A multigrid perspective on PFASST

For time-dependent PDEs, 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 non-linear multigrid ideas, PFASST allows to integrate multiple time-steps simultaneously using "multi-level spectral deferred corrections" with different coarsening strategies in space and time. In numerous studies, this approach has been successfully coupled to space-parallel solvers which use finite differences, spectral methods or even particles for discretization in space. While many use cases and benchmarks exist, a solid and reliable mathematical foundation is still missing. In this talk, we briefly discuss previous results and benefits of PFASST and then formulate this algorithm as a specialized FAS multigrid method. We use spectral deferred corrections for the definition of block smoothers and define the appropriate coarse grid correction to establish a tight link between PFASST and standard multigrid methods, providing an easy access to the mathematical analysis and algorithmic optimization. For linear problems, we can state the iteration matrix and use this formulation to prove asymptotic convergence of PFASST for CFL numbers going to zero or to infinity.

After his studies on applied math and informatics at the University of Trier Robert Speck did his PhD thesis from 2007 to 2011 at Juelich Supercomputing Centre (JSC) and University of Wuppertal on massively parallel vortex particle methods. He then joined the group of Prof. Rolf Krause at ICS in Lugano as a postdoc and started working on parallel-in-time integration methods. Since 2012, he is a researcher within the cross-sectional team "Mathematical Methods and Algorithms" at JSC and leads a small group of PhD and Master students. He is currently working on spectral deferred corrections and multigrid methods in time.