Localization errors in solving stochastic partial differential equations in the whole space
Provides a theoretical error bound for localizing SPDEs, enabling finite-dimensional approximation for practitioners solving SPDEs on unbounded domains.
The authors localize Cauchy problems for SPDEs on the whole space to a ball of radius R, proving the error is exponentially small. They present a fully implementable numerical scheme combining localization with space and time discretization.
Cauchy problems with SPDEs on the whole space are localized to Cauchy problems on a ball of radius $R$. This localization reduces various kinds of spatial approximation schemes to finite dimensional problems. The error is shown to be exponentially small. As an application, a numerical scheme is presented which combines the localization and the space and time discretisation, and thus is fully implementable.