Methods for the solution of differential equations with uncertainties in parameters

Abstract

The main topic of this thesis is the solution of partial differential equations (PDEs) with uncertainties in input data using the stochastic Galerkin method (SGM).

A significant part of the thesis is devoted to the theoretical background. This includes probability theory, with a focus on random fields and their regularity, variational formulations of PDEs with uncertainties, with a focus on differences to the deterministic counterpart, SGM discretization spaces, the efficient assembly of stochastic matrices, and the use of the resulting solution for sampling.

The rest of the thesis is devoted to efficient computations and numerical testing and covers two main topics. The first topic is the efficient computation of Karhunen-Loève (KL) decomposition of isotropic Gaussian random fields This includes the optimization approach to the separable approximation of covariance function, Galerkin approximation using polynomial basis and specific integration scheme for efficient construction of Galerkin matrices. The second topic is the reduced basis (RB) solver for SGM systems, including two methods for constructing the RB: the reduced rational Krylov subspace method (RRKS) and the Monte Carlo (MC) greedy method. Additionally, the efficient implementation of the RB solver includes a scheme for adaptive precision for the solution of the reduced system and the deflated conjugate gradient (DCG) method to speed up the construction of the RB.

The main contributions of the thesis can be divided into two parts. The first one is the compilation of the theoretical background with some minor additions. The second one is the improvements and modifications of methods for efficient computation. Shortly, the second one includes efficient computation of KL decomposition, improvements of RRKS construction of RB, a scheme for adaptive precision for the solution of reduced systems, and the use of DCG for speeding up the RB construction. Lastly, my own implementation in MATLAB and comprehensive numerical experiments inspired by geosciences.