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Research on parallel-in-time integration methods

The efficient use of modern high performance computing (HPC) systems has become one of the key challenges in computational science. Top HPC architectures already provide million-way concurrency, and current trends suggest that processor counts will continue to grow rapidly. Exploiting these levels of parallelism using traditional techniques for spatial parallelism becomes problematic when, for example, for a fixed problem size communication costs begin to dominate (“strong scaling barrier”) or for increased spatial resolution more time-steps are necessary due to stability constraints (“weak scaling barrier”).

For the numerical solution of time-dependent differential equations, parallel-in-time integration (PinT) methods have recently been shown to provide a promising way to extend prevailing scaling limits. To overcome the seemingly inherent serial dependence in the time direction and to enable integration of multiple time-steps simultaneously, one idea of time-parallel methods is to introduce a space/time hierarchy, where integrators with different costs are coupled in an iterative fashion. Serial dependencies are shifted to the coarsest level, allowing the computationally expensive parts on finer levels to be treated in parallel (“Parareal-based approaches”).

One promising PinT algorithm, the "parallel full approximation scheme in space and time" (PFASST), is an iterative, multilevel strategy for the temporal parallelization of ODEs and discretized PDEs. As the name suggests, PFASST is similar in spirit to a space-time FAS multigrid method performed over multiple timesteps in parallel. The key for optimal parallel efficiency is a careful choice of coarsening strategies in space and time. Besides straightforward approaches like reducing the number of degrees-of-freedom and/or integration nodes, recent works focused on the reduction of the spatial discretization order as well as inexact solves of systems arising in implicit steps on the coarse levels.

This group at JSC, being part of the cross-sectional team Mathematical Methods and Algorithms, primarily focuses on

  • large-scale applications with PFASST on HPC systems
  • mathematical analysis of spectral deferred corrections (SDC) and PFASST
  • application-tailored coarsening strategies for PFASST
  • space-time-parallel particle simulations

and is actively participating in the development of