{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T16:24:22Z","timestamp":1772295862219,"version":"3.50.1"},"reference-count":0,"publisher":"Walter de Gruyter GmbH","issue":"1","funder":[{"name":"Royal Institute of Technology in Stockholm","award":["Dahlquist fellowship"],"award-info":[{"award-number":["Dahlquist fellowship"]}]},{"name":"Department of Scientific Computing in Florida State University"},{"name":"University of Austin Subcontract","award":["024550"],"award-info":[{"award-number":["024550"]}]},{"name":"VR project","award":["\u201cEffektiva numeriska metoder f\u00f6r stokastiska differentialekvationer med till\u00e4mpningar\u201d"],"award-info":[{"award-number":["\u201cEffektiva numeriska metoder f\u00f6r stokastiska differentialekvationer med till\u00e4mpningar\u201d"]}]},{"name":"Center for Industrial and Applied Mathematics (CIAM) at the Royal Institute of Technology"},{"DOI":"10.13039\/501100004052","name":"King Abdullah University of Science and Technology","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100004052","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2014,3,1]]},"abstract":"Abstract.<\/jats:title>\n We present an adaptive multilevel Monte Carlo (MLMC) method for\nweak approximations of solutions to It\u00f4 stochastic\ndifferential equations (SDE). The work [Oper. Res. 56 (2008), 607\u2013617]\nproposed and analyzed an MLMC method based on a hierarchy of uniform time\ndiscretizations and control variates to reduce the computational\neffort required by a single level Euler\u2013Maruyama Monte\nCarlo method from \n \n \n \n \n \ud835\udcaa<\/m:mi>\n (<\/m:mo>\n \n \n TOL <\/m:mi>\n <\/m:mrow>\n \n -<\/m:mo>\n 3<\/m:mn>\n <\/m:mrow>\n <\/m:msup>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:math>\n ${{{\\mathcal {O}}({\\mathrm {TOL}}^{-3})}}$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> to\n\n \n \n \n \n \ud835\udcaa<\/m:mi>\n (<\/m:mo>\n \n \n TOL <\/m:mi>\n <\/m:mrow>\n \n -<\/m:mo>\n 2<\/m:mn>\n <\/m:mrow>\n <\/m:msup>\n log<\/m:mo>\n \n \n (<\/m:mo>\n \n \n TOL <\/m:mi>\n <\/m:mrow>\n \n -<\/m:mo>\n 1<\/m:mn>\n <\/m:mrow>\n <\/m:msup>\n )<\/m:mo>\n <\/m:mrow>\n 2<\/m:mn>\n <\/m:msup>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:math>\n ${{{\\mathcal {O}}({\\mathrm {TOL}}^{-2}\\log ({\\mathrm {TOL}}^{-1})^{2})}}$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> for a mean\nsquare error of \n \n \n \n \n \ud835\udcaa<\/m:mi>\n (<\/m:mo>\n \n \n TOL <\/m:mi>\n <\/m:mrow>\n 2<\/m:mn>\n <\/m:msup>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:math>\n ${{{\\mathcal {O}}({\\mathrm {TOL}}^2)}}$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>.\nLater, the work [Lect. Notes Comput. Sci. Eng. 82, Springer-Verlag, Berlin (2012), 217\u2013234] presented an MLMC method using a hierarchy of adaptively\nrefined, non-uniform time discretizations, and, as such, it may be\nconsidered a generalization of the uniform time discretization MLMC method.\nThis work improves the adaptive MLMC algorithms presented\nin [Lect. Notes Comput. Sci. Eng. 82, Springer-Verlag, Berlin (2012), 217\u2013234]\nand it also provides mathematical analysis of the improved algorithms.\nIn particular, we show that under some assumptions our\nadaptive MLMC algorithms are asymptotically accurate and\nessentially have the correct complexity\nbut with improved control of the complexity constant factor in the asymptotic\nanalysis.\nNumerical tests include one case with singular drift and\none with stopped diffusion, where the complexity of a uniform single\nlevel method is \n \n \n \n \n \ud835\udcaa<\/m:mi>\n (<\/m:mo>\n \n \n TOL <\/m:mi>\n <\/m:mrow>\n \n -<\/m:mo>\n 4<\/m:mn>\n <\/m:mrow>\n <\/m:msup>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:math>\n ${{{\\mathcal {O}}({\\mathrm {TOL}}^{-4})}}$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>. For both these cases the results\nconfirm the theory, exhibiting savings in the computational cost\nfor achieving the accuracy \n \n \n \n \n \ud835\udcaa<\/m:mi>\n (<\/m:mo>\n TOL <\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:math>\n ${{{\\mathcal {O}}({\\mathrm {TOL}})}}$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> from \n \n \n \n \n \ud835\udcaa<\/m:mi>\n (<\/m:mo>\n \n \n TOL <\/m:mi>\n <\/m:mrow>\n \n -<\/m:mo>\n 3<\/m:mn>\n <\/m:mrow>\n <\/m:msup>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:math>\n ${{{\\mathcal {O}}({\\mathrm {TOL}}^{-3})}}$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> for the\nadaptive single level algorithm to essentially\n\n \n \n \n \n \ud835\udcaa<\/m:mi>\n (<\/m:mo>\n \n \n TOL <\/m:mi>\n <\/m:mrow>\n \n -<\/m:mo>\n 2<\/m:mn>\n <\/m:mrow>\n <\/m:msup>\n log<\/m:mo>\n \n \n (<\/m:mo>\n \n \n TOL <\/m:mi>\n <\/m:mrow>\n \n -<\/m:mo>\n 1<\/m:mn>\n <\/m:mrow>\n <\/m:msup>\n )<\/m:mo>\n <\/m:mrow>\n 2<\/m:mn>\n <\/m:msup>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:math>\n ${{{\\mathcal {O}}({\\mathrm {TOL}}^{-2}\\log ({\\mathrm {TOL}}^{-1})^2)}}$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> for the adaptive\nMLMC algorithm.<\/jats:p>","DOI":"10.1515\/mcma-2013-0014","type":"journal-article","created":{"date-parts":[[2013,11,13]],"date-time":"2013-11-13T20:06:03Z","timestamp":1384373163000},"page":"1-41","source":"Crossref","is-referenced-by-count":35,"title":["Implementation and analysis of an adaptive multilevel Monte Carlo algorithm"],"prefix":"10.1515","volume":"20","author":[{"given":"H\u00e5kon","family":"Hoel","sequence":"first","affiliation":[{"name":"Applied Mathematics and Computational Sciences, KAUST, Thuwal, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Erik","family":"von Schwerin","sequence":"additional","affiliation":[{"name":"CSQI-MATHICSE, \u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne, Switzerland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Anders","family":"Szepessy","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Royal Institute of Technology (KTH), Stockholm, Sweden"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ra\u00fal","family":"Tempone","sequence":"additional","affiliation":[{"name":"Applied Mathematics and Computational Sciences, KAUST, Thuwal, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2013,11,13]]},"container-title":["mcma"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/mcma-2013-0014\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/mcma-2013-0014\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,4,1]],"date-time":"2023-04-01T22:12:18Z","timestamp":1680387138000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/mcma-2013-0014\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,11,13]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2014,3,1]]},"published-print":{"date-parts":[[2014,3,1]]}},"alternative-id":["10.1515\/mcma-2013-0014"],"URL":"https:\/\/doi.org\/10.1515\/mcma-2013-0014","relation":{},"ISSN":["0929-9629","1569-3961"],"issn-type":[{"value":"0929-9629","type":"print"},{"value":"1569-3961","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,11,13]]}}}