A Milstein Scheme for SPDEs

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 in...

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Bibliographic Details
Published inFoundations of computational mathematics Vol. 15; no. 2; pp. 313 - 362
Main Authors Jentzen, Arnulf, Röckner, Michael
Format Journal Article
LanguageEnglish
Published Boston Springer US 01.04.2015
Springer
Springer Nature B.V
Subjects
Algorithms
Analysis
Applications of Mathematics
Approximation
Burgers equation
Computation
Computational mathematics
Computer Science
Computer simulation
Differential equations
Differential equations, Partial
Economics
Heat equations
Linear and Multilinear Algebras
Math Applications in Computer Science
Mathematical analysis
Mathematical models
Mathematics
Mathematics and Statistics
Matrix Theory
Numerical Analysis
Stochastic differential equations
Stochastic models
Stochasticity
United States
Numerical approximation
65C30
SDE
SPDE
Stochastic partial differential equation
Milstein scheme
Higher-order approximation
Stochastic differential equation
60H35
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ISSN1615-3375
1615-3383
DOI10.1007/s10208-015-9247-y

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Summary: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.
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ISSN:1615-3375
1615-3383
DOI:10.1007/s10208-015-9247-y