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

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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.