A Milstein scheme for SPDEs
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 and stochastic reaction diffusion equations showing signifficant computational savings.
 Publication:

arXiv eprints
 Pub Date:
 January 2010
 arXiv:
 arXiv:1001.2751
 Bibcode:
 2010arXiv1001.2751J
 Keywords:

 Mathematics  Numerical Analysis;
 Mathematics  Analysis of PDEs;
 Mathematics  Probability;
 65C30;
 60H35;
 60H15;
 35R60
 EPrint:
 The article is slightly revised and shortened. In particular, some numerical simulations are removed