Algorithms for stochastic Galerkin projections: Solvers, basis adaptation and multiscale modeling and reduction

This dissertation focuses on facilitating the analysis of probabilistic models for physical systems. To that end, novel contributions are made to various aspects of the problem, namely, 1) development of efficient algorithms for solving stochastic system of equations governing the physical problem, 2) stochastic basis adaptation methods to compute the solution in reduced stochastic dimensional space and 3) stochastic upscaling methods to find coarse-scale models from the fine-scale stochastic solutions. In particular, algorithms are developed for stochastic systems that are governed by partial differential equations (PDEs) with random coefficients. Polynomial chaos-based stochastic Galerkin and stochastic collocation methods are employed for solving these equations. Solvers and preconditioners based on Gauss-Seidel and Jacobi algorithms are explored for solving system of linear equations arising from stochastic Galerkin discretization of PDEs with random input data. Gauss-Seidel and Jacobi algorithms are formulated such that the existing software is leveraged in the computational effort. These algorithms are also used to develop preconditioners to Krylov iterative methods. These solvers and preconditioners are tested by solving a steady state diffusion equation and a steady state advection-diffusion equation. Upon discretization, the former PDE results in a symmetric positive definite matrix on left-hand-side, whereas the latter results in a non-symmetric positive definite matrix. The stochastic systems face significant computational challenge due the curse of dimensionality as the solution often lives in very high dimensional space. This challenge is addressed in the present work by recognizing the low dimensional structure of many quantities of interest (QoI) even in problems that have been embedded, via parameterization, in very high-dimensional settings. A new method for the characterization of subspaces associated with low-dimensional QoI is presented here. The probability density function of these QoI is found to be concentrated around one-dimensional subspaces for which projection operators are developed. This approach builds on the properties of Gaussian Hilbert spaces and associated tensor product spaces.;For many physical problems, the solution lives in multiple scales, and it is important to capture the physics at all scales. To address this issue, a stochastic upscaling methodology is developed in which the above developed algorithms and basis adaptation methods are used. In particular upscaling methodology is demonstrated by developing a coarse scale stochastic porous medium model that replaces a fine-scale which consists of flow past fixed solid inclusions. The inclusions have stochastic spatially varying thermal conductivities and generate heat that is transported by the fluid. The permeability and conductivity of the effective porous medium are constructed as statistically dependent stochastic processes that are both explicitly dependent on the fine scale random conductivity.;Another contribution of this thesis is development of a probabilistic framework for synthesizing high resolution micrographs from low resolution ones using a parametric texture model and a particle filter. Information contained in high resolution micrographs is relevant to the accurate prediction of microstructural behavior and the nucleation of instabilities. As these micrographs may be tedious and uneconomical to obtain over an extended spatial domain, A statistical approach is proposed for interpolating fine details over a whole computational domain starting with a low resolution prior and high resolution micrographs available only at a few spatial locations. As a first step, a small set of high resolution micrographs are decomposed into a set of multi-scale and multi-orientation subbands using a complex wavelet transform. Parameters of a texture model are computed as the joint statistics of the decomposed subbands. The synthesis algorithm then generates random micrographs satisfying the parameters of the texture model by recursively updating the gray level values of the pixels in the input micrograph. A density-based Monte Carlo filter is used at each step of the recursion to update the generated micrograph, using a low resolution micrograph at that location as a measurement. The process is continued until the synthesized micrograph has the same statistics as those from the high resolution micrographs. The proposed method combines a texture synthesis procedure with a particle filter and produces good quality high resolution micrographs.