A framework for residual-based a posteriori error estimation and adaptive mesh refinement and polynomial chaos expansion for general second order linear elliptic PDEs with random coefficients is presented. A parametric, deterministic elliptic boundary value problem on an infinite-dimensional parameter space is discretized by means of a Galerkin projection onto finite generalized polynomial chaos (gpc) expansions, and by discretizing each gpc coefficient by a FEM in the physical domain.An anisotropic residual-based a posteriori error estimator is developed. It contains bounds for both contributions to the overall error: the error due to gpc discretization and the error due to Finite Element discretization of the gpc coefficients in the expansion. The reliability of the residual estimator is established.Based on the explicit form of the residual estimator, an adaptive refinement strategy is presented which allows to steer the polynomial degree adaptation and the dimension adaptation in the stochastic Galerkin discretization, and, embedded in the gpc adaptation loop, also the Finite Element mesh refinement of the gpc coefficients in the physical domain. Asynchronous mesh adaptation for different gpc coefficients is permitted, subject to a minimal compatibility requirement on meshes used for different gpc coefficients.Details on the implementation in the software environment FEniCS are presented; it is generic, and is based on available stiffness and mass matrices of a FEM for the deterministic, nonparametric nominal problem.Preconditioning of the resulting matrix equation and iterative solution are discussed. Numerical experiments in two spatial dimensions for membrane and plane stress boundary value problems on polygons are presented. They indicate substantial savings in total computational complexity due to FE mesh coarsening in high gpc coefficients.

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We analyze a-posteriori error estimation and adaptive refinement algorithms for stochastic Galerkin Finite Element methods for countablyparametric, elliptic boundary value problems. A residual error estimator which separates the effects of gpc-Galerkin discretization in parameter space and of the Finite Element discretization in physical space in energy norm is established. It is proved that the adaptive algorithm converges, and to this end we establish a contraction property satisfied by its iterates. It is shown that the sequences of triangulations which are produced by the algorithm in the FE discretization of the active gpc coefficients are asymptotically optimal. Numerical experiments illustrate the theoretical results.

In this article we introduce new methods for the analysis of high dimensional data in tensor formats, where the underling data come from the stochastic elliptic boundary value problem. After discretisation of the deterministic operator as well as the presented random fields via KLE and PCE, the obtained high dimensional operator can be approximated via sums of elementary tensors. This tensors representation can be effectively used for computing different values of interest, such as maximum norm, level sets and cumulative distribution function. The basic concept of the data analysis in high dimensions is discussed on tensors represented in the canonical format, however the approach can be easily used in other tensor formats. As an intermediate step we describe efficient iterative algorithms for computing the characteristic and sign functions as well as pointwise inverse in the canonical tensor format. Since during majority of algebraic operations as well as during iteration steps the representation rank grows up, we use lower-rank approximation and inexact recursive iteration schemes.

In a Bayesian setting, inverse problems and uncertainty quantification (UQ) -the propagation of uncertainty through a computational (forward) model -are strongly connected. In the form of conditional expectation the Bayesian update becomes computationally attractive. We give a detailed account of this approach via conditional approximation, various approximations, and the construction of filters. Together with a functional or spectral approach for the forward UQ there is no need for time-consuming and slowly convergent Monte Carlo sampling. The developed sampling-free non-linear Bayesian update in form of a filter is derived from the variational problem associated with conditional expectation. This formulation in general calls for further discretisation to make the computation possible, and we choose a polynomial approximation. After giving details on the actual computation in the framework of functional or spectral approximations, we demonstrate the workings of the algorithm on a number of examples of increasing complexity. At last, we compare the linear and nonlinear Bayesian update in form of a filter on some examples.