Abstract
The numerical approximation of the solution to a stochastic partial differential equation with additive spatial white noise on a bounded domain is considered. The differential operator is assumed to be a fractional power of an integer order elliptic differential operator. The solution is approximated by means of a finite element discretization in space and a quadrature approximation of an integral representation of the fractional inverse from the Dunford–Taylor calculus. For the resulting approximation, a concise analysis of the weak error is performed. Specifically, for the class of twice continuously Fréchet differentiable functionals with second derivatives of polynomial growth, an explicit rate of weak convergence is derived, and it is shown that the component of the convergence rate stemming from the stochasticity is doubled compared to the corresponding strong rate. Numerical experiments for different functionals validate the theoretical results.
Highlights
The representation of Gaussian random fields as solutions to stochastic partial differential equations (SPDEs) has become a popular approach in spatial statistics in recent years. It was observed already in [21] and [22] that a Gaussian random field u on Rd with a covariance function of Matérn type [13] solves an SPDE of the form (κ2 − Δ)β u = W
In [4] we showed that this restriction can be avoided by combining a finite element discretization in space with a quadrature approximation based on an integral representation of the inverse fractional power operator from the Dunford–Taylor calculus
Gaussian random fields are of great importance as models in spatial statistics
Summary
The representation of Gaussian random fields as solutions to stochastic partial differential equations (SPDEs) has become a popular approach in spatial statistics in recent years. In [12] it was shown that by restricting the value of β to 2β ∈ N and by solving the stochastic problem (1.1) by means of a finite element method, the computational costs of many operations, which are needed for statistical inference, such as sampling and likelihood evaluations can be significantly reduced This decrease in computing time is one of the main reasons for the popularity of the SPDE approach in spatial statistics. Fréchet differentiable real-valued functions φ, whose second derivatives are of polynomial growth Functions of this form occur in many applications, e.g., when integral means of the solution with respect to a certain subdomain of D are of interest, or when a transformation of the model is used as a component in a hierarchical model.
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