{F}^{C}

and

{boldT}_{C}^{F}

, which coarsen the problem in space and reduce the number of quadrature nodes but do not coarsen in time, that is, the number of time steps is not reduced.…”

Section: A Multigrid Perspective On Pfasstmentioning

confidence: 99%

“…Thus, we need coarsening strategies in place to reduce the costs on the coarser levels. 27 To this end, we introduce blockwise restriction and interpolation T C F and T F C , which coarsen the problem in space and reduce the number of quadrature nodes but do not coarsen in time, that is, the number of time steps is not reduced. We note that the latter is also possible in this formal notation, but so far, no PFASST implementation is working with this.…”

Section: The Composite Collocation Problem and Pfasstmentioning

confidence: 99%

Asymptotic convergence of the parallel full approximation scheme in space and time for linear problems

Bolten

Moser

Speck

2018

Numerical Linear Algebra App

Self Cite

View full text Add to dashboard Cite

Summary For time‐dependent partial differential equations, parallel‐in‐time integration using the “parallel full approximation scheme in space and time” (PFASST) is a promising way to accelerate existing space‐parallel approaches beyond their scaling limits. Inspired by the classical Parareal method and multigrid ideas, PFASST allows to integrate multiple time steps simultaneously using a space–time hierarchy of spectral deferred correction sweeps. While many use cases and benchmarks exist, a solid and reliable mathematical foundation is still missing. Very recently, however, PFASST for linear problems has been identified as a multigrid method. In this paper, we will use this multigrid formulation and, in particular, PFASST's iteration matrix to show that, in the nonstiff and stiff limit, PFASST indeed is a convergent iterative method. We will provide upper bounds for the spectral radius of the iteration matrix and investigate how PFASST performs for increasing numbers of parallel time steps. Finally, we will demonstrate that the results obtained here indeed relate to actual PFASST runs.

show abstract

A multi-level spectral deferred correction method

Cited by 62 publications

References 41 publications

50 Years of Time Parallel Time Integration

50 Years of Time Parallel Time Integration

Analysis of a New Space-Time Parallel Multigrid Algorithm for Parabolic Problems

Asymptotic convergence of the parallel full approximation scheme in space and time for linear problems

Contact Info

Product

Resources

About