| A new methodology for uncertainty quantification in practical applications is developed. The method, termed as 'generalized polynomial chaos' or 'Wiener-Askey polynomial chaos', is an extension of the mathematical theory of Nobert Wiener (1938). The original Wiener's polynomial chaos employs Hermite orthogonal polynomials in terms of Gaussian random variables to represent stochastic processes. This approach was adopted by Ghanem and his co-workers, who have conducted extensive research on uncertainty quantification via the Wiener-Hermite expansions in various areas. The generalized polynomial chaos is a broader framework which includes the Wiener-Hermite polynomial chaos as a subset. In addition to Hermite polynomials, more orthogonal polynomials from the Askey scheme are employed as the expansion bases in random space. Accordingly, the random variables in the basis functions are not necessarily Gaussian, and are determined by the random inputs to achieve fast convergence. Several types of discrete expansions are also incorporated that increase further the flexibility of generalized polynomial chaos. In the first part of this thesis, the construction of generalized polynomial chaos is presented and its mathematical properties examined. We then apply it to various differential equations subject to random inputs, including elliptic equations, parabolic equations, advection-diffusion equations, and Navier-Stokes equations. The results of generalized polynomial chaos are examined in model problems, and exponential convergence is demonstrated when the exact solutions are known and the appropriate type of chaos is employed. For model problems without explicit exact solutions, we validate the results by conducting Monte Carlo simulations. It is shown that the cost of generalized polynomial chaos is, in many cases, significantly lower than that of Monte Carlo methods, and that the generalized polynomial chaos can serve as an effective means for uncertainty quantification in real systems. |
