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. However, wavelets being local waves are known for expressing discontinuities or steep gradients more accurately than the global basis. • Wavelet methods are generally known for their self adaptive nature which makes it a good choice for the numerical solution of a stochastic PDEs. The self adaptivity property comes from the good localization properties of wavelets which are seen both in space and frequency. This makes B-spline wavelets an interesting tool for adaptivity to explore it in stochastic PDEs. • Although there is wide research available for wavelets, yet the theory of wavelets for numerically solving stochastic PDEs in UQ is in its emerging phase. The beauty of semi-orthogonal compactly supported B-spline wavelets is that they have finite support, both even and odd symmetry and simple analytical expressions, ideal attributes of a basis function. Adaptive schemes perform refinements where most needed in order to reduce the computational effort. Such local refinements plays a vital role when the system dynamics indicate steep dependences on the random parameters. One such method known as wavelet optimized finite difference method is employed in collaboration with B-spline wavelet based gPC to find the adaptive solutions of stochastic PDEs with periodic BCs. Moreover, we introduce another method known as adaptive wavelet collocation gPC which involves the concept of autocorrelation functions of compactly supported Daubechies scaling functions. In particular, this collocation method seems well suited to the treatment of Dirichlet’s BCs. We have implemented the above developed methods on number of differential equations from engineering problems. In this thesis, we have illustrated simple second order ODEs with discontinuous solution, coupled ODEs such as stochastic linear oscillator problem, stochastic Kraichan-Orszag problem, linear as well as non-linear PDEs. Moreover, we have practiced our method on real life epidemic situation consisting of influenza in boys boarding school in England, 1978 and Ebola in Liberia, 2014. Real data is considered from British Medical Journal and World Health Organization. Moreover, in this, we have compared the linear and cubic B-spline wavelets and it has been observed that linear B-spline wavelets show faster results as compared to cubic B-spline wavelets. The wavelet optimized finite difference B-spline wavelet gPC is tested on stochastic heat equation and stochastic Burger’s equation. Further, the adaptive wavelet collocation based gPC is tested on elliptic problem, advection equation and non-linear Burger’s equation. The CPU time analysis of the methods reveals that the methods are quite efficient. The postprocessing step which includes first order moment (mean) and second order moment (variance) is performed for each test problem.