Performance study of gradient-enhanced Kriging

Abstract

The use of surrogate models for approximating computationally expensive simulations has been on the rise for the last two decades. Krigingbased surrogate models are popular for approximating deterministic computer models. In this work, the performance of Kriging is investigated when gradient information is introduced for the approximation of computationally expensive black-box simulations. This approach, known as Gradient Enhanced Kriging, is applied to various benchmark functions of varying dimensionality (2D-20D). As expected, results from the benchmark problems show that additional gradient information can significantly enhance the accuracy of Kriging. Gradient Enhanced Kriging provides a better approximation even when gradient information is only partially available. Further comparison between Gradient Enhanced Kriging and an alternative formulation of Gradient Enhanced Kriging, called indirect Gradient Enhanced Kriging, highlights various advantages of directly employing gradient information, such as improved surrogate model accuracy, better conditioning of the correlation matrix, etc. Finally, Gradient Enhanced Kriging is used to model 6-and 10-variable Fluid-Structure Interaction problems from bio-mechanics to identify the arterial wall's sti↵ness.Keywords Kriging · Surrogate Modelling · Gradient Enhancement · Fluid Structure Interaction S. Ulaganathan · I. Couckuyt · T. Dhaene · E. Laermans Ghent University -iMINDS, Department of Information Technology (INTEC), Gaston Crommenlaan 8, 9050 Ghent, Belgium E-mail: {selvakumar.ulaganathan, ivo.couckuyt, tom.dhaene, eric.laermans}@ugent.be J. Degroote Ghent University, Department of Flow, Heat and Combustion Mechanics, Sint -Pietersnieuwstraat 41, 9000 Ghent, Belgium E-mail: Joris.Degroote@ugent.be 2 S e l v a k u m a r U l a g a n a t h a n e t a l .

Data‐driven model based design and analysis of antenna structures

“…GEK requires only half (or even less) of the training sample points demanded by OK to accurately approximate the actual |S 11 | curve ( Figures 3-5 and Table 1). GEK benefits from the additional gradient data which allow the model to accurately capture the covariance structure of the training data [35,36]. This is achieved by the fact that the GEK model is forced to interpolate both the function and the gradient values at the training samples whereas OK interpolates the function values only.…”

“…In addition, data of various levels of accuracy can be obtained by executing simulations on various mesh resolutions or with various convergence criteria. Various variable-fidelity metamodeling procedures have been reported in [34][35][36][37].…”

Variance-based interaction index measuring heteroscedasticity

“…The simulation code we use was developed by Degroote et al [21]. This code has recently been used as an example problem for Gradient Enhanced Kriging [22] since the function exposes the gradient as well as the objective value. For our purpose, we will ignore the gradients and treat it as a scalar function with vector input consisting of the parameters we want to identify.…”

Kernel Parameter Optimization for Kriging Based on Structural Risk Minimization Principle

“…Kriging surrogate model is simple and stable compared to other methods and has been widely used in many fields [9][10][11][12]. Some improved Kriging models have 2 Mathematical Problems in Engineering also been studied, such as Gradient-Enhanced Kriging [13], CoKriging [14], and Hierarchical Kriging [15].…”