Variants of Polynomial Chaos Methods in Uncertainty Quantification

Abstract

The area of uncertainty quantification (UQ) has acquired a lot of importance in the past few years. The eagerness to achieve precision has driven today s world to quantify the uncertainties present in various physical and engineering problems. In order to have a better understanding of the stochastic approaches, a current state-of-the-art review of the numerical methods for stochastic computations is presented. In this thesis, a brief account of the related work of various authors to numerically solve stochastic partial differential equations (PDEs) by using several approaches is reviewed. The framework of the methods is discussed along with their algorithms, literature, comparison analysis, strengths, and weaknesses. An illustrative example of a simple ordinary differential equations (ODEs) with uncertain parameter is discussed and is compared with three main methods-Monte Carlo, polynomial chaos and stochastic collocation method. We initially started with the traditional polynomial chaos method which involves Hermite polynomials and united it with the summation by parts-simultaneous approximation terms (SBP-SAT) operators in order to acquire the stability conditions for the Dirichlet boundary conditions (BCs). Spatial derivatives are approximated by SBP operators and SATs are used to enforce BCs by ensuring stable solutions. As our aim was to develop variants of polynomial chaos methods in engineering problems, so, a new class of wavelets known as B-spline wavelets is introduced into the area of UQ. On the basis of order of B-spline wavelets, linear and cubic Wiener semi-orthogonal B-spline generalized polynomial chaos (gPC) is developed. The advantages of B-spline wavelet based gPC are Usually, gPC may have slow convergence or fails to converge in problems which consists of discontinuities or sharp dependence on the random space even in shorttime integration.