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

Abstract

We propose a modified parallel-in-time -parareal -multi-level time integration method which, in contrast to previously proposed methods, employs a coarse solver based on a reduced model, built from the information obtained from the fine solver at each iteration. This approach is demonstrated to offer two substantial advantages: it accelerates convergence of the original parareal method for similar problems and the reduced basis stabilizes the parareal method for purely advective problems where instabilities are known to arise. When combined with empirical interpolation methods (EIM), we develop this approach to solve both linear and nonlinear problems and highlight the minimal changes required to utilize this algorithm to accelerate existing implementations. We illustrate the advantages through algorithmic design, through analysis of stability, convergence, and computational complexity, and through several numerical examples.

Time-parallel gravitational collapse simulation

“…This results in a total of 30,144 compute cores and a peak performance of 1.254 PFlops; it occupies position 56 in the Top500 November, 2014 list. 7 On Piz Dora, we used the GNU compiler collection 8 version 4.9.2 and the runtimes we provide do not include the cost of I/O operations. Some simulations measuring convergence were performed on a machine located at the Università della Svizzera italiana that is maintained by members of the Institute of Computational Science of the Faculty of Informatics.…”

An index-aware parametric model order reduction method for parameterized quadratic differential–algebraic equations

“…Substituting x P = p 0 ξ p and x Q = q 0 ξ q into (11), and simplifying using (12) leads to a decoupled system given by…”

An Exponential Time Integrator for the Incompressible Navier--Stokes Equation

“…One advantage of Paraexp is that its parallel performance does not deteriorate for hyperbolic PDEs or parabolic PDEs with little dissipation. Although the well-known Parareal method is known to work successfully for some hyperbolic problems, e.g., [9,14,28,41,51], convection-dominated flows were generally found to be highly challenging for Parareal [57]. The EBK method allows for a natural extension of the Paraexp method to nonlinear PDEs.…”

“…However, in [18] speedup was achieved for high Reynolds simulations for which a turbulence model was deployed. For some hyperbolic problems a stabilization of Parareal can be applied [9,14,28,51]. There are also reports of cases in which Parareal succeeds, even for problems with little or no dissipation.…”