#### Conference abstracts

Session A1 - Approximation Theory

July 11, 15:30 ~ 16:20 - Room B3

## Space-parameter-adaptive approximation of affine-parametric elliptic PDEs

### Markus Bachmayr

### Universität Bonn, Germany - bachmayr@ins.uni-bonn.de

We consider the approximation of PDEs with parameter-dependent coefficients by sparse polynomial approximations in the parametric variables, combined with suitable discretizations in the spatial domain. Here we focus on problems with countably many parameters, as they arise when coefficients with uncertainties are modelled as random fields. For the resulting fully discrete approximations of the corresponding solution maps, we obtain convergence rates in terms of the total number of degrees of freedom. In particular, in the case of affine parametrizations, we find that independent adaptive spatial approximation for each term in the polynomial expansion yields improved convergence rates. Moreover, we discuss new operator compression results showing that standard adaptive solvers for finding such approximations can be made to converge at near-optimal rates.

Joint work with Albert Cohen (UPMC Paris VI), Wolfgang Dahmen (RWTH Aachen), Dinh Dung (Vietnam National University), Giovanni Migliorati (UPMC Paris VI) and Christoph Schwab (ETH Zurich).