Simulation of Stochastic Partial Differential Equations and Stochastic Active Contours

Abstract

This thesis discusses several aspects of the simulation of stochastic partial differential equations. First, two fast algorithms for the approximation of infinite dimensional Gaussian random fields with given covariance are introduced. Later Hilbert space-valued Wiener processes are constructed out of these random fields. A short introduction to infinite-dimensional stochastic analysis and stochastic differential equations is given. Furthermore different definitions of numerical stability for the discretization of stochastic partial differential equations are presented and the numerical stability of the heat equation with additive and multiplicative noise is explicitely computed using semigroup theory. Finally stochastic active contours are used for segmentation. This thesis generalizes work done by Juan et al. and does the simulation of different stochastic partial differential equations. The results are compared to equations without stochastics