{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:34:36Z","timestamp":1787330076448,"version":"build-2736575974"},"reference-count":28,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","funder":[{"DOI":"10.13039\/100018708","name":"Euratom Research and Training Programme","doi-asserted-by":"crossref","award":["633053"],"award-info":[{"award-number":["633053"]}],"id":[{"id":"10.13039\/100018708","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["EP\/S022244\/1"],"award-info":[{"award-number":["EP\/S022244\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100008517","name":"Culham Centre for Fusion Energy","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100008517","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2023,10,31]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>Stochastic Parareal (SParareal) is a probabilistic variant of the popular parallel-in-time algorithm known as Parareal. Similarly to Parareal, it combines fine- and coarse-grained solutions to an ODE using a predictor-corrector (PC) scheme. The key difference is that carefully chosen random perturbations are added to the PC to try to accelerate the location of a stochastic solution to the ODE. In this paper, we derive superlinear and linear mean-square error bounds for SParareal applied to nonlinear systems of ODEs using different types of perturbations. We illustrate these bounds numerically on a linear system of ODEs and a scalar nonlinear ODE, showing a good match between theory and numerics.<\/jats:p>","DOI":"10.1137\/22m1533062","type":"journal-article","created":{"date-parts":[[2023,10,6]],"date-time":"2023-10-06T04:30:41Z","timestamp":1696566641000},"page":"A2657-A2678","source":"Crossref","is-referenced-by-count":2,"title":["Error Bound Analysis of the Stochastic Parareal Algorithm"],"prefix":"10.1137","volume":"45","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9530-1890","authenticated-orcid":true,"given":"Kamran","family":"Pentland","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Warwick, Coventry CV4 7AL, UK."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4661-8071","authenticated-orcid":true,"given":"Massimiliano","family":"Tamborrino","sequence":"additional","affiliation":[{"name":"Department of Statistics, University of Warwick, Coventry CV4 7AL, UK."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4081-8301","authenticated-orcid":true,"given":"T. J.","family":"Sullivan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Warwick, Coventry CV4 7AL, UK."},{"name":"Alan Turing Institute, British Library, London NW1 2DB, UK."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2023,10,6]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.1007\/3-540-26825-1_43"},{"key":"ref2","unstructured":"G. Bal   (2006), Parallelization in Time of (Stochastic) Ordinary Differential Equations, https:\/\/citeseerx.ist.psu.edu\/viewdoc\/download;jsessionid=7CAF738B3BD4696A3F993C2A0E8AD547?doi=10.1.1.332.4585&rep=rep1&type=pdf."},{"key":"ref3","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-56118-4_12"},{"key":"ref4","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-75199-1_51"},{"key":"ref5","doi-asserted-by":"crossref","unstructured":"B. Carrel , \nM. J. Gander , and \nB. Vandereycken   (2022), Low-Rank Parareal: A Low-Rank Parallel-in-Time Integrator, preprint, arXiv:2203.08455.","DOI":"10.1007\/s10543-023-00953-3"},{"key":"ref6","doi-asserted-by":"publisher","DOI":"10.1007\/s11222-016-9671-0"},{"key":"ref7","doi-asserted-by":"publisher","DOI":"10.1051\/m2an\/2012046"},{"key":"ref8","doi-asserted-by":"crossref","unstructured":"W. R. Elwasif , \nS. S. Foley , \nD. E. Bernholdt , \nL. A. Berry , \nD. Samaddar , \nD. E. Newman , and \nR. Sanchez   (2011), A dependency-driven formulation of parareal: Parallel-in-time solution of PDEs as a many-task application, in \u2014MTAGS\u201911 Proceedings of the 2011 ACM International Workshop on Many Task Computing on Grids and Supercomputers, Co-Located with SC\u201911, ACM Press, New York, pp. 15\u201324, https:\/\/doi.org\/10.1145\/2132876.2132883.","DOI":"10.1145\/2132876.2132883"},{"key":"ref9","doi-asserted-by":"publisher","DOI":"10.1137\/080733723"},{"key":"ref10","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-23321-5_3"},{"key":"ref11","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-75199-1_4"},{"key":"ref12","doi-asserted-by":"publisher","DOI":"10.1051\/proc:082508"},{"key":"ref13","doi-asserted-by":"publisher","DOI":"10.1137\/05064607X"},{"key":"ref14","doi-asserted-by":"crossref","unstructured":"M. J. Gander , \nT. Lunet , \nD. Ruprecht , and \nR. Speck   (2022), A Unified Analysis Framework for Iterative Parallel-in-Time Algorithms, preprint, arXiv:2203.16069.","DOI":"10.1137\/22M1487163"},{"key":"ref15","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2021.110282"},{"key":"ref16","series-title":"Springer Series in Computational Mathematics 8","volume-title":"Solving Ordinary Differential Equations I: Nonstiff Problems","author":"Hairer E.","year":"1993","edition":"2"},{"key":"ref17","doi-asserted-by":"publisher","DOI":"10.1007\/s00791-020-00329-y"},{"key":"ref18","doi-asserted-by":"publisher","DOI":"10.1137\/21M1412979"},{"key":"ref19","doi-asserted-by":"publisher","DOI":"10.1007\/s11222-019-09898-6"},{"key":"ref20","doi-asserted-by":"publisher","DOI":"10.1007\/s10092-022-00457-6"},{"key":"ref21","doi-asserted-by":"publisher","DOI":"10.1016\/S0764-4442(00)01793-6"},{"key":"ref22","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2020.112915"},{"key":"ref23","doi-asserted-by":"publisher","DOI":"10.1145\/355588.365137"},{"key":"ref24","doi-asserted-by":"publisher","DOI":"10.1007\/s00791-020-00331-4"},{"key":"ref25","first-page":"S82","author":"Pentland K.","year":"2022","journal-title":"SIAM J. Sci. Comput."},{"key":"ref26","doi-asserted-by":"publisher","DOI":"10.1007\/s11222-022-10195-y"},{"key":"ref27","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2010.05.012"},{"key":"ref28","doi-asserted-by":"publisher","DOI":"10.1016\/j.cpc.2018.08.007"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/22M1533062","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:46:45Z","timestamp":1787327205000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/22M1533062"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,10,6]]},"references-count":28,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2023,10,31]]}},"alternative-id":["10.1137\/22M1533062"],"URL":"https:\/\/doi.org\/10.1137\/22m1533062","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,10,6]]}}}