Adaptive construction of locally anisotropic spectral discretization for stochastic PDEs

Abstract

Many stochastic problems are anisotropic in nature in that at certain spatial locations the solution may depend only on a couple of random variables and have negligible dependence on others. This anisotropic nature of stochastic problems can be exploited by employing a stochastic discretization that is locally anisotropic (i.e., in a spatial sense). In this thesis, we formulate and apply an anisotropic adaptive approach for stochastic discretization. In our approach, we use finite element basis in the physical/spatial domain and spectral basis (based on generalized polynomial chaos) in the stochastic domain. We employ the stochastic variational multiscale (VMS) method that has been developed for this basis setting, where the effect of missing or fine scales on resolved or coarse scales is modeled as an algebraic approximation within each element. In addition, a model term of a similar form has been developed to estimate the error in the numerical solution in a local/element-wise fashion. We make use of these developments and propose a variance-based sensitivity for the element-wise error to devise an anisotropic indicator. We then use the anisotropic indicator in conjunction with the estimated measure of the local error to adaptively control the order of the spectral basis in each stochastic direction independently, resulting in anisotropic stochastic adaptivity. We demonstrate the effectiveness of our approach for two examples involving scalar transport in which the stochastic behavior is anisotropic and varying over the physical domain.

As the field of computational mechanics is growing, deterministic modeling and computation are being augmented by their stochastic counterpart to account for any non-trivial uncertainty present in the input data. However, as the number of uncertain parameters and associated random variables grows, a stochastic analysis becomes exponentially more expensive to solve and suffers from the so-called curse-of-dimensionality. Therefore, an automated approach to create efficient discretizations for stochastic problems is highly desired. Specifically, we seek an adaptive approach to construct an efficient approximation in the stochastic domain.