Barth, Andrea; Lang, Annika Multilevel Monte Carlo method with applications to stochastic partial differential equations. (English) Zbl 1270.65003 Int. J. Comput. Math. 89, No. 18, 2479-2498 (2012). To approximate the expectation (or generally a functional) of a solution \(Y\) of a stochastic partial differential equation, taking as a Hilbert valued random variable, the multilevel Monte Carlo method is: by using a finite-dimensional subspace sequence of random variables \(Y_{\ell}\) to approximate \(Y\) first, then to take a sample of size of \(N_{\ell}\) of \(Y_{\ell}-Y_{{\ell}-1}\) to form a Monte Carlo estimate of \(E(Y_{\ell}-Y_{{\ell}-1})\), and finally to sum to get the \(\ell\)-level Monte Carlo estimate of \(E(Y_{\ell})\). Here \(N_{\ell}\) is chosen balanced to different levels to make the approximate rate better. In the present paper, the finite element approximation of the heat equation with white noise and the linearized backward Euler scheme of a first-order hyperbolic equation with white noise are treated to obtain better rates. Numerical examples are illustrated. Reviewer: Gong Guanglu (Beijing) Cited in 30 Documents MSC: 65C30 Numerical solutions to stochastic differential and integral equations 65C05 Monte Carlo methods 60G60 Random fields 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 35R60 PDEs with randomness, stochastic partial differential equations 35K05 Heat equation 35L02 First-order hyperbolic equations 60H35 Computational methods for stochastic equations (aspects of stochastic analysis) Keywords:multilevel Monte Carlo method; stochastic partial differential equation; stochastic finite element methods; stochastic parabolic equation; multilevel approximation; heat equation; linearized backward Euler scheme; first-order hyperbolic equation; numerical examples PDFBibTeX XMLCite \textit{A. Barth} and \textit{A. Lang}, Int. J. Comput. Math. 89, No. 18, 2479--2498 (2012; Zbl 1270.65003) Full Text: DOI References: [1] Barth A., Comm. Stoch. Anal. 4 pp 355– (2010) [2] Barth A., Stochastics 84 pp 217– (2012) [3] DOI: 10.1007/s00211-011-0377-0 · Zbl 1230.65006 · doi:10.1007/s00211-011-0377-0 [4] DOI: 10.1016/j.cma.2004.01.026 · Zbl 1067.76563 · doi:10.1016/j.cma.2004.01.026 [5] DOI: 10.1007/s00791-011-0160-x · Zbl 1241.65012 · doi:10.1007/s00791-011-0160-x [6] DOI: 10.1017/CBO9780511666223 · doi:10.1017/CBO9780511666223 [7] DOI: 10.1016/j.spa.2011.03.015 · Zbl 1234.60067 · doi:10.1016/j.spa.2011.03.015 [8] Engel K.-J., One-Parameter Semigroups for Linear Evolution Equations (2000) [9] DOI: 10.1287/opre.1070.0496 · Zbl 1167.65316 · doi:10.1287/opre.1070.0496 [10] Giles M. B., SIAM J. Financ. Math. (2012) [11] DOI: 10.1016/j.cam.2010.08.011 · Zbl 1208.65017 · doi:10.1016/j.cam.2010.08.011 [12] DOI: 10.1007/s10543-011-0344-2 · Zbl 1242.65010 · doi:10.1007/s10543-011-0344-2 [13] DOI: 10.1515/mcma.2010.007 · Zbl 1197.65012 · doi:10.1515/mcma.2010.007 [14] DOI: 10.1007/978-1-4612-5561-1 · Zbl 0516.47023 · doi:10.1007/978-1-4612-5561-1 [15] DOI: 10.1017/CBO9780511721373 · Zbl 1205.60122 · doi:10.1017/CBO9780511721373 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.