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Metamodel Building on the Basis of Globally Weighted High Order Polynomials

By Jānis Auziņš, Edgars Labans, Kaspars Kalniņš, Rihards Priedītis and Anatolijs Meļņikovs


In the paper, the method of weighted polynomials for model building on the basis of numerical or physical experiments is proposed. Traditionally metamodels for multidisciplinary optimization are built using global polynomial approximations (mainly second order) and nonparametric methods: kriging, radial basis functions, locally weighted polynomials. High order multivariate polynomials are relatively easy to use, but by growth the degree of polynomial and the number of variables the number of polynomial terms quickly exceeds the number of experimental observations, which makes the conventional Least Squares method unusable for coefficient estimation. In this paper the generalized thin-plate energy functional is used for penalization of the functional to be minimized for the case of parametric approximation. The optimal choice of smoothing parameter is implemented using the cross-validation method. The proposed algorithm is implemented in the software tool EDAOpt that has been developed in the Riga Technical University for more than 13 years [4, 5, 3, 12]. The simple method of elimination of low significance term is used for the simplification of polynomial model. The method was tested for known optimization test problems with 2-5 variables and showed prediction accuracy comparable with kriging. For the noisy responses the proposed approach gives better prediction accuracy of approximate model. The usability of the method is demonstrated by several practical optimization problems, including shape optimization and material elasticity parameter identification by vibration method. The method can be used in metamodel building of systems with highly correlated inputs

Topics: metamodels, penalized least squares, multivariate polynomials, thin-plate energy.
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