Model Order Reduction Techniques for Uncertainty Quantification Problems

The last few years have witnessed a tremendous development of the computational field of uncertainty quantification (UQ), which includes statistical, sensitivity and reliability analyses, stochastic or robust optimal control/design/optimization, parameter estimation, data assimilation, to name just a few. In all these problems, the solution of stochastic partial differential equations (PDEs) is commonly faced, for which many computational methods have been proposed, such as the extensively used Monte Carlo method and its several variants, the fast convergent stochastic Galerkin projection method and the nonintrusive stochastic collocation method. The large advancement of these computational methods with sparse and adaptive techniques has enabled efficient solution of the aforementioned UQ problems that feature high dimensionality, low regularity and arbitrary probability measures. However, when it becomes very expensive to solve the underlying deterministic PDEs, e.g., only a few tens or hundreds of full solutions are affordable in practice, these computational methods can not be applied directly since they may need millions of full solutions, or even beyond, in order to achieve a certain accuracy. In this thesis, we develop, analyze and demonstrate novel stochastic computational strategies and algorithms based on model order reduction techniques, in particular based on reduced basis methods, to tackle this challenge in solving several typical UQ problems. We first compare the convergence properties and computational costs of the reduced basis method and the sparse grid stochastic collocation method, and demonstrate that the former is much more efficient than the latter without loss of accuracy in solving large-scale and high-dimensional UQ problems. In dealing with arbitrary probability measures, we propose a weighted reduced basis method inspired by the generalized polynomial chaos, and establish explicitly a priori error estimates for both one-dimensional and multidimensional stochastic/parametrized problems. A weighted empirical interpolation method with improved convergence property is proposed in order to decompose nonaffine random fields, which paves the way for effective application of the reduced basis method in solving more general UQ problems. A hybrid and goal-oriented adaptive reduced basis method with certification is proposed to efficiently and accurately solve a large class of UQ problems, involving pointwise evaluation, in particular failure probability for reliability analysis. Moreover, taking advantage of the sparsity and reducibility of UQ problems, we develop an adaptive and reduced computational framework that enables precise detection of the distinctive importance and the interaction of different dimensions, as well as automatic construction of a generalized sparse grid and reduced basis approximation of the quantities of interest. Besides the development and demonstration of the model order reduction techniques in solving various demanding forward UQ problems, a large effort of this thesis has been devoted to the analysis and the efficient solution of inverse UQ problems, in particular stochastic optimal control problems. We succeed in proving not only the existence but also the uniqueness of the optimal solution via a stochastic saddle point formulation in the case of elliptic and Stokes constraints. A detailed analysis is carried out for the stochastic regularity of the optimal solution w.r.t. the random input data under certain smoothness hypothesis. We tailor the main ingredients of the developed adaptive and reduced computational strategy to solve stochastic optimal control problems with several different PDE constraints. The efficiency and accuracy of this strategy demonstrate its potentials in solving more general large-scale and high-dimensional inverse UQ problems with arbitrary probability measures.