A generalized active subspace for dimension reduction in mixed aleatory-epistemic uncertainty quantification

Abstract

Aleatory and epistemic uncertainties are being increasingly incorporated in verification, validation, and uncertainty quantification (UQ). However, the crucial UQ of high efficiency and confidence remains challenging for mixed multidimensional uncertainties. In this study, a generalized active subspace (GAS) for dimension reduction is presented and the characteristics of GAS are investigated by interval analysis. An adaptive response surface model can then be employed for uncertainty propagation. Since the precise eigenvalues of interval matrix are difficult to solve in mathematics, three alternative estimate methods, i.e. interval eigenvalue analysis (IEA), empirical distribution function (EDF), and Taylor expansions, are developed for the GAS computation and practical use. The efficacy of the GAS and the estimate methods is demonstrated on three test examples: a three-dimensional response function, a standard NASA test of six-dimensional mixed uncertainties, and a NACA0012 airfoil design case of ten epistemic uncertainties. The IEA estimate is comparatively more suitable, but needs more computational cost due to the requirement of bound matrices. When the uncertainty level is small, the three methods are all applicable and the estimate based on EDF can be more efficient. The methodology exhibits high accuracy and strong adaptability in dimension reduction, thus providing a potential template for tackling a wide variety of multidimensional mixed aleatory-epistemic UQ problems.