Recent results on spectral deferred corrections, parallel and otherwise (Daniel Ruprecht, Technische Universität Hamburg)

24.09.2024 14:00

Spectral deferred corrections (SDC) remain an underutilized approach for solving initial value problems. SDC can be considered a preconditioned, iterative solver for the collocation problem arising in a fully implicit Runge-Kutta method with a dense Butcher table. The key trick in SDC is that each iteration, or sweep, only requires stepping through the nodes with a simple, low order method like implicit Euler. In that way, SDC provide a framework to build methods of arbitrary high order out of simple, low order methods.

SDC-based methods have many advantages. They typically come with large stability domains, easily tunable order of accuracy and are often more accurate than RKM of similar order when using comparable time step sizes. However, they require more right hand side evaluations per step than RKM (sometimes a lot more) and can only be computationally efficient if they allow a user to take much larger step sizes than RKM. In the talk, I will present recent work aiming to increase the computational efficiency of SDC. This will include approaches to parallelize SDC "across-the-method" to allow the use of small-scale parallelism to reduce the wallclock time of iterations. I will also show how the iterative structure of SDC can easily be exploited to construct a form of embedded method that can be used for step size control. Finally, I will comment one ongoing work regarding the use of reduced asymptotic coarse models to speed up SDC convergence as well as attempts to use physics-informed neural operators (PINO) to generate accurate starting values.

This is joint work with J. Fregin, T. Lunet, I. Akramov, P. Freese, S. Götschel (TUHH), G. Čaklovic (KIT), T. Baumann (FZ Jülich) and M. Schreiber (Grenoble).

Lieu

Conseil Général 7-9, Room 1-07 (unusual room!), Séminaire d'analyse numérique

entrée libre