  The cubature on Wiener space method, a high-order weak approximation scheme,
is established for SPDEs in the case of unbounded characteristics and unbounded
payoffs. We first introduce a recently described 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
assumption on weak symmetry of the cubature formula. In finite dimensions, we
use the UFG condition to obtain optimal rates of convergence on non-uniform
meshes for nonsmooth payoffs with exponential growth.
