2021/03/23 by Gayatri Caklovic, Caklovic, Gayatri, Robert Speck +3 · 1 citation
Computer Science · Engineering · Mathematics · #Electromagnetic Simulation and Numerical Methods #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.2103.12571
openalex publication_date 2021/03/23 · openalex created_date 2022/09/23 · openalex updated_date 2026/07/28
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.