EAS PHD Defense: Jiahua Jiang
When: Wednesday,
May 10, 2017
10:00 AM
-
12:00 PM
Where: Textiles Building 105
Description: Title: THE REDUCED BASIS METHOD AND ITS APPLICATION TO UNCERTAINTY QUANTIFICATION PROBLEMS
Abstract:The last few years have witnessed a tremendous development of the computational field of uncertainty quantification (UQ), which includes sensitivity analyses, parameter estimation, data assimilation, etc. In all these areas, the solution of parameterized partial differential equations (PDEs) whose parameters can be considered random variables is commonly faced, for which many computational methods have been proposed, such as the non-intrusive generalized Polynomial Chaos (gPC) method. Even with the large advancement of these computational methods, it is still challenging to efficiently solve the UQ problems that feature high dimensionality. In this thesis, we tackle this challenge from two aspects.
First, we propose a weighted version of a mathematically rigorous and computationally efficient model reduction strategy, the reduced basis method (RBM), and synergistically integrate it with the non-intrusive gPC framework in a goal-oriented fashion. As a result, the hybrid gPC is orders of magnitude more efficient than the classical version. Moreover, this efficiency gain becomes more pronounced as the dimension of the randomness increases.
Second, we design an offline-enhancing strategy to mitigate the well-known computational difficulty faced by RBM when the parameter dimension is high. Indeed, the greedy algorithm critically employed by RBM requires maximization of an error estimate over parameter space which is substituted with a discrete "training" set in practice. When the dimension of parameter space is large, it is necessary to significantly increase the size of this training set in order to effectively search parameter space. As a result, the cost of the offline phase increase proportionally diminishing the effectiveness of RBM. To break this barrier, our offline-enhancing strategy adopts a multi-fidelity approach. It increases the RBM efficiency by 3 to 6 times without degrading any accuracy.
Advisor: Prof. Yanlai Chen
Committee members: Prof. Akil Narayan, Prof. Sigal Gottlieb, Prof. Bo Dong, Prof. Mazdak Pour A Tootkaboni
Abstract:The last few years have witnessed a tremendous development of the computational field of uncertainty quantification (UQ), which includes sensitivity analyses, parameter estimation, data assimilation, etc. In all these areas, the solution of parameterized partial differential equations (PDEs) whose parameters can be considered random variables is commonly faced, for which many computational methods have been proposed, such as the non-intrusive generalized Polynomial Chaos (gPC) method. Even with the large advancement of these computational methods, it is still challenging to efficiently solve the UQ problems that feature high dimensionality. In this thesis, we tackle this challenge from two aspects.
First, we propose a weighted version of a mathematically rigorous and computationally efficient model reduction strategy, the reduced basis method (RBM), and synergistically integrate it with the non-intrusive gPC framework in a goal-oriented fashion. As a result, the hybrid gPC is orders of magnitude more efficient than the classical version. Moreover, this efficiency gain becomes more pronounced as the dimension of the randomness increases.
Second, we design an offline-enhancing strategy to mitigate the well-known computational difficulty faced by RBM when the parameter dimension is high. Indeed, the greedy algorithm critically employed by RBM requires maximization of an error estimate over parameter space which is substituted with a discrete "training" set in practice. When the dimension of parameter space is large, it is necessary to significantly increase the size of this training set in order to effectively search parameter space. As a result, the cost of the offline phase increase proportionally diminishing the effectiveness of RBM. To break this barrier, our offline-enhancing strategy adopts a multi-fidelity approach. It increases the RBM efficiency by 3 to 6 times without degrading any accuracy.
Advisor: Prof. Yanlai Chen
Committee members: Prof. Akil Narayan, Prof. Sigal Gottlieb, Prof. Bo Dong, Prof. Mazdak Pour A Tootkaboni
Contact: EAS Seminar Series
Topical Areas: University Community, Mathematics, College of Engineering, Physics