Numerical methods for boundary value problems on random domains

In this thesis, we consider the numerical solution of
elliptic boundary value problems on random domains.
The underlying domain is modelled
via a random vector field which is given by its mean
and its covariance.
Having these statistics of the random perturbation at
hand, we aim at determining the related statistics of
the random solution.
To that end, we propose the domain mapping method
on the one hand and the perturbation method on the
other hand.
For the domain mapping method, we have to compute the
random vector field's Karhunen-Loève expansion.
For this purpose, we compare cluster methods, namely
the adaptive cross approximation and the fast multipole
method, and the pivoted Cholesky decomposition.
After this, we show regularity results for the random
solution dependent on the decay of the random vector
field's Karhunen-Loève expansion. These results are used
to employ a Quasi-Monte Carlo quadrature for the
approximation of mean and variance.
For the perturbation method, we linearize the random
solution's dependence on the vector field by means of
a shape Taylor expansion. This approach yields a single
partial differential equation for the approximation of
the mean and a tensor product partial differential
equation for the approximation of the covariance. The latter
is solved efficiently with the aid of the sparse tensor
product combination technique.