Numerical methods for boundary value problems on random domains

摘要

In this thesis, we consider the numerical solution ofudelliptic boundary value problems on random domains.udThe underlying domain is modelled udvia a random vector field which is given by its meanudand its covariance.udHaving these statistics of the random perturbation atudhand, we aim at determining the related statistics ofudthe random solution.udTo that end, we propose the domain mapping methodudon the one hand and the perturbation method on the udother hand.udFor the domain mapping method, we have to compute the udrandom vector field's Karhunen-Loève expansion. udFor this purpose, we compare cluster methods, namelyudthe adaptive cross approximation and the fast multipoleudmethod, and the pivoted Cholesky decomposition.udAfter this, we show regularity results for the random udsolution dependent on the decay of the random vector udfield's Karhunen-Loève expansion. These results are used udto employ a Quasi-Monte Carlo quadrature for the udapproximation of mean and variance.udFor the perturbation method, we linearize the randomudsolution's dependence on the vector field by means ofuda shape Taylor expansion. This approach yields a singleudpartial differential equation for the approximation of udthe mean and a tensor product partial differential udequation for the approximation of the covariance. The latterudis solved efficiently with the aid of the sparse tensor udproduct combination technique.ud