We review the recent results of [21, 22], and establish the exponential convergence of hp-version discontinuous Galerkin finite element methods for the numerical approximation of linear second-order elliptic boundary-value problems with homogeneous Dirichlet boundary conditions and constant coefficients in threedimsional and axiparallel polyhedra. The exponential rates are confirmed in a series of numerical tests.

Based on the parametric deterministic formulation of Bayesian inverse problems with unknown input parameter from infinite dimensional, separable Banach spaces proposed in [28], we develop a practical computational algorithm whose convergence rates are provably higher than those of Monte-Carlo (MC) and Markov-Chain Monte-Carlo methods, in terms of the number of solutions of the forward problem. In the formulation of [28], the forward ...