Computational Aspects of Lévy-Driven SPDE Approximations

Licentiate thesis, 2017

In order to simulate solutions to stochastic partial differential equations (SPDE) they must be approximated in space and time. In this thesis such fully discrete approximations are considered, with an emphasis on finite element methods combined with rational semigroup approximations. There are several notions of the error resulting from this. One of them is the weak error, measured in terms of the mean of a functional applied to the solution. To approximate the mean, one typically employs Monte Carlo and multilevel Monte Carlo methods that are based on generating a large number of realizations of the approximate solution to the SPDE. The thesis consists of two papers. In Paper 1 the additional error caused by Monte Carlo and multilevel Monte Carlo methods when one attempts to simulate the weak error is analysed Upper and lower bounds are derived for the different methods and simulations illustrate the results. When using multilevel Monte Carlo methods to estimate the weak error, along with other properties of the SPDE, it is important that the discretizations used are sufficiently stable in a mean square sense. In Paper 2 a framework for the analysis of the asymptotic mean square stability of a general stochastic recursion scheme is set up. This framework is then applied to several discretizations of an SPDE, which results in a series of sufficient conditions for stability. Some of these results are found to be sharp in simulations.