Analysis of the Parareal Time‐Parallel Time‐Integration Method

Integrating an N-Body Problem with SDC and PFASST

“…While PFASST(8, 7, 2) is still slightly slower than SDC(12, 7), PFASST (8,7,4) and PFASST (8,7,8) show significant speedup. The cost of enlarging the problem, i. e. switching from SDC to a multi-level scheme, is compensated by the fact that this multi-level approach is amenable to parallelization while SDC itself is not.…”

“…When run on one time-processors, PFASST reduces to a multi-level SDC scheme. Comparing SDC (12,7) and PFASST (8,7,1) we note that by switching from a single-level SDC scheme of a given order to a multi-level SDC scheme with comparable accuracy, the number of iterations are reduced (from 12 to 8 in our case). On the other hand, significant additional costs are introduced due to the additional f evaluations required by the coarse step and the transfer operations between fine and coarse quadrature nodes.…”

“…Hence, if the workload of PFASST (8, 7) is distributed across sufficiently many processors, then the total runtime becomes smaller than the time-to-solution of the serial SDC (12,7) method. This is highlighted by (12, 7)) and PFASST (8,7, Z) with Z = 1, 2, 4, 8 time-processes (abbrev. by P(8, 7, Z)).…”

“…A very general scheme is Parareal [15], which allows arbitrary integration schemes to be used in a black-box fashion. A detailed mathematical analysis of Parareal is conducted in [8] and comprehensive lists of references can be found e. g. in [17,20]. The drawback of Parareal is that the parallel efficiency is formally bounded by 1/K where K is the number of iterations required for convergence.…”

On the Use of Reduced Basis Methods to Accelerate and Stabilize the Parareal Method

“…For parabolic and diffusion dominated problems, stability is well understood and observed in many applications [12]. However, for hyperbolic and convection dominated problems, the question of stability is considerably more complex and generally remains open [23,8,3].…”