An optimal transport method for the PC representation of non-Gaussian fields

In the last decade, the class of polynomial chaos (PC) methods for non-Gaussian random field modeling has received considerable attention. However, these methods have been limited to low random dimension problems due to the curse of dimensionality in the Rosenblatt transformation. In this paper, we develop an optimal transport method for the PC representation of non-Gaussian fields. Our method firstly generates the samples of the non-Gaussian field by the present random field simulation method. Then, the obtained samples are represented by the Karhunen–Loeve (KL) expansion, resulting in the samples of KL variables. By solving an optimal transport problem with a proposed transport cost, a new map from Gaussian variables to non-Gaussian KL variables is further determined. With the constructed optimal transport map, the PC expansion of the non-Gaussian field is readily determined. In our method, since the optimal transport permits to move the Gaussian measure to the measure of KL variables without requiring its absolute continuity, the map can be readily constructed from samples of KL variables, bypassing the curse of dimensionality in the Rosenblatt transformation. By further incorporating the structure of the KL expansion of a random field into the Euclidean cost, the resulting transport cost enables to induce to the PC expansion of non-Gaussian field with a reasonable precision. In this way, the current work serves as a general method for PC representation of non-Gaussian random fields. Three illustrative examples, including the stochastic dynamic analysis of a bridge–vehicle system, are presented to highlight the effectiveness of proposed PC representation of non-Gaussian field and its application in the efficient stochastic response analysis.