Towards probabilistic time-parallel algorithms for solving initial value problems

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Abstract

This thesis concerns the development of probabilistic time-parallel algorithms for solving initial value problems (IVPs) that are computationally expensive to simulate using traditional (serial) time-stepping methods. We begin by considering Parareal, a well-studied deterministic time-parallel algorithm that combines solutions from cheap (coarse) and expensive (fine) time-steppers within a predictor-corrector (PC) scheme, to solve the IVP in parallel. Our goal is to derive, analyse, and test our own probabilistic time-parallel algorithms that incorporate sampling- and learning-based techniques from the _eld of probabilistic numerics into Parareal. These techniques enable us to exploit valuable information contained within the _ne and coarse solution data generated during a Parareal simulation. We aim to accelerate the convergence of Parareal (i.e. increase numerical speedup), generate probabilistic solutions to the IVPs (to quantify numerical uncertainty explicitly), and verify the accuracy of these solutions both numerically and analytically.

We first propose SParareal, a sampling-based algorithm that provides the PC with candidate solution values drawn from probability distributions constructed using the most recent fine and coarse solution data. Increased sampling in SParareal leads to accelerated convergence vs. Parareal for low-dimensional IVPs, returning stochastic solutions that are accurate (in the mean-square sense) with respect to the (exact) serially obtained _ne solver solution. Next, we propose GParareal, a learning-based algorithm that models part of the PC using a Gaussian process emulator, trained on all previously collected _ne and coarse solution data. GParareal achieves accelerated convergence for low to moderately sized IVPs, attains accurate solutions, and has the ability to re-use legacy solution data from prior simulations|something that existing time-parallel methods do not do. After introducing both algorithms, we investigate their performance and analyse their limitations, assessing whether or not they are viable methods for solving large-scale IVPs in parallel and discussing what can be done to improve them in their current form.

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