Combining Polynomial Chaos Expansions and Kriging

Authors

Schöbi, R., Kersaudy, P., Sudret., B. and Wiart, J.

Abstract

Computer simulation has emerged as a key tool for designing and assessing engineering systems in the last two decades. Uncertainty quantification has become popular more recently as a way to model all the uncertainties affecting the system and their impact onto its performance. In this respect meta-models (a.k.a. surrogate models) have gained interest. Indeed dealing with uncertainties requires running the computer model many times, which may not be affordable for complex models. Surrogate models mimic the behaviour of the original model while being cheap to evaluate.

Polynomial chaos expansion (PCE) and Kriging are two popular techniques, which have been developed with very little interaction so far. In this report we present a new approach, called PC-Kriging, that combines the two tools. The algorithm is based on the universal Kriging model where the trend is represented by a set or orthonormal polynomials. Various aspects of the new metamodelling technique are presented and investigated in details. The discussion starts with a survey on methods for generating an optimal design of experiments (DOE). The PC-Kriging algorithm inherits many parameters and sub-methods such as the number of polynomial terms and the choice of the autocorrelation kernel. A variety of kernels are presented and discussed.

The methods are compared on analytical benchmark functions. The conclusion of this report is that PC-Kriging performs better or at least as well as PCE or Kriging taken separately in terms of relative generalized error (L₂ error).