Multilevel Monte Carlo method for parabolic stochastic partial differential equations

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• 作者：Andrea Barth (1)
  Annika Lang (1)
  Christoph Schwab (1)
  
• 关键词：Multilevel Monte Carlo ; Stochastic partial differential equations ; Stochastic Finite Element Methods ; Stochastic parabolic equation ; Multilevel approximations ; 60H15 ; 60H35 ; 65C30 ; 41A25 ; 65C05 ; 65N30
• 刊名：BIT Numerical Mathematics
• 出版年：2013
• 出版时间：March 2013
• 年：2013
• 卷：53
• 期：1
• 页码：3-27
• 全文大小：775KB
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• 作者单位：Andrea Barth (1)
  Annika Lang (1)
  Christoph Schwab (1)
  
  1. Seminar für Angewandte Mathematik, ETH, R?mistrasse 101, 8092, Zürich, Switzerland
  
• ISSN：1572-9125

文摘

We analyze the convergence and complexity of multilevel Monte Carlo discretizations of a class of abstract stochastic, parabolic equations driven by square integrable martingales. We show under low regularity assumptions on the solution that the judicious combination of low order Galerkin discretizations in space and an Euler–Maruyama discretization in time yields mean square convergence of order one in space and of order?1/2 in time to the expected value of the mild solution. The complexity of the multilevel estimator is shown to scale log-linearly with respect to the corresponding work to generate a single path of the solution on the finest mesh, resp. of the corresponding deterministic parabolic problem on the finest mesh.