Derivative-based new upper bound of Sobol’ sensitivity measure

Global sensitivity (also called “uncertainty importance measure”)can reflect the effect of input variables on output response. The variance-based importance measure proposed by Sobol’ has highly general applicability. The Sobol’ total sensitivity index S_i ^totcan estimate the total contribution of input variables to the model output, including the self-influence of variable and the intercross influence of variable vectors. However, the computational load of S_i ^tot is extremely heavy for double-loop simulation. The main sensitivity index S_i is the lower bound of S_i ^tot, and new upper bounds of S_i ^tot based derivative are derived and proposed. New upper bounds of S_i ^tot for different variable distribution types (such as uniform, normal, exponential, triangular, beta and gamma)are analyzed, and the process and formulas are presented comprehensively according to functional analysis and the Euler–Lagrange equation. Derivative-based upper bounds are easy to implement and evaluate numerically. Several numerical and engineering examples are adopted to verify the efficiency and applicability of the presented upper bounds, which can effectively estimate the S_i ^tot value.