Jay D. Martin

Use of kriging models to approximate deterministic computer models

The use of kriging models for approximation and metamodel-based design and optimization has been steadily on the rise in the past decade. The widespread usage of kriging models appears to be hampered by (1) the lack of guidance in selecting the appropriate form of the kriging model, (2) computationally efficient algorithms for estimating the model’s parameters, and (3) an effective method to assess the resulting model’s quality. In this paper, we compare (1) Maximum Likelihood Estimation (MLE) and Cross-Validation (CV) parameter estimation methods for selecting a kriging model’s parameters given its form and (2) and an R 2 of prediction and the corrected Akaike Information Criterion for assessing the quality of the created kriging model, permitting the comparison of different forms of a kriging model. These methods are demonstrated with six test problems. Finally, different forms of kriging models are examined to determine if more complex forms are more accurate and easier to fit than simple forms of kriging models for approximating computer models.

A STUDY ON THE USE OF KRIGING MODELS TO APPROXIMATE DETERMINISTIC COMPUTER MODELS

The use of kriging models for approximation and global optimization has been steadily on the rise in the past decade. The standard approach used in the Design and Analysis of Computer Experiments (DACE) is to use an Ordinary kriging model to approximate a deterministic computer model. Universal and Detrended kriging are two alternative types of kriging models. In this paper, a description on the basics of kriging is given, highlighting the similarities and differences between these three different types of kriging models and the underlying assumptions behind each. A comparative study on the use of three different types of kriging models is then presented using six test problems. The methods of Maximum Likelihood Estimation (MLE) and Cross-Validation (CV) for model parameter estimation are compared for the three kriging model types. A one-dimension problem is first used to visualize the differences between the different models. In order to show applications in higher dimensions, four two-dimension and a 5-dimension problem are also given.Copyright

On the Use of Kriging Models to Approximate Deterministic Computer Models

The use of kriging models for approximation and metamodel-based design and optimization has been steadily on the rise in the past decade. The widespread usage of kriging models appears to be hampered by (1) the lack of guidance in selecting the appropriate form of the kriging model, (2) computationally efficient algorithms for estimating the model’s parameters, and (3) an effective method to assess the resulting model’s quality. In this paper, we compare (1) Maximum Likelihood Estimation (MLE) and Cross-Validation (CV) parameter estimation methods for selecting a kriging model’s parameters given its form and (2) and an R2 of prediction and the corrected Akaike Information Criterion for assessing the quality of the created kriging model, permitting the comparison of different forms of a kriging model. These methods are demonstrated with six test problems. Finally, different forms of kriging models are examined to determine if more complex forms are more accurate and easier to fit than simple forms of kriging models for approximating computer models.Copyright

USE OF ADAPTIVE METAMODELING FOR DESIGN OPTIMIZATION

This paper describes a method to implement an adaptive metamodeling procedure during simulation- based design. Metamodels can be used for design space visualization and design optimization applications when model evaluation performance is critical. The proposed method uses a sequential technique to update a kriging metamodel. This sequential technique will determine the point of the metamodels design space with the maximum mean square error and select this as the next point to use to update the metamodel. At each iteration the quality of the metamodel is assessed using a leave- k-out cross-validation technique with three different values for k. The method is intended to permit continuous updating of the metamodel to investigate the entire design space without concern of finding an optimal value in the metamodel or model.

A Methodology to Manage System-level Uncertainty During Conceptual Design

Current design decisions must be made while considering uncertainty in both models of the design and inputs to the design, In most cases, high fidelity models are used with the assumption that the resulting model uncertainties are insignificant to the decision making process. This paper presents a methodology for managing uncertainty during system-level conceptual design of complex multidisciplinary systems. This methodology is based upon quantifying the information available in a set of observations of computationally expensive subsystem models with more computationally efficient kriging models. By using kriging models, the computational expense of a Monte Carlo simulation to assess the impact of the sources of uncertainty on system-level performance parameters becomes tractable. The use of a kriging model as an approximation to an original computer model introduces model uncertainty, which is included as part of the methodology. The methodology is demonstrated as a decision-making tool for the design of a satellite system.

