Modified Gram-Charlier Method for Analytical PDF

This paper studies the viability of the Gram-Charlier Series (GCS) method for the probability density function (PDF) construction of a stochastic response. The PDF represents the relative likelihood of a specific output value occurring and is often used to represent the likelihood of a continuous random variable taking on a particular range of values. The GCS method creates an analytical function representing the PDF for an arbitrary dataset, which allows more tractable integration for uncertainty quantification (UQ) and optimization. A non-intrusive polynomial chaos expansion (PCE) approach using ��1-minimization is implemented to generate samples for arbitrary objective functions. Once the PCE of a stochastic response is constructed, the standard GCS method employing Hermite polynomials and the Gaussian distribution parent function is tested using analytical functions to validate the process initially. Furthermore, a novel extension to the GCS utilizing monomials and the Beta distribution parent function is proposed. The standard GCS and proposed modified GCS have been applied to analytical test functions and for the preliminary conceptual design of an aircraft to quantify cruise fuel burn (operating cost) under epistemic and aleatoric uncertainties.