Date of Award
Doctor of Philosophy (PhD)
Uncertainty quantification (UQ) is an emerging research area that aims to develop methods for accurate predictions of quantities of interest (QoI's) from complex engineering systems, as well as quantitative validation of the associated mathematical models, with presence of random inputs. To perform a comprehensive UQ analysis, polynomial chaos expansion (PCE) is now a commonly used approach in which the QoI is represented in a series of multi-variate polynomials that are orthogonal with respect to the measure of the inputs. Traditional methods for PCE, such as Monte Carlo, stochastic collocation, least-squares regression, are known to suffer from either slow convergence rate or rapid growth of the computational cost (as the number of random inputs increases) in identifying the PCE coefficients. When the PCE coefficients are sparse, i.e., many of them are negligible, it has been shown that compressive sampling is an effective technique to identify the coefficients with smaller number of system simulations.
In the context of compressive sampling, this thesis presents new approaches which improve the accuracy of identifying PCE coefficients, and therefore the PCE itself. In detail, a weighted L_1-minimization including a priori information about the PCE coefficients, a bi-fidelity L_1-minimization, a bi-fidelity orthogonal matching pursuit (OMP), and an L_1-minimization including the derivatives of QoI with respect to the random inputs are proposed. Both theoretical analyses and numerical experiments are presented to demonstrate that all the proposed approaches reduce the cost of computing a PCE.
% We use various numerical experiments to show that all the proposed approaches improve the accuracy in PCE approximation.
For a QoI whose PCE with respect to the measure of the underlying random inputs is not sparse, a polynomial basis design is proposed where, in addition to the coefficients, the basis functions are also learned from the simulation data. The approach has been empirically shown to find the optimal basis which makes the PCE converge more rapidly, and enhances the accuracy of the PCE approximation.
Peng, Ji, "Uncertainty Quantification via Sparse Polynomial Chaos Expansion" (2015). Mechanical Engineering Graduate Theses & Dissertations. 112.