Collaborative Research: Uncertainty Quantification, Optimal Designs and Calibration in Computer Experiments

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From aerospace designs to material science to biomedical studies, today's practice in engineering and physical sciences has made increasing use of computer simulations. How to design the computer simulations, analyze the computer output data, as well as to enhance the accuracy of the computer models are fundamental challenges in computer simulations. This project will focus on the development of a statistical framework for computer experiments, with the goal of developing both theoretical and methodological tools that cover the typical computer simulation pipeline from data collection to modeling and analysis to verification and validation. Specifically, the team plans to establish a new uncertainty quantification theory and foster novel methodologies for data mining, interpretation and decision making. The project will provide accurate, efficient and robust approaches that would make an impact on contemporary computer simulation practice. A graduate student will work on both the theoretical derivations and numerical verification for the project.The project has three major objectives: (i) establish a statistically and computationally efficient uncertainty quantification framework for Gaussian process regression, (ii) propose a general experimental design scheme for multi-fidelity computer experiments, (iii) study the statistical properties and suggest efficient algorithms for novel calibration methods for computer models. The proposed work should lead to methodological development of a generic nature in the design, uncertainty quantification and calibration in computer experiments. The improved uniform error bounds can potentially lead to the use of fewer experimental runs for the same precision. Their significance can go beyond computer experiments such as in spatial statistics, which heavily uses kriging method. The optimal designs for nonstationary Gaussian Process models can help stimulate further development of experimental design theory in more complex situations. Standard approaches in experimental design do not pay much attention to the nonstationary situations. The proposed algorithm can substantially enhance the value of the projected kernel calibration (PKC) method. Although PKC is known to be theoretically superior, there is no known algorithm that can effectively calculate the PKC estimates. Because calibration is used to bridge the gap between computer simulations and physical experiments, this work can be potentially significant. Theoretical and technical advances made in this project can help facilitate further interactions between statistics, applied mathematics and probability theory, through journal publications, student exchange visits and presentations in interdisciplinary conferences, etc.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.