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

在计算机实验的统计方法中，经常采用高斯过程模型来描述模拟输出与输入变量之间的关系。有三种类型的输入变量：控制变量，调整参数和校准参数。调整参数可以是有限元分析中的网格密度。校准参数也是计算机代码的一部分，但不属于物理实验。利用计算机和物理实验的数据对计算机模型进行了标定。这两种类型在文献中受到的关注要少得多。本计画的主要目的是研究在电脑实验中，调整与校正参数的建模、估计与设计等问题。提出了一类非平稳高斯过程模型，该模型可以有效地连接来自具有不同调谐参数值的仿真数据。协方差建模和竞争模型的比较问题进行了研究。为了设计计算机实验，空间填充设计的典型应用被非均匀设计所取代，非均匀设计可以更好地反映数据中信息的非平稳性质。对于校准参数，标准的估计过程是渐近不一致的。一个新的理论框架，提出了研究的估计性能，包括修改和新的估计程序，以实现一致性和最佳的收敛rate.The过去的十年中已经看到了快速的进步，在现实的物理建模和有效的数值方法，这使得人们有可能使用复杂的数学模型来模拟物理现实。计算机模拟可以比运行物理实验快得多或成本更低。此外，物理实验可能很难或不可行。因此，计算机模拟现在通常用来代替物理实验。计算机建模和实验在科学和工程研究中已经变得流行。他们帮助获得了从缩短开发周期时间、更好的产品到降低成本等一系列好处。考虑到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中的各种问题产生广泛的影响。它将被整合到像R这样的公开发布的软件中，从而直接使行业从业者和研究人员受益。

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.