POLYNOMIAL-CHAOS-BASED KRIGING

Abstract

Computer simulation has become the standard tool in many engineering fields for designing and optimizing systems, as well as for assessing their reliability. Optimization and uncertainty quantification problems typically require a large number of runs of the computational model at hand, which may not be feasible with high-fidelity models directly. Thus surrogate models (a.k.a meta-models) have been increasingly investigated in the last decade. Polynomial chaos expansions (PCE) and Kriging are two popular nonintrusive meta-modeling techniques. PCE surrogates the computational model with a series of orthonormal polynomials in the input variables where polynomials are chosen in coherency with the probability distributions of those input variables. A least-square minimization technique may be used to determine the coefficients of the PCE. Kriging assumes that the computer model behaves as a realization of a Gaussian random process whose parameters are estimated from the available computer runs, i.e., input vectors and response values. These two techniques have been developed more or less in parallel so far with little interaction between the researchers in the two fields. In this paper, PC-Kriging is derived as a new nonintrusive meta-modeling approach combining PCE and Kriging. A sparse set of orthonormal polynomials (PCE) approximates the global behavior of the computational model whereas Kriging manages the local variability of the model output. An adaptive algorithm similar to the least angle regression algorithm determines the optimal sparse set of polynomials. PC-Kriging is validated on various benchmark analytical functions which are easy to sample for reference results. From the numerical investigations it is concluded that PC-Kriging performs better than or at least as good as the two distinct meta-modeling techniques. A larger gain in accuracy is obtained when the experimental design has a limited size, which is an asset when dealing with demanding computational models.