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A parallel-in-time collocation method using diagonalization: theory and implementation for linear problems

Gayatri Čaklović, Robert Speck and Martin Frank

Vol. 18 (2023), No. 1, 55–85
Abstract

We present and analyze a parallel implementation of a parallel-in-time collocation method based on α-circulant preconditioned Richardson iterations. While many papers explore this family of single-level, time-parallel “all-at-once” integrators from various perspectives, performance results of actual parallel runs are still scarce. This leaves a critical gap, because the efficiency and applicability of any parallel method heavily rely on the actual parallel performance, with only limited guidance from theoretical considerations. Further, challenges like selecting good parameters, finding suitable communication strategies, and performing a fair comparison to sequential time-stepping methods can be easily missed. In this paper, we first extend the original idea of these fixed point iterative approaches based on α-circulant preconditioners to high-order collocation methods, adding yet another level of parallelization in time “across the method”. We derive an adaptive strategy to select a new α-circulant preconditioner for each iteration during runtime for balancing convergence rates, round-off errors, and inexactness of inner system solves for the individual time-steps. After addressing these more theoretical challenges, we present an open-source space- and time-parallel implementation and evaluate its performance for two different test problems.

Keywords
parallel-in-time integration, iterative methods, diagonalization, collocation, high-performance computing, petsc4py
Mathematical Subject Classification
Primary: 65F10, 65G50, 65M70, 65Y05
Secondary: 65M22, 65Y20
Milestones
Received: 20 September 2022
Revised: 24 May 2023
Accepted: 2 June 2023
Published: 21 December 2023
Authors
Gayatri Čaklović
Jülich Supercomputing Centre
Jülich
Germany
Department of Mathematics
Karlsruhe Institute of Technology
Karlsruhe
Germany
Robert Speck
Jülich Supercomputing Centre
Jülich
Germany
Martin Frank
Department of Mathematics
Karlsruhe Institute of Technology
Karlsruhe
Germany