On the use of derivatives in the polynomial chaos based global sensitivity and uncertainty analysis applied to the distributed parameter models

Abstract

Sensitivity Analysis (SA) and Uncertainty Quantification (UQ) are important components of modern numerical analysis. However, solving the relevant tasks involving large-scale fluid flow models currently remains a serious challenge. The difficulties are associated with the computational cost of running such models and with large dimensions of the discretized model input. The common approach is to construct a metamodel – an inexpensive approximation to the original model suitable for the Monte Carlo simulations. The polynomial chaos (PC) expansion is the most popular approach for constructing metamodels. Some fluid flow models of interest are involved in the process of variational estimation/data assimilation. This implies that the tangent linear and adjoint counterparts of such models are available and, therefore, computing the gradient (first derivative) and the Hessian (second derivative) of a given function of the state variables is possible. New techniques for SA and UQ which benefit from using the derivatives are presented in this paper. The gradient-enhanced regression methods for computing the PC expansion have been developed recently. An intrinsic step of the regression method is a minimization process. It is often assumed that generating ‘data’ for the regression problem is significantly more expensive that solving the regression problem itself. This depends, however, on the size of the importance set, which is a subset of ‘influential’ inputs. A distinguishing feature of the distributed parameter models is that the number of the such inputs could still be large, which means that solving the regression problem becomes increasingly expensive. In this paper we propose a derivative-enhanced projection method, where no minimization is required. The method is based on the explicit relationships between the PC coefficients and the derivatives, complemented with a relatively inexpensive filtering procedure. The method is currently limited to the PC expansion of the third order. Besides, we suggest an improved derivative-based global sensitivity measure. The numerical tests have been performed for the Burgers' model. The results confirm that the low-order PC expansion obtained by our method represents a useful tool for modeling the non-gaussian behavior of the chosen quantity of interest (QoI).