Quantile-based optimization under uncertainties using adaptive Kriging surrogate models

Uncertainties are inherent to real-world systems. Taking them into account is crucial in industrial design problems and this might be achieved through reliability-based design optimization (RBDO) techniques. In this paper, we propose a quantile-based approach to solve RBDO problems. We first transform the safety constraints usually formulated as admissible probabilities of failure into constraints on quantiles of the performance criteria. In this formulation, the quantile level controls the degree of conservatism of the design. Starting with the premise that industrial applications often involve high-fidelity and time-consuming computational models, the proposed approach makes use of Kriging surrogate models (a.k.a. Gaussian process modeling). Thanks to the Kriging variance (a measure of the local accuracy of the surrogate), we derive a procedure with two stages of enrichment of the design of computer experiments (DoE) used to construct the surrogate model. The first stage globally reduces the Kriging epistemic uncertainty and adds points in the vicinity of the limit-state surfaces describing the system performance to be attained. The second stage locally checks, and if necessary, improves the accuracy of the quantiles estimated along the optimization iterations. Applications to three analytical examples and to the optimal design of a car body subsystem (minimal mass under mechanical safety constraints) show the accuracy and the remarkable efficiency brought by the proposed procedure

Quantile forecast discrimination ability and value

While probabilistic forecast verification for categorical forecasts is well established, some of the existing concepts and methods have not found their equivalent for the case of continuous variables. New tools dedicated to the assessment of forecast discrimination ability and forecast value are introduced here, based on quantile forecasts being the base product for the continuous case (hence in a nonparametric framework). The relative user characteristic (RUC) curve and the quantile value plot allow analysing the performance of a forecast for a specific user in a decision-making framework. The RUC curve is designed as a user-based discrimination tool and the quantile value plot translates forecast discrimination ability in terms of economic value. The relationship between the overall value of a quantile forecast and the respective quantile skill score is also discussed. The application of these new verification approaches and tools is illustrated based on synthetic datasets, as well as for the case of global radiation forecasts from the high resolution ensemble COSMO-DE-EPS of the German Weather Service