We consider the tensorized operator for the Maxwell cavity source problem in frequency domain. We establish a discrete inf-sup condition for its Ritz-Galerkin discretization on sparse tensor product edge element spaces built on nested sequences of meshes. Our main tool is a generalization of the edge element Fortin projector to a tensor product setting. The techniques extend to the surface boundary edge element discretization of tensorized ...

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 ...

Several classes of MCMC methods for the numerical solution of Bayesian Inverse Problems for partial differential equations (PDEs) with unknown random field coefficients are considered. A general framework for their numerical analysis is presented. The complexity of MCMC sampling for the unknown fields from the posterior density, as well as the convergence of the discretization error of the PDE of interest in the forward response map, is ...

Let $D \subset \mathbb{R}^d, d=2,3$, be a bounded domain with piecewise smooth boundary $\partial D$ and let $U$ be an open subset of Banach space $Y$. We consider a parametric family $P_y$ of uniformly strongly elliptic, parametric second order partial differential operators $P_y$ on $D$ in divergence form, where the parameter $y$ ranges in the parameter domain $U$ so that, for a given set of data $f_y$, the solution $u$ and the ...