Dörsek, Philipp; Teichmann, Josef; Velušček, Dejan Cubature methods for stochastic (partial) differential equations in weighted spaces. (English) Zbl 1315.60075 Stoch. Partial Differ. Equ., Anal. Comput. 1, No. 4, 634-663 (2013). Summary: The cubature on Wiener space method, a high-order weak approximation scheme, is established for SPDEs in the case of unbounded characteristics and unbounded test functions. We first describe a recently introduced flexible functional analytic framework, so called weighted spaces, where Feller-like properties hold. A refined analysis of vector fields on weighted spaces then yields optimal convergence rates of cubature methods for stochastic partial differential equations of Da Prato-Zabczyk type. The ubiquitous stability for the local approximation operator within the functional analytic setting is proved for SPDEs, however, in the infinite dimensional case we need a newly introduced technical assumption on weak symmetry of the cubature formula. Computational results for a cubature discretization of a spatially extended stochastic FitzHugh-Nagumo model, an SPDE model from mathematical biology, are shown, illustrating the applicability of our theory. Cited in 6 Documents MSC: 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 60H35 Computational methods for stochastic equations (aspects of stochastic analysis) 65C30 Numerical solutions to stochastic differential and integral equations 65C35 Stochastic particle methods 46N30 Applications of functional analysis in probability theory and statistics Keywords:stochastic partial differential equations; cubature methods; Wiener space; higher order weak approximation scheme Software:sobol.cc PDFBibTeX XMLCite \textit{P. Dörsek} et al., Stoch. Partial Differ. Equ., Anal. 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