We discuss several numerical aspects of uncertainty quantification in elliptic partial differential equations.
We begin by representing random data in terms of a Karhunen-Loeve expansion. The latter can be computed efficiently by the pivoted Cholesky decomposition. 
Having this representation at hand, the computation of statistics of the system response leads to high dimensional quadrature problems which we address by the anisotropic sparse quadrature.
Employing multilevel techniques, we can speed up the computations even further.
Finally, we show how to incorporate measurement data into our model by means of Bayesian inversion.

Michael Peters is a Mathematician who works on uncertainty quantification for partial differential equations. He is interested in developing and implementing efficient algorithms to address the resulting high dimensional problems.  Michael received his PhD in Applied Mathematics at the University of Basel in 2014.
Currently, he has a PostDoc position at the Department of Biosystems Science and Engineering of the ETH Zurich.