Comparative study on gradient and Hessian estimation using the Kriging method and neural network approximation

This paper discusses the accuracy of a gradient or Hessian estimation of an approximated surface for an unknown nonlinear function using the Kriging method and a hierarchical neural network. These methods will give a good global approximated response surface for an unknown function, and gradient and Hessian components of the approximated surface can be also estimated directly without using finite differences of estimated function values. However, those components may often include a large estimation error even if an approximated surface for a function value can be constructed well. In this paper, therefore, the accuracy of the estimated results of gradient and Hessian components is investigated in the case that the approximated surface for a function value is well constructed. Numerical examples illustrate a characteristic of the gradient and Hessian estimation using these methods.