26 research outputs found
Performance of parallel-in-time integration for Rayleigh Bénard convection
Rayleigh–Bénard convection (RBC) is a fundamental problem of fluid dynamics, with many applications to geophysical, astrophysical, and industrial flows. Understanding RBC at parameter regimes of interest requires complex physical or numerical experiments. Numerical simulations require large amounts of computational resources; in order to more efficiently use the large numbers of processors now available in large high performance computing clusters, novel parallelisation strategies are required. To this end, we investigate the performance of the parallel-in-time algorithm Parareal when used in numerical simulations of RBC. We present the first parallel-in-time speedups for RBC simulations at finite Prandtl number. We also investigate the problem of convergence of Parareal with respect to statistical numerical quantities, such as the Nusselt number, and discuss the importance of reliable online stopping criteria in these cases
A parallel implementation of a diagonalization-based parallel-in-time integrator
We present and analyze a parallel implementation of a parallel-in-time method
based on -circulant preconditioned Richardson iterations. While there
are a lot of papers exploring this new class of single-level, time-parallel
integrators from many perspectives, performance results of actual parallel runs
are still missing. This leaves a critical gap, because the efficiency and
applicability heavily rely on the actual parallel performance, with only
limited guidance from theoretical considerations. Also, many 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 by using a collocation
method of arbitrary order, which adds another level of parallelization in time.
We derive an adaptive strategy to select a new -circulant
preconditioner for each iteration during runtime for balancing convergence
rates, round-off errors and inexactness in the individual time-steps. After
addressing these more theoretical challenges, we present an open-source space-
and doubly-time-parallel implementation and evaluate its performance for two
different test problems
Parallel-in-Time Multi-Level Integration of the Shallow-Water Equations on the Rotating Sphere
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
Numerical wave propagation aided by deep learning
We propose a deep learning approach for wave propagation in media with
multiscale wave speed, using a second-order linear wave equation model. We use
neural networks to enhance the accuracy of a given inaccurate coarse solver,
which under-resolves a class of multiscale wave media and wave fields of
interest. Our approach involves generating training data by the given
computationally efficient coarse solver and another sufficiently accurate
solver, applied to a class of wave media (described by their wave speed
profiles) and initial wave fields. We find that the trained neural networks can
approximate the nonlinear dependence of the propagation on the wave speed as
long as the causality is appropriately sampled in training data. We combine the
neural-network-enhanced coarse solver with the parareal algorithm and
demonstrate that the coupled approach improves the stability of parareal
algorithms for wave propagation and improves the accuracy of the enhanced
coarse solvers