Foundations of Computational Mathematics

, Volume 15, Issue 2, pp 313–362 | Cite as

A Milstein Scheme for SPDEs

Article

Abstract

This article studies an infinite-dimensional analog of Milstein’s scheme for finite-dimensional stochastic ordinary differential equations (SODEs). The Milstein scheme is known to be impressively efficient for SODEs which fulfill a certain commutativity type condition. This article introduces the infinite-dimensional analog of this commutativity type condition and observes that a certain class of semilinear stochastic partial differential equation (SPDEs) with multiplicative trace class noise naturally fulfills the resulting infinite-dimensional commutativity condition. In particular, a suitable infinite-dimensional analog of Milstein’s algorithm can be simulated efficiently for such SPDEs and requires less computational operations and random variables than previously considered algorithms for simulating such SPDEs. The analysis is supported by numerical results for a stochastic heat equation, stochastic reaction diffusion equations and a stochastic Burgers equation, showing significant computational savings.

Keywords

Milstein scheme Stochastic partial differential equation SPDE Stochastic differential equation SDE  Numerical approximation Higher-order approximation 

Mathematics Subject Classification

60H35 65C30 

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© SFoCM 2015

Authors and Affiliations

  1. 1.Seminar for Applied Mathematics, Department of MathematicsETH ZurichZurichSwitzerland
  2. 2.Faculty of MathematicsBielefeld UniversityBielefeldGermany

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