Computer Experiments with Tuning or Calibration Parameters: Modeling, Estimation and Design

在计算机实验的统计方法中，经常使用高斯过程模型来描述仿真输出和输入变量之间的关系。有三种类型的输入变量：控制变量、调谐参数和校准参数。在有限元分析中，调整参数可以是网格密度。校准参数也是计算机代码的一部分，但不是物理实验的一部分。利用计算机和物理实验相结合的数据对计算机模型进行了校正。这两种类型在文献中受到的关注要少得多。本文的主要目的是研究计算机实验中调谐和校准参数的建模、估计和设计问题。提出了一类非平稳高斯过程模型，可以有效地将具有不同调优参数值的仿真数据链接起来。研究了协方差建模和竞争模型的比较问题。在设计计算机实验时，用非均匀设计代替了典型的填充设计，这种设计能更好地反映数据中信息的非平稳性。对于校准参数，标准估计过程是渐近不一致的。提出了一种新的理论框架来研究估计性质，包括改进和新的估计过程，以达到一致性和最优收敛速度。近十年来，现实物理建模和有效的数值方法取得了迅速进展，这使得使用复杂的数学模型来模拟物理现实成为可能。计算机模拟可以比物理实验更快或更便宜。此外，物理实验很难或不可行。因此，现在通常使用计算机模拟来代替物理实验。计算机建模和实验在科学和工程研究中已经变得很流行。它们帮助我们获得了从缩短开发周期时间、提高产品质量到降低成本等诸多好处。鉴于复杂系统模拟的广泛应用，所提出的工作应该对汽车和航空航天、计算材料设计、地质和大气研究以及绿色能源模拟等各种问题产生广泛的影响。它将被整合到像R这样公开发布的软件中，从而直接使工业界的从业者和学术界的研究人员受益。

In the statistical approach to computer experiments, Gaussian process models are often employed to describe the relationship between the simulation output and the input variables. There are three types of input variables: control variables, tuning parameters and calibration parameters. The tuning parameter can be the mesh density in finite element analysis. Calibration parameters are also part of the computer code but not part of the physical experiment. The combined data from computer and physical experiments are used to calibrate the computer model. These two types have received much less attention in the literature. The main goal of this proposal is to study some issues in modeling, estimation and design for tuning and calibration parameters in computer experiments. A class of nonstationary Gaussian process models is proposed, which can be used to efficiently link data from simulations with different tuning parameter values. Issues on covariance modeling and comparisons of competing models are studied. For designing computer experiments, typical use of space-filling designs is replaced by non-uniform designs that can better reflect the nonstationary nature of information in the data. For calibration parameters, the standard estimation procedure is shown to be asymptotically inconsistent. A new theoretical framework is proposed for studying the estimation properties, including modification and new estimation procedures to achieve consistency and optimal convergence rates.The last decade has seen rapid advances in realistic physical modeling and efficient numerical methods, which make it possible to use complex mathematical models to mimic physical realities. Computer simulations can be much faster or less costly than running physical experiments. Furthermore, physical experiments can be difficult or infeasible to conduct. Therefore computer simulations are now routinely used in lieu of physical experimentations. Computer modeling and experiments have become popular in scientific and engineering investigations. They have helped reap benefits ranging from reduced development cycle time, better product, to cost reduction. In view of the wide range of applications of complex system simulations, the proposed work should have broad-based impacts on a variety of problems in autos and aerospace, computational material design, geological and atmospheric studies, and green energy simulations. It will be incorporated into publicly released software like R, thus directly benefiting practitioners in industries and researchers in academe.