Hessian-based dimension reduction for uncertainty propagation and robust design optimization

Abstract

This dissertation describes a dimension reduction method and its application to problems in uncertainty propagation and robust design optimization. For a given problem, the nonlinear behavior of a quantity of interest (QoI) with respect to the random variables is identified by computing the dominant eigenmodes of the Hessian of a QoI. A modified Arnoldi's method is used to estimate these dominant eigenmodes of the Hessian by utilizing only the gradient information of the QoI, thereby avoiding the explicit computation of the Hessian matrix. The eigenvectors corresponding to these dominant eigenmodes can then be used for approximating the statistical moments, such as the mean and the variance, by integrating the QoI in the stochastic space only along these dominant directions, therefore mitigating the potentially intractable cost of computing the high-dimensional integrals associated with these statistical moments. The dimension reduction method is first applied to simple quadratic functions and its performance in estimating the statistical moments is examined. Subsequently, this method is applied to two practical problems. The first problem demonstrates the application of the dimension reduction method to the aerostructural robust design optimization of a tailless aircraft. The second problem investigates the impact of uncertainty in the atmospheric density when solving an aircraft minimum-time-to-climb problem. It is observed that the proposed dimension reduction method performs well when accurate gradient information is available and the eigenmodes of the Hessian decay rapidly.

If accurate gradient information is not available, the results show that an interpolation-based surrogate, for which gradients are readily available, should be adopted.