A stencil-based implementation of Parareal in the C++ domain specific embedded language STELLA

Abstract

In view of the rapid rise of the number of cores in modern supercomputers, time-parallel methods that introduce concurrency along the temporal axis are becoming increasingly popular. For the solution of time-dependent partial differential equations, these methods can add another direction for concurrency on top of spatial parallelization. The paper presents an implementation of the time-parallel Parareal method in a C++ domain specific language for stencil computations (STELLA). STELLA provides both an OpenMP and a CUDA backend for a shared memory parallelization, using the CPU or GPU inside a node for the spatial stencils. Here, we intertwine this node-wise spatial parallelism with the time-parallel Parareal. This is done by adding an MPI-based implementation of Parareal, which allows us to parallelize in time across nodes. The performance of Parareal with both backends is analyzed in terms of speedup, parallel efficiency and energy-to-solution for an advection-diffusion problem with a time-dependent diffusion coefficient.

A stable parareal-like method for the second order wave equation

“…In a context of linear evolutionary problemu(t) = Au(t) for t ∈ {0, ∆t, ...N ∆t = T } and A : R → R linear function, let us denote the fine propagator/solver Fu n → u n+1 and the coarse propagator/solver Cu n → u n+1 . Then the plain parareal iteration k + 1 can be written as a recurrence relation (3) u k+1 n+1 = Cu k+1 n + Fu k n − Cu k n . Starting solution k = 1 is the serial coarse solution u k=1 n = C n u 0 .…”

Numerical simulation of skin transport using Parareal

“…For the spatial solver and parallelization of UG4 applied to the benchmark used here, this has been demonstrated successfully in [34]. However, when the number of fine time steps N f is also doubled, twice as many time steps have to be computed in order to keep the ratio δt/δx constant, which leads to a doubling of time-to-solution (see also the discussion in [4]). Time parallelization can provide some mitigation by also doubling the number of subintervals and thus of cores used to parallelize along time -unless this leads to a massive increase in the number of required iterations.…”

Shared Memory Pipelined Parareal

“…While this a rather specific setup, finite difference stencils are a widely used motif in computational science: even though specifics and runtimes will be different if more complex problems are solved, the general concepts are used in complex application codes, e.g. by means of domain specific embedded languages (DSEL) [24] and the possibility to use Parareal with such a DSEL has been shown [25]. Thus, conclusions in terms of performance of different implementations of Parareal can be generalised to some extent, in particular because the Parareal routines only operate on linear arrays and do not see any specifics of the underlying time steppers or spatial discretisation.…”

“…The implementation of Parareal with MPI is straightforward and has been illustrated before [25]. However, it is repeated here for the sake of completeness and sketched in Algorithm 1.…”

“…Parareal without pipelining in MPI would therefore be more complex and deliver reduced performance, so that this strategy seems rather senseless. Previous benchmarks did not show a significant difference between blocking and non-blocking communication [25] and for the sake of simplicity, blocking communication is used here.…”