Computational Improvements to Estimating Kriging Metamodel Parameters

The details of a method to reduce the computational burden experienced while estimating the optimal model parameters for a Kriging model are presented. A Kriging model is a type of surrogate model that can be used to create a response surface based a set of observations of a computationally expensive system design analysis. This Kriging model can then be used as a computationally efficient surrogate to the original model, providing the opportunity for the rapid exploration of the resulting tradespace. The Kriging model can provide a more complex response surface than the more traditional linear regression response surface through the introduction of a few terms to quantify the spatial correlation of the observations. Implementation details and enhancements to gradient-based methods to estimate the model parameters are presented. It concludes with a comparison of these enhancements to using maximum likelihood estimation to estimate Kriging model parameters and their potential reduction in computational burden. These enhancements include the development of the analytic gradient and Hessian for the log-likelihood equation of a Kriging model that uses a Gaussian spatial correlation function. The suggested algorithm is similar to the SCORING algorithm traditionally used in statistics.

A Monte Carlo Simulation of the Kriging Model

In this paper, we investigate the resulting probability distribution of a kriging model when the values of the model parameters must be estimated from observations of the process being modeled. The output of a kriging model defines a Gaussian probability distribution when the values of the model parameters are given. In practice, these model parameters must be estimated from observations of the process being modeled (typically a computationally expensive computer model). We found that when the model parameters are treated as random variables instead of known values, the resulting probability distribution of the kriging model can be well approximated by a Student-t distribution, a distribution with fatter tails than the Gaussian or normal distribution. The Markov chain Monte Carlo (MCMC) method was used to determine the probability distributions of the model parameters and the output of the kriging model given the observations. The resulting model parameters were validated against the results of a Bayesian analysis of a simple one- dimensional test problem. The results were also compared to the standard method of Maximum Likelihood Estimation as an alternative method to estimate model parameters.

On Using Kriging Models as Probabilistic Models in Design

Kriging models are frequently used as metamodels during system design optimization. In many applications, a kriging model is used as a deterministic model of a computationally expensive analysis or simulation. In this paper, a kriging model is employed as a probabilistic model on a one-dimensional and two two-dimensional test problems. A probabilistic model is a model in which the parameters are random variables resulting in a probability distribution of the output rather than a deterministic value. A probabilistic model can be used in design to quantify the knowledge designers have about a subsystem and the lack of knowledge or uncertainty in the model. Using a kriging model as a probabilistic model requires that the correlation of observations is only a function of the distance between the observations and that the observations have a Gaussian probability distribution. This paper will provide some methods to satisfy these requirements when using kriging models as probabilistic models.

A Methodology to Manage Uncertainty During System-Level Conceptual Design

Current design decisions must be made while considering uncertainty in both models and inputs to the design. In most cases this uncertainty is ignored in the hope that it is not important to the decision making process. This paper presents a methodology for managing uncertainty during system-level conceptual design of complex multidisciplinary systems. The methodology is based upon quantifying the information available in computationally expensive subsystem models with more computationally efficient kriging models. By using kriging models, the computational expense of a Monte Carlo simulation to assess the impact of the sources of uncertainty on system-level performance parameters becomes tractable. The use of a kriging model as an approximation to an original computer model introduces model uncertainty, which is included as part of the methodology. The methodology is demonstrated as a decision making tool for the design of a satellite system.© 2005 ASME

Update Strategies for Kriging Models for Use in Variable Fidelity Optimization

Many optimization methods for simulation-based design rely on the sequential use of metamodels to reduce the associated computational burden. In particular, kriging models are frequently used in variable fidelity optimization. Nevertheless, such methods may become computationally inefficient when solving problems with large numbers of design variables and/or sampled data points due to the expensive process of optimizing the kriging model parameters each iteration. One solution to this problem would be to replace the kriging models with traditional Taylor series response surface models. Kriging models, however, have been shown to provide good approximations of computer simulations that incorporate larger amounts of data, resulting in better global accuracy. In this paper two metamodel update management schemes (MUMS) are proposed to reduce the cost of using kriging models sequentially by updating the kriging model parameters only when they produce a poor approximation. The two schemes differ in how they determine when the parameters should be updated. The first method uses ratios of likelihood values (LMUMS), which are computed based on the model parameters and the data points used to construct the kriging model. The second scheme uses the trust region ratio (TR-MUMS), which is a ratio that compares the approximation to the true model. Two demonstration problems are used to evaluate the proposed methods: an internal combustion engine sizing problem and a control-augmented structural design problem. The results indicate that the L-MUMS approach does not perform well. The TR-MUMS approach, however, was found to be very effective; on the demonstration problems, it reduced the number of likelihood evaluations by three orders of magnitude compared to using a global optimizer to find the kriging parameters every iteration. It was also found that in trust region-based methods, the kriging model parameters need not be updated using a global optimizer–local methods perform just as well in terms of providing a good approximation without effecting the overall convergence rate, which, in turn, results in a faster execution time.