A global sensitivity index based on Frechet derivative and its efficient numerical analysis

Sensitivity analysis plays an important role in reliability evaluation, structural optimization and structural design, etc. The local sensitivity, i.e., the partial derivative of the quantity of interest in terms of parameters or basic variables, is inadequate when the basic variables are random in nature. Therefore, global sensitivity such as the Sobol' indices based on the decomposition of variance and the moment-independent importance measure, among others, have been extensively studied. However, these indices are usually computationally expensive, and the information provided by them has some limitations for decision making. Specifically, all these indices are positive, and therefore they cannot reveal whether the effects of a basic variable on the quantity of interest are positive or adverse. In the present paper, a novel global sensitivity index is proposed when randomness is involved in structural parameters. Specifically, a functional perspective is firstly advocated, where the probability density function (PDF) of the output quantity of interest is regarded as the output of an operator on the PDF of the source basic random variables. The Frechet derivative is then naturally taken as a measure for the global sensitivity. In some sense such functional perspective provides a unified perspective on the concepts of global sensitivity and local sensitivity. In the case the change of the PDF of a basic random variable is due to the change of parameters of the PDF of the basic random variable, the computation of the Frechet-derivative-based global sensitivity index can be implemented with high efficiency by incorporating the probability density evolution method (PDEM) and change of probability measure (COM). The numerical algorithms are elaborated. Several examples are illustrated, demonstrating the effectiveness of the proposed method.