Multilevel Monte Carlo method with applications to stochastic partial differential equations

References

• Barth , A. 2010 . A finite element method for martingale-driven stochastic partial differential equations . Comm. Stoch. Anal. , 4 : 355 – 375 .
   Google Scholar
• A. Barth and A. Lang, Milstein approximation for advection–diffusion equations driven by multiplicative noncontinuous martingale noises, Seminar für Angewandte Mathematik (SAM)-Report 2011-36, ETH Zürich, preprint (2011), submitted for publication.
   Google Scholar
• Barth , A. and Lang , A. 2012 . Simulation of stochastic partial differential equations using finite element methods . Stochastics , 84 : 217 – 231 .
   Google Scholar
• A. Barth, A. Lang, and C. Schwab, Multilevel Monte Carlo finite element method for parabolic stochastic partial differential equations, Seminar für Angewandte Mathematik (SAM)-Report 2011-30, ETH Zürich, preprint (2011), submitted for publication.
   Google Scholar
• Barth , A. , Schwab , C. and Zollinger , N. 2011 . Multi-level Monte Carlo finite element method for elliptic PDEs with stochastic coefficients . Numer. Math. , 119 : 123 – 161 . (doi:10.1007/s00211-011-0377-0)
   Web of Science ®Google Scholar
• Bochev , P. , Gunzburger , M. and Shadid , J. 2004 . Stability of the SUPG finite element method for transient advection–diffusion problems . Comput. Meth. Appl. Mech. Eng. , 193 : 2301 – 2323 . (doi:10.1016/j.cma.2004.01.026)
   Web of Science ®Google Scholar
• Cliffe , K. , Giles , M. , Scheichl , R. and Teckentrup , A. 2011 . Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients . Comput. Vis. Sci. , 14 : 3 – 15 . (doi:10.1007/s00791-011-0160-x)
   Google Scholar
• Da Prato , G. and Zabczyk , J. 1992 . Stochastic Equations in Infinite Dimensions , Cambridge : Cambridge University Press . Vol. 44 of Encyclopedia of Mathematics and Its Applications
   Google Scholar
• Dereich , S. and Heidenreich , F. 2011 . A multilevel Monte Carlo algorithm for Lévy-driven stochastic differential equations . Stochastic Process. Appl. , 121 : 1565 – 1587 . (doi:10.1016/j.spa.2011.03.015)
   Web of Science ®Google Scholar
• Engel , K.-J. and Nagel , R. 2000 . One-Parameter Semigroups for Linear Evolution Equations , Berlin : Springer . Graduate Texts in Mathematics Vol. 194
   Google Scholar
• Giles , M. B. 2008 . Multilevel Monte Carlo path simulation . Oper. Res. , 56 : 607 – 617 . (doi:10.1287/opre.1070.0496)
   Web of Science ®Google Scholar
• Giles , M. B. and Reisinger , C. 2012 . Stochastic finite differences and multilevel Monte Carlo for a class of SPDEs in finance . SIAM J. Financ. Math. , accepted for publication
   PubMed Web of Science ®Google Scholar
• S. Heinrich, Multilevel Monte Carlo methods, Large-Scale Scientific Computing, Third International Conference, LSSC 2001, Sozopol, Bulgaria, June 6–10, 2001, Revised Papers, S. Margenov, J. Wasniewski, and P. Y. Yalamov, eds., Vol. 2179 of Lecture Notes in Computer Science, Springer, Berlin, 2001, pp. 58–67.
   Google Scholar
• H. Hoel, E. von Schwerin, A. Szepessy, and R. Tempone, Adaptive multilevel Monte Carlo simulation, in Numerical Analysis of Multiscale Computations, B. Engquist, O. Runborg, and Y.-H.R. Tsai, eds., Vol. 82 of Lecture Notes in Computational Science and Engineering, Springer, Berlin, Heidelberg, 2012, pp. 217–234.
   Google Scholar
• Hutzenthaler , M. , Jentzen , A. and Kloeden , P. E. Divergence of the multilevel Monte Carlo method . May 2011. Available at arXiv: math.PR.1105.0226
   Google Scholar
• Kloeden , P. , Lord , G. , Neuenkirch , A. and Shardlow , T. 2011 . The exponential integrator scheme for stochastic partial differential equations: Pathwise error bounds . J. Comput. Appl. Math. , 235 : 1245 – 1260 . (doi:10.1016/j.cam.2010.08.011)
   Web of Science ®Google Scholar
• Kovács , M. , Larsson , S. and Lindgren , F. 2012 . Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise . BIT Numer. Math , 52 : 85 – 108 . (doi:10.1007/s10543-011-0344-2)
   Web of Science ®Google Scholar
• Marxen , H. 2010 . The multilevel Monte Carlo method used on a Lévy driven SDE . Monte Carlo Methods Appl. , 16 : 167 – 190 . (doi:10.1515/mcma.2010.007)
   Google Scholar
• Mishra , S. and Schwab , C. Sparse tensor multi-level Monte Carlo finite volume methods for hyperbolic conservation laws with random initial data, Math. Comp. (2012). Available at http://www.ams.org/journals/mcom/0000-000-00/S0025-5718-2012-02574-9/.
   Google Scholar
• Pazy , A. 1983 . Semigroups of Linear Operators and Applications to Partial Differential Equations , New York, Berlin, Heidelberg , , Tokyo : Springer . Applied Mathematical Sciences Vol. 44
   Google Scholar
• Peszat , S. and Zabczyk , J. 2007 . Stochastic Partial Differential Equations with Lévy Noise , Cambridge : Cambridge University Press . Vol. 113 of Encyclopedia of Mathematics and Its Applications
   Google Scholar