Data Analytic UQ Cascade

This contribution gathers some of the ingredients presented at Erice during the third workshop on “Variational Analysis and Aerospace Engineering.” It is a collection of several previous publications on how to set up an uncertainty quantification (UQ) cascade with ingredients of growing computational complexity for both forward and reverse uncertainty propagation. It uses data analysis ingredients in a context of existing deterministic simulation platforms. It starts with a complexity-based splitting of the independent variables and the definition of a parametric optimization problem. Geometric characterization of global sensitivity spaces through their dimensions and relative positions through principal angles between vector spaces bring a first set of information on the impact of uncertainties of the functioning parameters on the optimal solution. Joining the multi-point descent direction and probability density function quantiles of the optimization parameters permits to define the notion of directional extreme scenarios (DES) without sampling of large dimension design spaces. One goes beyond DES with ensemble Kalman filters (EnKF) after the multi-point optimization algorithm is cast into an ensemble simulation environment. This formulation accounts for the variability in large dimension. The UQ cascade continues with the joint application of the EnKF and DES leading to the concept of ensemble directional extreme scenarios which provides a more exhaustive description of the possible extreme scenarios. The different ingredients developed for this cascade also permit to quantify the impact of state uncertainties on the design and provide confidence bounds for the optimal solution. This is typical of inverse designs where the target should be assumed uncertain. Our proposal uses the previous DES strategy applied this time to the target data. We use these scenarios to define a matrix having the structure of the covariance matrix of the optimization parameters. We compare this construction to another one using available adjoint-based gradients of the functional. Eventually, we go beyond inverse design and apply the method to general optimization problems. The ingredients of the paper have been applied to constrained aerodynamic performance analysis problems.