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

从航空航天设计到材料科学再到生物医学研究，今天在工程和物理科学领域的实践越来越多地使用计算机模拟。如何设计计算机仿真，分析计算机输出数据，提高计算机模型的准确性是计算机仿真的基本挑战。该项目将侧重于开发计算机实验的统计框架，其目标是开发理论和方法工具，涵盖从数据收集到建模和分析再到验证和确认的典型计算机模拟管道。具体来说，该团队计划建立一种新的不确定性量化理论，并培养数据挖掘、解释和决策的新方法。该项目将提供准确、高效和稳健的方法，对当代计算机模拟实践产生影响。研究生将对该项目进行理论推导和数值验证。该项目有三个主要目标：(i)为高斯过程回归建立一个统计和计算效率高的不确定性量化框架；（ii）为多保真度计算机实验提出一个通用的实验设计方案；（iii）研究统计特性并为计算机模型的新校准方法提出有效的算法。所建议的工作应导致在设计、不确定度量化和计算机实验校准方面具有通用性质的方法学发展。改进的均匀误差范围可能导致使用更少的实验运行相同的精度。它们的意义可以超越计算机实验，例如大量使用克里格方法的空间统计学。非平稳高斯过程模型的优化设计有助于促进实验设计理论在更复杂情况下的进一步发展。标准的实验设计方法不太注意非平稳情况。该算法可以大大提高投影核校准（PKC）方法的价值。虽然已知PKC在理论上是优越的，但没有已知的算法可以有效地计算PKC估计。由于校准用于弥合计算机模拟和物理实验之间的差距，因此这项工作可能具有潜在的重要意义。该项目取得的理论和技术进步可以通过期刊发表、学生互访和跨学科会议上的演讲等方式，帮助促进统计学、应用数学和概率论之间的进一步互动。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，认为值得支持。

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.