Bernoullis Tafelrunde: Marc Schmidlin
Uncertainty Quantification for PDEs with Anisotropic Random Diffusion
We will consider elliptic diffusion problems with an anisotropic random diffusion coefficient.
More specifically, we will model diffusion in a medium comprised of very thin fibres,
where the diffusion strength in fibre direction is notably different to the diffusion strength
perpendicular to the fibres -- thus we may describe the diffusion strength in fibre direction
and the actual fibre direction by a vector field V. Any uncertainty regarding the vector field V then
propagates, yielding uncertainty in the diffusion coefficient and therefore also in the solution
of our elliptic diffusion problem.
Using the vector field V’s Karhunen-Loève expansion we can reformulate the elliptic diffusion
problem into a parametric from, with a random parameter. We then derive that the regularity
of the solution’s dependence on the random parameter is entirely determined by the decay
of the vector field V’s Karhunen-Loève expansion. This result allows for sophisticated quadrature
methods, such as the quasi-Monte Carlo method or the anisotropic sparse grid quadrature,
to be used to approximate quantities of interest, like the solution’s mean or its variance.
Numerical examples will supplement the presented theory.