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A Milstein scheme for SPDEs. (English) Zbl 1318.60072

Summary: This article studies an infinite-dimensional analogue 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 analogue 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 analogue 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.

MSC:

60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
65C30 Numerical solutions to stochastic differential and integral equations
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