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

TL;DR

This paper develops and analyzes a parallel-in-time collocation method with adaptive preconditioning, extending previous approaches to high-order methods, and provides an open-source implementation with performance evaluation.

Contribution

It extends parallel-in-time collocation methods with adaptive preconditioning to high-order schemes and offers a practical, open-source implementation with performance analysis.

Findings

Effective adaptive preconditioning improves convergence.

Parallel implementation achieves promising speedups.

Method is applicable to different test problems.

Abstract

We present and analyze a parallel implementation of a parallel-in-time collocation method based on $\alpha$-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 $\alpha$-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 $\alpha$-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.