Numerical solutions of stochastic partial differential equations
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The main part of this thesis is devoted to the analysis of stochastic partial differential equations of Wick-type. In particular, we study numerical approximations of solutions to boundary value problems for such equations. The problems are formulated in a weak sense over a Hilbert space of stochastic distributions. The numerical approximations are then obtained using a Galerkin type of approximation, where elements of the stochastic Hilbert space are approximated by cutting off their chaos expansion and approximating the remaining coefficients in a suitable finite element space. This leads to a sequence of deterministic variational problems, each giving one chaos coeÆcient of the approximation. We establish, to some extent, optimal convergence results for these approximations. These theoretical results have been motivated and supported through numerical experiments on some particular examples. Another aspect of this research has been the practical implementation of the described methods on a digital computer. During our investigations we have developed a software library, which has been used in the implementation of the various examples given. The library is not included in the thesis, but is available on request. We have throughout this thesis given special attention to the stochastic pressure equation of Wick-type. This equation models the pressure in a flow through a stochastic porous medium, with applications, for example, in the field of oil-recovery. We have established better stochastic regularity of the solution than what was previously known. Moreover, we develop a new numerical method for this equation based on a mixed finite element approach. We derive optimal error estimates, both for the pressure and for the velocity of the fluid. In the final part in this thesis we consider the somewhat different problem of estimating parameters in a stochastic differential equation on the basis of partial and noisy observations. We treat this classical problem in the context of nonlinear filtering theory, and derive a robust estimator, based on a Feynman-Kac representation of the solution to the associated Zakai measure equation.