A Bayesian approach for quantile optimization problems with high-dimensional uncertainty sources

• Published in: Computer methods in applied mechanics and engineering 2021-04, Vol.376, p.113632, Article 113632
• Main Authors: Sabater, Christian, Le Maître, Olivier, Congedo, Pietro Marco, Görtz, Stefan
• Format: Article
• Language: English
• Subjects: Aerodynamics; Applications; Bayesian analysis; Bayesian quantile regression; Computational Fluid Dynamics; Engineering Sciences; Fluids mechanics; High dimensional problems; Mathematics; Mechanics; Optimization; Optimization and Control; Optimization under uncertainty; Parameterization; Probability; Redevelopment; Regression analysis; Regression models; Robust design; Samples; Statistical analysis; Statistical methods; Statistics; Uncertainty
• ISSN: 0045-7825; 1879-2138
• DOI: 10.1016/j.cma.2020.113632

Robust optimization strategies typically aim at minimizing some statistics of the uncertain objective function and can be expensive to solve when the statistic is costly to estimate at each design point. Surrogate models of the uncertain objective function can be used to reduce this computational cost. However, such surrogate approaches classically require a low-dimensional parametrization of the uncertainties, limiting their applicability. This work concentrates on the minimization of the quantile and the direct construction of a quantile regression model over the design space, from a limited number of training samples. A Bayesian quantile regression procedure is employed to construct the full posterior distribution of the quantile model. Sampling this distribution, we can assess the estimation error and adjust the complexity of the regression model to the available data. The Bayesian regression is embedded in a Bayesian optimization procedure, which generates sequentially new samples to improve the determination of the minimum of the quantile. Specifically, the sample infill strategy uses optimal points of a sample set of the quantile estimator. The optimization method is tested on simple analytical functions to demonstrate its convergence to the global optimum. The robust design of an airfoil’s shock control bump under high-dimensional geometrical and operational uncertainties serves to demonstrate the capability of the method to handle problems with industrial relevance. Finally, we provide recommendations for future developments and improvements of the method. •Bayesian optimization framework that can handle a large number of uncertainties.•Sampling the quantile posterior distribution characterizes the uncertainty in the quantile estimator.•Bayesian model is enriched using points from the current distribution of the estimated optimum.•Application to aerodynamic robust design of shock control bump subject to many uncertainties.