{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,18]],"date-time":"2026-09-18T08:59:43Z","timestamp":1789721983691,"version":"4.0.1"},"reference-count":39,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","funder":[{"DOI":"10.13039\/501100001809","name":"National Science Foundation of China","doi-asserted-by":"publisher","award":["11971414"],"award-info":[{"award-number":["11971414"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Hunan National Applied Mathematics Center","award":["2020ZYT003"],"award-info":[{"award-number":["2020ZYT003"]}]},{"DOI":"10.13039\/501100001809","name":"National Science Foundation of China","doi-asserted-by":"publisher","award":["12171080"],"award-info":[{"award-number":["12171080"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Science Foundation of China","doi-asserted-by":"publisher","award":["12292982"],"award-info":[{"award-number":["12292982"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100007847","name":"Natural Science Foundation of Jilin Province","doi-asserted-by":"publisher","award":["JC010284408"],"award-info":[{"award-number":["JC010284408"]}],"id":[{"id":"10.13039\/100007847","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>We propose and analyze a mixed precision parareal algorithm that uses for the fine propagator [Formula: see text] and the coarse propagator [Formula: see text] a high precision [Formula: see text] and a low [Formula: see text], respectively. This paradigm potentially provides faster and more energy efficient coarse grid correction for parareal, compared to the original paradigm, which uses a uniform precision for both [Formula: see text] and [Formula: see text]. Low precision is also beneficial to reduce communication and memory costs, since we have to move and store fewer bits. Let [Formula: see text] and [Formula: see text] be, respectively, the decaying rate of the error of the parareal algorithm in the mixed and uniform precision modes. We perform a convergence analysis for the mixed precision parareal algorithm. The derived convergence rates are dependent on the precision of the coarse propagator and the number of coarse time steps [Formula: see text]. Numerical results indicate that the converged solution of the mixed precision parareal algorithm attains the desired high precision [Formula: see text], while degeneration of the convergence rate is indeed observed in some worst case scenario, such as the situation that we use a half precision for [Formula: see text] and [Formula: see text] is very large.<\/jats:p>","DOI":"10.1137\/22m1510169","type":"journal-article","created":{"date-parts":[[2023,9,22]],"date-time":"2023-09-22T07:03:45Z","timestamp":1695366225000},"page":"A2483-A2510","source":"Crossref","is-referenced-by-count":5,"title":["Convergence Analysis of a Mixed Precision Parareal Algorithm"],"prefix":"10.1137","volume":"45","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0097-7017","authenticated-orcid":true,"given":"Xiaoqiang","family":"Yue","sequence":"first","affiliation":[{"name":"National Center for Applied Mathematics in Hunan, Key Laboratory of Intelligent Computing & Information Processing of Ministry of Education, Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Xiangtan, 411105, China."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zhiyong","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 611731, China."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Shu-Lin","family":"Wu","sequence":"additional","affiliation":[{"name":"Corresponding author. School of Mathematics and Statistics, Northeast Normal University, Changchun, 130024, China."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2023,9,22]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.1017\/S0962492916000015"},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1177\/10943420211003313"},{"key":"ref3","unstructured":"Advanpix, Multiprecision Computing Toolbox, http:\/\/www.advanpix.com."},{"key":"ref4","volume-title":"Five-Precision GMRES-Based Iterative Refinement, MIMS EPrint 2021.5","author":"Amestoy P.","year":"2021"},{"key":"ref5","unstructured":"J. Angel , \nS. G\u00f6tschel , and \nD. Ruprecht  , Impact of Spatial Coarsening on Parareal Convergence, http:\/\/arxiv.org\/abs\/2111.10228v1, 2021."