Weak convergence of Galerkin approximations for fractional elliptic stochastic PDEs with spatial white noise
Journal article, 2018

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.

Matérn covariances

Finite element methods

Galerkin methods

Spatial statistics

Fractional operators

Stochastic partial differential equations

Weak convergence

Gaussian white noise

Author

David Bolin

Chalmers, Mathematical Sciences, Applied Mathematics and Statistics

Kristin Kirchner

Chalmers, Mathematical Sciences, Applied Mathematics and Statistics

Mihaly Kovacs

Chalmers, Mathematical Sciences, Applied Mathematics and Statistics

BIT (Copenhagen)

0006-3835 (ISSN)

Vol. 58 4 881-906

Latent jump fields for spatial statistics

Swedish Research Council (VR), 2017-01-01 -- 2020-12-31.

Subject Categories

Computational Mathematics

Probability Theory and Statistics

Mathematical Analysis

DOI

10.1007/s10543-018-0719-8

More information

Latest update

12/19/2018