AIAA 2002–0317 Using Gradients to Construct Cokriging Approximation Models for High-Dimensional Design Optimization Problems

The Kriging method is an interpolation scheme that can be used for modeling deter-ministic computer analyses as the realization of a stochastic process. The technique has been recognized as an alternative to the traditional Response Surface method in generating approximation models of computationally expensive CFD analyses. This is due to its ability to interpolate sample data and to model a function with multiple local extrema. To fully exploit the advantage of the Kriging method, however, a large number of sample data points should be spread out to fill the design space. This can be very costly and even impractical in high-dimensional design optimization. In this work, the Cokriging method, an extension of Kriging, which can incorporate secondary information such as values of gradients in addition to primary function values of the sample points has been utilized for constructing approximation models in a realistic design optimization process. This approach improves on the accuracy and efficiency of using the Kriging method for high-dimensional design problems. Provided that gradient information is available through inexpensive algorithms such as the adjoint method, Cokriging significantly reduces the large computational cost needed for the original Kriging method to accurately capture multiple local extrema of the unknown response function within a relatively large design space. After validating the feasibility of the Cokriging method using simple one-and two-dimensional analytic functions, the approach is applied to the aerodynamic design of a supersonic business jet. The results of these 2-and 5-variable test design problems indicate that great improvements on the efficiency and applicability of the Kriging method in high-dimensional design optimization problems can be achieved. Nomenclature β constant underlying global portion of Kriging model CD drag coefficient f constant vector used in Kriging model fc constant vector used in Cokriging model k number of design variables n s number of sample points r vector of correlation values for Kriging model r c vector of correlation values for Cokriging model R(.) correlation function for Kriging model R correlation matrix for Kriging model Rc correlation matrix for Cokriging model x scalar component of x x vector denoting all locations (sites) in the design space x p vector denoting the p th location in the design space y(.) unknown functionˆy(.) estimated model of y(.) vector of correlation parameters for Kriging modeî σ 2 estimated sample variance