Parallel Time Integration with Multigrid

Abstract

Abstract. We consider optimal-scaling multigrid solvers for the linear systems that arise from the discretization of problems with evolutionary behavior. Typically, solution algorithms for evolution equations are based on a time-marching approach, solving sequentially for one time step after the other. Parallelism in these traditional time-integration techniques is limited to spatial parallelism. However, current trends in computer architectures are leading towards systems with more, but not faster, processors. Therefore, faster compute speeds must come from greater parallelism. One approach to achieve parallelism in time is with multigrid, but extending classical multigrid methods for elliptic operators to this setting is not straightforward. In this paper, we present a non-intrusive, optimal-scaling time-parallel method based on multigrid reduction (MGR). We demonstrate optimality of our multigrid-reduction-in-time algorithm (MGRIT) for solving diffusion equations in two and three space dimensions in numerical experiments. Furthermore, through both parallel performance models and actual parallel numerical results, we show that we can achieve significant speedup in comparison to sequential time marching on modern architectures.Key words. parabolic problems, reduction-based multigrid, multigrid-in-time, parareal AMS subject classifications. 65F10, 65M22, 65M551. Introduction. One of the major challenges facing the computational science community with future architectures is that faster compute speeds must come from increased concurrency, since clock speeds are no longer increasing but core counts are going up sharply. As a consequence, traditional time marching is becoming a huge sequential bottleneck in time integration simulations in the following way: improving simulation accuracy by scaling up the spatial resolution requires a similar (or greater) increase in the temporal resolution, which is also required to maintain stability in explicit methods. As a result, numerical time integration involves many more time steps leading to long overall compute times, since parallelizing only in space limits concurrency. Solving for multiple time steps in parallel and, therefore, increasing concurrency would remove this time integration bottleneck.Because time is sequential in nature, the idea of simultaneously solving for multiple time steps is not intuitive. Yet it is possible, with work on this topic going back to as early as 1964 [33]. However, most research on this subject has been done within the past 30 years including [2, 7-10, 14-22, 25, 28, 31, 32, 38, 40-44]. One approach to achieve parallelism in time is with multigrid methods. The parareal in time method, introduced by Lions, Maday, and Turinici in [25], can be interpreted as a two-level multigrid method [16], even though the leading idea came from a spatial domain decomposition approach. The algorithm is optimal, but concurrency is limited since the coarse-grid solve is still sequential. Considering true multilevel (not two-level) schemes, only a few method...

A generalized predictive analysis tool for multigrid methods

“…The multigrid‐reduction‐in‐time (MGRIT) algorithm is based on applying multigrid reduction techniques to time integration. The method uses block smoothers for relaxation and employs a semicoarsening strategy that, in contrast to waveform relaxation, coarsens only in the temporal dimension.…”

“…Current trends in computer architectures are leading towards systems with more, but not faster processors. As a consequence, faster compute speeds must come from greater parallelism, resurging the interest in multigrid algorithms for parabolic problems that allow temporal parallelism . Regarding this development, a predictive analysis tool for these methods becomes highly relevant.…”

“…In §2, we describe two multigrid schemes for a parabolic model problem. One method based on waveform relaxation as in and one based on temporal semicoarsening as in . In §3, we illustrate the failure of LFA to generate accurate predictions of the convergence behavior for the waveform relaxation method applied to the parabolic model problem.…”

Parallel-In-Space-Time, Adaptive Finite Element Framework for Nonlinear Parabolic Equations

“…These factors include (a) hitting the limits of scaling when using purely sequential approaches, (b) the ability to store and solve the full time horizon with significant implications to solving adjoint problems, (c) the ability to only perform spatial adaptivity in blocks greatly simplifying storage, and, finally, (d) as a first step to simultaneous adaptivity in space and in time [19], which would prove very significant. 10 Several other researchers have also made this case [12,6,20]. We next look at how increasing the number of time steps, N , in a time block impacts performance of the space-time approach.…”

A Posteriori Error Analysis of Two-Stage Computation Methods with Application to Efficient Discretization and the Parareal Algorithm

“…The two stages are used to devise algorithms that provide a way to exploit parallelization for time integration [35, 15, 28], e.g. the Parareal Algorithm [39, 37, 38, 41, 29, 30].…”