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Efficient Uncertainty quantification with high dimensionality
Uncertainty exists everywhere in scientific and engineering applications. To avoid potential risk, it is critical to understand the impact of uncertainty on a system by performing uncertainty quantification (UQ) and reliability analysis (RA). However, the computational cost may be unaffordable using current UQ methods with high-dimensional input. Moreover, current UQ methods are not applicable when numerical data and image data coexist.
To decrease the computational cost to an affordable level and enable UQ with special high dimensional data (e.g. image), this dissertation develops three UQ methodologies with high dimensionality of input space. The first two methods focus on high-dimensional numerical input. The core strategy of Methodology 1 is fixing the unimportant variables at their first step most probable point (MPP) so that the dimensionality is reduced. An accurate RA method is used in the reduced space. The final reliability is obtained by accounting for the contributions of important and unimportant variables. Methodology 2 addresses the issue that the dimensionality cannot be reduced when most of the variables are important or when variables equally contribute to the system. Methodology 2 develops an efficient surrogate modeling method for high dimensional UQ using Generalized Sliced Inverse Regression (GSIR), Gaussian Process (GP)-based active learning, and importance sampling. A cost-efficient GP model is built in the latent space after dimension reduction by GSIR. And the failure boundary is identified through active learning that adds optimal training points iteratively. In Methodology 3, a Convolutional Neural Networks (CNN) based surrogate model (CNN-GP) is constructed for dealing with mixed numerical and image data. The numerical data are first converted into images and the converted images are then merged with existing image data. The merged images are fed to CNN for training. Then, we use the latent variables of the CNN model to integrate CNN with GP to quantify the model error using epistemic uncertainty. Both epistemic uncertainty and aleatory uncertainty are considered in uncertainty propagation.
The simulation results indicate that the first two methodologies can not only improve the efficiency but also maintain adequate accuracy for the problems with high-dimensional numerical input. GSIR with active learning can handle the situations that the dimensionality cannot be reduced when most of the variables are important or the importance of variables are close. The two methodologies can be combined as a two-stage dimension reduction for high-dimensional numerical input. The third method, CNN-GP, is capable of dealing with special high-dimensional input, mixed numerical and image data, with the satisfying regression accuracy and providing an estimate of the model error. Uncertainty propagation considering both epistemic uncertainty and aleatory uncertainty provides better accuracy. The proposed methods could be potentially applied to engineering design and decision making.