Uncertainty Quantification and Reduction Using Sensitivity Analysis and Hessian Derivatives

We study the use of Hessian interaction terms to quickly identify design variables that reduce variability of system performance. To start we quantify the uncertainty and compute the variance decomposition to determine noise variables that contribute most, all at an initial design. Minimizing the uncertainty is next sought, though probabilistic optimization becomes computationally difficult, whether by including distribution parameters as an objective function or through robust design of experiments. Instead, we consider determining the more easily computed Hessian interaction matrix terms of the variance-contributing noise variables and the variables of any proposed design change. We also relate the Hessian term coefficients to subtractions in Sobol indices and reduction in response variance. Design variable changes that can reduce variability are thereby identified quickly as those with large Hessian terms against noise variables. Furthermore, the Jacobian terms of these design changes can indicate which design variables can shift the mean response, to maintain a desired nominal performance target. Using a combination of easily computed Hessian and Jacobian terms, design changes can be proposed to reduce variability while maintaining a targeted nominal. Lastly, we then recompute the uncertainty and variance decomposition at the more robust design configuration to verify the reduction in variability. This workflow therefore makes use of UQ/SA methods and computes design changes that reduce uncertainty with a minimal 4 runs per design change. An example is shown on a Stirling engine design where the top four variance-contributing tolerances are matched with two design changes identified through Hessian terms, and a new design found with 20% less variance.