Polynomial chaos expansion for dependent inputs

Abstract

In PCE metamodelling, the random model response is approximated using orthonormal polynomials. For this purpose, the input variables are usually assumed to be independent, which leads to very efficient and easy ways to construct the orthonormal polynomial basis. However, often the assumption of independence is not valid and input dependencies have to be considered. This report aims to present and discuss strategies for the construction of orthonormal polynomial PCE bases, a crucial step in PCE metamodeling, for dependent input variables.

Five main approaches for the construction of the orthonormal basis discussed in the literature (Jakeman et al., 2019; Torre et al., 2019; Rahman, 2018; Paulson et al., 2017; Sudret, 2007) are presented. The first approach uses transformations to map dependent input variables to independent ones and the second approach uses a dominating independent measure as alternative to the original dependent PDF to construct the PCE basis. In contrast to the first two approaches, the third approach does not take this workaround of an independent space but directly constructs an orthonormal polynomial basis in the dependent space. For this purpose, a Gram-Schmidt orthogonalization (GSO) scheme is used. However, also a more general formulation of the orthonormalization in the dependent space, which includes the GSO scheme as a special case, is explained briefly. The fourth approach follows the GSO approach. However, instead of calculating the GSO coefficients with the whole dependent joint PDF, the coefficients are only calculated with a finite number of its moments. The last approach presented uses some manipulations of the orthonormality condition to deduce new non-polynomial orthonormal basis terms.

The first three approaches from above (mapping approach, domination approach, GSO approach) are implemented and tested on two benchmarks from literature (Torre et al., 2019) with the software UQLab (Marelli and Sudret, 2014). The results of these experiments indicate that the mapping approach performs worst and the domination approach performs best. However, these results are not representative in general because other results from literature (Jakeman et al., 2019) show that the GSO approach can also outperform the domination approach.

Suggestions for further research are made. These are: finding the root cause for the different performance of the approaches in this project compared to other experiments in literature, investigate the role of the ordering of the initial set of linear independent polynomials in the GSO approach, test alternative integration techniques for the GSO approach than the ones used in this project and implement and compare the remaining approaches not implemented in this project.