Pentland, Kamran, Tamborrino, Massimiliano, Samaddar, Debasmita and Appel, Lynton C. (2022) Stochastic parareal : an application of probabilistic methods to time-parallelization. SIAM Journal on Scientific Computing . S82-S102. doi:10.1137/21m1414231 ISSN 1064-8275.
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Abstract
Parareal is a well-studied algorithm for numerically integrating systems of time-dependent differential equations by parallelizing the temporal domain. Given approximate initial values at each temporal subinterval, the algorithm locates a solution in a fixed number of iterations using a predictor-corrector, stopping once a tolerance is met. This iterative process combines solutions located by inexpensive (coarse resolution) and expensive (fine resolution) numerical integrators. In this paper, we introduce a stochastic parareal algorithm aimed at accelerating the convergence of the deterministic parareal algorithm. Instead of providing the predictor-corrector with a deterministically located set of initial values, the stochastic algorithm samples initial values from dynamically varying probability distributions in each temporal subinterval. All samples are then propagated in parallel using the expensive integrator. The set of sampled initial values yielding the most continuous (smoothest) trajectory across consecutive subintervals are fed into the predictor-corrector, converging in fewer iterations than the deterministic algorithm with a given probability. The performance of the stochastic algorithm, implemented using various probability distributions, is illustrated on low-dimensional systems of ordinary differential equations (ODEs). We provide numerical evidence that when the number of sampled initial values is large enough, stochastic parareal converges almost certainly in fewer iterations than the deterministic algorithm, maintaining solution accuracy. Given its stochastic nature, we also highlight that multiple simulations of stochastic parareal return a distribution of solutions that can represent a measure of uncertainty over the ODE solution.
| Item Type: | Journal Article |
|---|---|
| Subjects: | Q Science > QA Mathematics > QA76 Electronic computers. Computer science. Computer software |
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics Faculty of Science, Engineering and Medicine > Science > Statistics |
| SWORD Depositor: | Library Publications Router |
| Library of Congress Subject Headings (LCSH): | Parallel algorithms, Parallel processing (Electronic computers), Differential equations -- Numerical solutions -- Data processing, Machine learning -- Mathematics, Computer algorithms |
| Journal or Publication Title: | SIAM Journal on Scientific Computing |
| Publisher: | Society for Industrial and Applied Mathematics |
| ISSN: | 1064-8275 |
| Official Date: | 7 July 2022 |
| Dates: | Date Event 7 July 2022 Published 30 March 2022 Accepted |
| Page Range: | S82-S102 |
| DOI: | 10.1137/21m1414231 |
| Status: | Peer Reviewed |
| Publication Status: | Published |
| Access rights to Published version: | Restricted or Subscription Access |
| Date of first compliant deposit: | 22 June 2023 |
| Date of first compliant Open Access: | 22 June 2023 |
| RIOXX Funder/Project Grant: | Project/Grant ID RIOXX Funder Name Funder ID 633053 Euratom Research and Training Programme UNSPECIFIED EP/S022244/1 [EPSRC] Engineering and Physical Sciences Research Council |
| Open Access Version: | |
| URI: | https://wrap.warwick.ac.uk/167539/ |
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