},{"key":"ref6","doi-asserted-by":"publisher","DOI":"10.1137\/19M1257780"},{"key":"ref7","doi-asserted-by":"publisher","DOI":"10.1137\/17M1140819"},{"key":"ref8","doi-asserted-by":"publisher","DOI":"10.1137\/17M1122918"},{"key":"ref9","unstructured":"E. Carson  and \nN. Khan  , Mixed Precision Iterative Refinement with Sparse Approximate Inverse Preconditioning, https:\/\/arxiv.org\/abs\/2202.10204, 2022."},{"key":"ref10","doi-asserted-by":"publisher","DOI":"10.1137\/20M1334796"},{"key":"ref11","doi-asserted-by":"publisher","DOI":"10.1137\/110861002"},{"key":"ref12","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0121009"},{"key":"ref13","doi-asserted-by":"publisher","DOI":"10.2140\/camcos.2012.7.105"},{"key":"ref14","doi-asserted-by":"publisher","DOI":"10.1137\/130944230"},{"key":"ref15","doi-asserted-by":"publisher","DOI":"10.1002\/nme.860"},{"key":"ref16","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-23321-5_3"},{"key":"ref17","doi-asserted-by":"publisher","DOI":"10.1137\/05064607X"},{"key":"ref18","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-019-01060-8"},{"key":"ref19","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-75199-1_4"},{"key":"ref20","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898718027"},{"key":"ref21","doi-asserted-by":"publisher","DOI":"10.1017\/S0962492922000022"},{"key":"ref22","doi-asserted-by":"crossref","unstructured":"A. Haidar , \nS. Tomov , \nJ. Dongarra , and \nN. J. Higham  , Harnessing GPU tensor cores for fast FP16 arithmetic to speed up mixed-precision iterative refinement solvers, in Proceedings of the International Conference for High Performance Computing, Networking, Storage, and Analysis, SC\u201918, IEEE Press, Piscataway, NJ, 2018, pp. 603\u2013613, https:\/\/doi.org\/10.1109\/SC.2018.00050.","DOI":"10.1109\/SC.2018.00050"},{"key":"ref23","doi-asserted-by":"crossref","unstructured":"A. Haidar , \nP. Wu , \nS. Tomov , and \nJ. Dongarra  , Investigating half precision arithmetic to accelerate dense linear system solvers, in Proceedings of the 8th Workshop on Latest Advances in Scalable Algorithms for Large-Scale Systems, ScalA\u201917, ACM, New York, 2017, pp. 1\u20138, https:\/\/doi.org\/10.1145\/3148226.3148237.","DOI":"10.1145\/3148226.3148237"},{"key":"ref24","doi-asserted-by":"publisher","DOI":"10.1137\/17M1144982"},{"key":"ref25","doi-asserted-by":"crossref","unstructured":"J. Langou , \nJ. Langou , \nP. Luszczek , \nJ. Kurzak , \nA. Buttari , and \nJ. Dongarra  , Exploiting the performance of 32 bit floating point arithmetic in obtaining 64 bit accuracy (revisiting iterative refinement for linear systems), in Proceedings of the ACM\/IEEE Conference on Supercomputing, 2006, https:\/\/doi.org\/10.1109\/SC.2006.30.","DOI":"10.1145\/1188455.1188573"},{"key":"ref26","first-page":"661","volume":"332","author":"Lions J.-L.","year":"2001","journal-title":"C. R. Math."},{"key":"ref27","doi-asserted-by":"publisher","DOI":"10.1137\/19M1309869"},{"key":"ref28","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2020.112915"},{"key":"ref29","doi-asserted-by":"publisher","DOI":"10.1137\/17M1149225"},{"key":"ref30","doi-asserted-by":"publisher","DOI":"10.1137\/16M1062016"},{"key":"ref31","doi-asserted-by":"publisher","DOI":"10.1137\/20M1348571"},{"key":"ref32","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2019.109156"},{"key":"ref33","doi-asserted-by":"publisher","DOI":"10.1007\/s00791-020-00331-4"},{"key":"ref34","first-page":"1","volume":"356","author":"Southworth B. S.","year":"2021","journal-title":"Springer Proc. Math. Stat."},{"key":"ref35","doi-asserted-by":"publisher","DOI":"10.1137\/20M1349230"},{"key":"ref36","doi-asserted-by":"publisher","DOI":"10.1137\/140970756"},{"key":"ref37","doi-asserted-by":"publisher","DOI":"10.1137\/17M1141102"},{"key":"ref38","doi-asserted-by":"publisher","DOI":"10.1093\/imanum\/dru031"},{"key":"ref39","doi-asserted-by":"publisher","DOI":"10.1007\/s10915-015-0100-x"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/22M1510169","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:46:26Z","timestamp":1787327186000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/22M1510169"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,9,22]]},"references-count":39,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2023,10,31]]}},"alternative-id":["10.1137\/22M1510169"],"URL":"https:\/\/doi.org\/10.1137\/22m1510169","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,9,22]]}}}