B-Spline based uncertainty quantification for stochastic analysis

Abstract

The consideration of uncertainties has become inevitable in state-of-the-art science and technology. Research in the field of uncertainty quantification has gained much importance in the last decades. The main focus of scientists is the identification of uncertain sources, the determination and hierarchization of uncertainties, and the investigation of their influences on system responses. Polynomial chaos expansion, among others, is suitable for this purpose, and has asserted itself as a versatile and powerful tool in various applications. In the last years, its combination with any kind of dimension reduction methods has been intensively pursued, providing support for the processing of high-dimensional input variables up to now. Indeed, this is also referred to as the curse of dimensionality and its abolishment would be considered as a milestone in uncertainty quantification. At this point, the present thesis starts and investigates spline spaces, as a natural extension of polynomials, in the field of uncertainty quantification. The newly developed method 'spline chaos', aims to employ the more complex, but thereby more flexible, structure of splines to counter harder real-world applications where polynomial chaos fails. Ordinarily, the bases of polynomial chaos expansions are orthogonal polynomials, which are replaced by B-spline basis functions in this work. Convergence of the new method is proved and emphasized by numerical examples, which are extended to an accuracy analysis with multi-dimensional input. Moreover, by solving several stochastic differential equations, it is shown that the spline chaos is a generalization of multi-element Legendre chaos and superior to it. Finally, the spline chaos accounts for solving partial differential equations and results in a stochastic Galerkin isogeometric analysis that contributes to the efficient uncertainty quantification of elliptic partial differential equations. A general framework in combination with an a priori error estimation of the expected solution is provided.