Fractional stochastic PDEs: From statistical applications to numerical analysis

16.06.2021, 14:00 - 15:45  –  Online Kolloquium
Institutskolloquium

David Bolin (King Abdullah University of Science and Technology, Saudi Arabia) and Mihály Kovács (Pázmány Péter Catholic University, Hungary and Chalmers Uni/University of Gothenburg, Sweden)

2pm-2:45pm David Bolin (King Abdullah University of Science and Technology, Saudi Arabia): Statistical models and methods based on fractional stochastic PDEs

3pm-3:45 pm Mihály Kovács (Pázmány Péter Catholic University, Hungary and Chalmers University/University of Gothenburg, Sweden): Numerical solution of fractional stochastic PDEs

Abstracts:

David Bolin (King Abdullah University of Science and Technology, Saudi Arabia):Statistical models and methods based on fractional stochastic PDEs

During the last ten years, the SPDE approach to statistical modeling has become highly popular in spatial statistics, a branch of statistics which analyses geo-referenced or spatial data. The SPDE approach is based on the representation of Gaussian random fields as solutions to stochastic partial differential equations (SPDEs) involving a fractional power of a second-order differential operator, the power determining the smoothness of the solution.

In this talk we review this method and its recent developments and extensions. We shall focus on two recent models motivated by applications in environmental statistics and longitudinal data analysis.

Mihály Kovács (Pázmány Péter Catholic University, Hungary and Chalmers University/University of Gothenburg, Sweden): Numerical solution of fractional stochastic PDEs

A stationary fractional stochastic partial differential equation involves a fractional power of an integer order elliptic differential operator which makes the solution not directly accessible. In this talk we discuss numerical approximation of solutions to such fractional stochastic PDEs with additive spatial white noise on a bounded domain. The inverse operator is represented by a Bochner integral in the Dunford--Taylor functional calculus. By applying a quadrature formula to this integral representation, the inverse fractional power is approximated by a weighted sum of non-fractional resolvents evaluated at appropriate quadrature points. This yields standard integer order problems and the resolvents are then discretized in space by a continuous finite element method. This approach is combined with an approximation of the white noise based on the mass matrix of the finite element discretization. In this way an efficient numerical algorithm for computing samples of the approximate solution is obtained. The method is particularly interesting for real-world applications in spatial statistics, where fractional order stochastic partial differential equations with spatial white noise play an important role owing to their connection to Gaussian Matérn fields.

This is a joint work with D. Bolin (Kaust) and K. Kirchner (Delft).

If you wish to attend the talks,  please contact Sylvie Paycha paycha@math.uni-potsdam.de for the login details.