An Overview of Gradient-Enhanced Metamodels with Applications

Abstract

Metamodeling, the science of modeling functions observed at a finite number of points, benefits from all auxiliary information it can account for. Function gradients are a common auxiliary information and are useful for predicting functions with locally changing behaviors. This article is a review of the main metamodels that use function gradients in addition to function values. The goal of the article is to give the reader both an overview of the principles involved in gradientenhanced metamodels while also providing insightful formulations. The following metamodels have gradient-enhanced versions in the literature and are reviewed here: classical, weighted and moving least squares, Shepard weighting functions, and the kernel-based methods that are radial basis functions, kriging and support vector machines. The methods are set in a common framework of linear combinations between a priori chosen functions and coefficients that depend on the observations. The characteristics common to all kernel-based approaches are underlined. A new ν-GSVR metamodel which uses gradients is given. Numerical comparisons of the metamodels are carried out for approximating analytical test functions. The experiments are replicable, as they are performed with an opensource available toolbox. The results indicate that there is a trade-off between the better computing time of least squares methods and the larger versatility of kernelbased approaches.

The Use of Kriging Within the SCF Procedure

“…After initial sample points are chosen and the function value of the respective points is obtained, a surrogate is created. Convergence of this is tested by using any kind of quality test such as R 2 or Q 3 criteria [15]. Depending on the type of function/problem one faces, different strategies can be utilized to find additional sample points, should the convergence test fail.…”

“…Depending on the type of function/problem one faces, different strategies can be utilized to find additional sample points, should the convergence test fail. This procedure is repeated until enough data is collected for the surrogate model to fulfill the above mentioned criteria and thus convergence is reached [15].…”

“…Thus it can then be used to predict the properties of this function without having to actually calculate it. Even though the creation of the model itself is time consuming, having an inexpensive approximate of the real function can easily make up for this [15]. This is especially true, when thinking about quantum mechanical calculations.…”

Coupling multi-fidelity kriging and model-order reduction for the construction of virtual charts

“…The major drawback of this method is the wrong variance associated to lowfidelity corrected data, which is treated as zero. Benchmarks of these methods can be found in [6,24]. With these methods, results from multiple solvers can be well exploited.…”

Layout Optimization Design of Two Vortex Induced Piezoelectric Energy Converters (VIPECs) Using the Combined Kriging Surrogate Model and Particle Swarm Optimization Method

“…However, due to money and/or time costs, it is practically infeasible to experiment or numerically simulate every feasible design point; thus, based on obtained results, different techniques to predict the outcome at a specified point have been developed, among which, one of the most widely used types is surrogate models, such as polynomial response surfaces [8], Kriging, gradient-enhanced Kriging (GEK) [9], radial basis function [10], support vector machine [11] et al With the constructed approximation models, an optimization procedure is consequently used to find the optimal result. Optimization methods arise from optimal objectives, and they are becoming essential in every field of research.…”