Uncertainty quantification (UQ) is an interesting, fast growing research area aiming at developing methods to address, characterize and minimize the impact of parameter, data and model uncertainty in complex systems. Applications of uncertainty quantification include all areas of engineering, environmental, physical and biological systems, e.g., groundwater flow problems, shape uncertainties in aerodynamic applications or nano-optics, biochemical networks and finance. The efficient treatment of uncertainties in mathematical models requires ideas and tools from various disciplines including numerical analysis, statistics, probability and computational science. In this talk, we will focus on the identification of parameters through observations of the response of the system - the inverse problem. The uncertainty in the solution of the inverse problem will be described via the Bayesian approach.
We will discuss a parametric deterministic formulation of Bayesian inverse problems with distributed parameter uncertainty from infinite dimensional, separable Banach spaces, with uniform prior probability measure on the uncertain parameter. For forward problems belonging to a certain sparsity class, we quantify analytic regularity of the Bayesian posterior and prove that the parametric, deterministic density of the Bayesian posterior belongs to the same sparsity class. These results suggest in particular dimension-independent convergence rates for data-adaptive Smolyak integration algorithms.