Sequential Sampling Framework for Metamodeling Uncertainty Reduction in Multilevel Optimization of Hierarchical Systems

Metamodels instead of computer simulations are often adopted to reduce the computational cost in the uncertainty-based multilevel optimization. However, metamodel techniques may bring prediction discrepancy, which is defined as metamodeling uncertainty, due to the limited training data. An unreliable solution will be obtained when the metamodeling uncertainty is ignored, while an overly conservative solution, which contradicts the original intension of the design, may be got when both parametric and metamodeling uncertainty are treated concurrently. Hence, an adaptive sequential sampling framework is developed for the metamodeling uncertainty reduction of multilevel systems to obtain a solution that approximates the true solution. Based on the Kriging model for the probabilistic analytical target cascading (ATC), the proposed framework establishes a revised objective-oriented sampling criterion and sub-model selection criterion, which can realize the location of additional samples and the selection of subsystem requiring sequential samples. Within the sampling criterion, the metamodeling uncertainty is decomposed by the Karhunen–Loeve expansion into a set of stochastic variables, and then polynomial chaos expansion (PCE) is used for uncertainty quantification (UQ). The polynomial coefficients are encoded and integrated in the selection criterion to obtain subset sensitivity indices for the sub-model selection. The effectiveness of the developed framework for metamodeling uncertainty reduction is demonstrated on a mathematical example and an application.