Computer Experiments: Multi-Layer Designs, Kriging, and Beyond

本项目旨在发展计算机实验设计与分析的新思路和新方法。在设计方面，研究人员提出了一种新的设计，称为多层设计。它们是由众所周知的两级分数阶乘设计构建的，通过将点移动到不同的层中。这增加了每个因素的层次数量，有助于更均匀地填充实验区域。多层设计具有与最佳拉丁超立方体设计相当的空间填充特性，但更容易构建。由于其独特的几何特征，所提出的多层设计可能导致对空间填充设计的结构和性质的重新思考。该方法为研究人员提供了一种新的工具，可以利用分数阶乘设计的大量知识来构建计算效率高的计算机实验设计。在分析方面，提出了两个主题，以解决使用克里格方法的潜在严重稳定性问题，克里格方法是分析此类数据的最常用工具。首先，通过相关矩阵的条件数，探讨了相关矩阵在克里金变换中数值稳定性的可能原因。二是将克里格法的预测精度与基于回归的逆距离加权法计算速度快、成本低相结合，提出了混合克里格法。由于克里格法在空间统计和计算机实验中都得到了广泛的应用，因此克里格法的研究也会对空间变异统计建模的研究产生影响。由于物理建模和数值方法的快速发展，复杂的数学模型现在可以可靠地用于模拟物理现实。它们的实际实现得益于快速算法和软件开发的出现。因此，现在通常使用复杂系统模拟来代替物理实验。例如，汽车中的安全气囊可以通过复杂的计算机模拟来设计，这种模拟可以在计算机中模拟汽车碰撞，而不是制造和碰撞真正的汽车。建议的工作应该导致设计和分析此类计算机实验的通用性质的方法开发，这反过来可以导致减少开发周期时间，更好的产品和降低成本。鉴于复杂系统模拟的广泛应用，所提出的实验设计和分析方法应该对地质和大气研究、计算材料设计、超级计算机热管理以及其他绿色能源应用等各种问题产生广泛的影响。

This project aims at developing new ideas and methods on the design and analysis of computer experiments. On the design side, the investigators propose a new class of designs, called multi-layer designs. They are constructed from the well-known two-level fractional factorial designs by moving the points into different layers. This increases the number of levels for each factor and helps in filling the experimental region more evenly. The multi-layer designs have comparable space-filling properties to those of the optimal Latin hypercube designs, but are much easier to construct. Because of its unique geometric features, the proposed multi-layer designs may lead to a rethinking about the construction and properties of space-filling designs. The proposed approach gives a new tool for the researchers to take advantage of the vast amount of knowledge on fractional factorial designs in the construction of computationally efficient experimental designs for computer experiments. On the analysis side, two topics are proposed to address the potentially serious issue of stability with the kriging method, which is the most common tool for analyzing such data. The first is to investigate the possible causes for the numerical stability in the inversion of the correlation matrix in kriging via its condition number. The second is to propose a new method, called hybrid kriging, by combining the prediction accuracy of kriging with the cheap/fast computation of the regression-based inverse distance weighting method. Since kriging has been commonly used in spatial statistics as well as in computer experiments, the proposed work on kriging can also influence the research on statistical modeling of spatial variation.Because of the rapid advances in physical modeling and numerical methods, complex mathematical models can now be reliably used to mimic physical realities. Their practical implementations benefit from the advent in fast algorithms and software development. Therefore, complex system simulations are now routinely used in lieu of physical experimentations. For example, airbags in a car can be designed through sophisticated computer simulation that mimics a car-crash in a computer instead of building and crashing real cars. The proposed work should lead to the methodological development of a generic nature for designing and analyzing such computer experiments, which in turn can lead to reduced development cycle time, better product, and cost reduction. In view of the wide ranges of applications of complex system simulations, the proposed experimental design and analysis methodology should have broad-based impacts on a variety of problems like geological and atmospheric studies, computational material design, thermal management of supercomputers, and other green energy applications.