On the implementation of generalized polynomial chaos in dynamic optimization under stochastic uncertainty: a user perspective

Abstract Throughout the past century, numerous frameworks have been presented to address different types of uncertainty in model-based (dynamic) optimization. One of the most successful and promising frameworks to address uncertainty in dynamic optimization is generalized polynomial chaos (gPC). This framework is applicable to uncertainties modeled as random variables with generic (e.g., correlated and bimodal) probability distributions. An accurate and efficient approximation of the mean and variances of the model responses can then be readily computed from the coefficients of the gPC expansion. Two types of formulations exist to compute the gPC coefficients: intrusive and non-intrusive. In this paper, a tutorial and critical comparison are presented on the implementation of gPC. More specifically, an intrusive Galerkin approach and two non-intrusive approaches (probabilistic collocation and least-squares regression) have been implemented on a continuously stirred tank reactor case study.