{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T13:31:48Z","timestamp":1787232708654,"version":"build-2736575974"},"reference-count":49,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Multiscale Model. Simul."],"published-print":{"date-parts":[[2013,1]]},"abstract":"<jats:p>In this paper Monte Carlo finite element approximations for elliptic homogenization problems with random coefficients, which oscillate on $n\\in\\mathbb{N}$ a priori known, separated microscopic length scales, are considered. The convergence of multilevel Monte Carlo finite element discretizations is analyzed. In particular, it is considered that the multilevel finite element discretization resolves the finest physical length scale, but the coarsest finite element mesh does not, so that the so-called resonance case occurs at intermediate multilevel Monte Carlo sampling levels. It is shown that for first order finite elements in two space dimensions, the multilevel Monte Carlo finite element method converges at the same rate as the corresponding single-level Monte Carlo finite element method, despite the majority of samples being underresolved in the multilevel Monte Carlo finite element estimator. It is proved that switching to a hierarchic multiscale finite element method such as the finite element heterogeneous multiscale method to compute the multilevel Monte Carlo finite element estimator, when only meshes are used which underresolve all physical length scales, implies optimal convergence. Specifically, both methods proposed here allow one to obtain estimates of the expectation of the random solution, with accuracy versus work that is identical to the solution of a single deterministic problem. In the case of the finite element heterogeneous multiscale method the estimate is, moreover, robust with respect to the physical length scales. Numerical experiments corroborate our analytical findings.<\/jats:p>","DOI":"10.1137\/120894725","type":"journal-article","created":{"date-parts":[[2013,10,22]],"date-time":"2013-10-22T15:40:22Z","timestamp":1382456422000},"page":"1033-1070","source":"Crossref","is-referenced-by-count":30,"title":["Multilevel Monte Carlo Methods for Stochastic Elliptic Multiscale PDEs"],"prefix":"10.1137","volume":"11","author":[{"given":"Assyr","family":"Abdulle","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Andrea","family":"Barth","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Christoph","family":"Schwab","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2013,10,22]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1137\/040607137"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1142\/S0218202506001285"},{"key":"atypb3","first-page":"135","author":"Abdulle A.","year":"2009","journal-title":"Tokyo"},{"key":"atypb4","first-page":"280","author":"Abdulle A.","year":"2011","journal-title":"Beijing"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-2011-02527-5"},{"key":"atypb6","unstructured":"A. Abdulle and Y. Bai,\n                      Fully Discrete Analysis of the Heterogeneous Multiscale Method for Elliptic Problems with Multiple Scales\n                      , Mathicse Report 47.2012."},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1017\/S0962492912000025"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2009.03.019"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1137\/030600771"},{"key":"atypb10","first-page":"355","volume":"4","author":"Barth A.","year":"2010","journal-title":"Commun. Stoch. Anal."},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1080\/17442508.2010.523466"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1016\/j.spa.2013.01.003"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-011-0377-0"},{"key":"atypb14","unstructured":"A. Bensoussan, J.L. Lions, and G. Papanicolaou,\n                      Asymptotic Analysis for Periodic Structures\n                      , Stud. Math. 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Math. 40, SIAM, Philadelphia, 2002.","DOI":"10.1137\/1.9780898719208"},{"key":"atypb20","doi-asserted-by":"crossref","unstructured":"P. Ciarlet and P. A. Raviart,\n                      The combined effect of curved boundaries and numerical integration in isoparametric finite element methods\n                      , in The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations (Proc. Sympos., Univ. Maryland, Baltimore, MD, 1972), Academic Press, New York, 1972, pp. 409-474.","DOI":"10.1016\/B978-0-12-068650-6.50020-4"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1007\/s10208-010-9072-2"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1142\/S0219530511001728"},{"key":"atypb23","doi-asserted-by":"crossref","unstructured":"G. Da Prato and J. Zabczyk,\n                      Stochastic Equations in Infinite Dimensions\n                      , Encyclopedia Math. Appl. 44, Cambridge University Press, Cambridge, UK, 1992.","DOI":"10.1017\/CBO9780511666223"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1145\/992200.992206"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1145\/992200.992205"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1145\/305658.287640"},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.4310\/CMS.2003.v1.n1.a8"},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1090\/S0894-0347-04-00469-2"},{"key":"atypb29","first-page":"97","author":"Gabriel E.","year":"2004","journal-title":"Hungary"},{"key":"atypb30","doi-asserted-by":"publisher","DOI":"10.1007\/s10543-009-0227-y"},{"key":"atypb31","unstructured":"M. B. Giles,\n                      Improved Multilevel Monte Carlo Convergence Using the Milstein Scheme\n                      , Preprint NA-06\/22, Oxford University Computing Laboratory, Parks Road, Oxford, UK, 2006."},{"key":"atypb32","doi-asserted-by":"publisher","DOI":"10.1287\/opre.1070.0496"},{"key":"atypb33","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-013-0537-5"},{"key":"atypb34","unstructured":"P. Grisvard,\n                      Elliptic Problems in Nonsmooth Domains\n                      , Monogr. Stud. Math. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985."},{"key":"atypb35","doi-asserted-by":"crossref","unstructured":"S. Heinrich,\n                      Multilevel Monte Carlo methods\n                      , in Large-Scale Scientific Computing, Third International Conference, LSSC 2001 (Sozopol, Bulgaria, 2001), Revised Papers, Lecture Notes in Comput. Sci. 2179, S. Margenov, J. Wasniewski, and P. Y. 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