Martin Eigel: Thursday 1 June at 10:45 am, A415 Inria Paris.
We consider a class of linear PDEs with stochastic coefficients which depend on a countable (inifinite) number of random parameters. As an alternative to classical Monte Carlo sampling techniques, a functional discretisation of the stochastic space in generalised polynomial chaos may lead to significantly improved (optimal) convergence rates. However, when employed in the context of Galerkin methods, the arising algebraic systems are very high-dimensional and quickly become intractable to computations. As a matter of fact, this is an exemplary example for the curse of dimensionality with exponential growth of complexity which makes model reduction techniques inevitable. In the first part, we discuss two approaches for this: (1) a posteriori adaptivity and exploitation of sparsity of the solution, and (2) low-rank compression in a hierarchical tensor format.
In the second part, the low-rank discretisation is used as an efficient representation of the stochastic model for Bayesian inversion. This is an important application in Uncertainty Quantification where one is interested in determining the (statistics of) parameters of the model based on a set of noisy measurements. In contrast to popular sampling techniques such as MCMC, we derive an explicit representation of the posterior densities. The examined sampling-free Bayesian inversion is adaptive in all discretisation parameters. Moreover, convergence of the method is shown.
