Multi-Element Stochastic Galerkin Method Based on Edge Detection for Uncertainty Quantification of Discontinuous Responses

We propose a new multi-element generalized polynomial chaos (MEgPC) method to minimize the computational costs required for the existing MEgPC to circumvent the Gibbs phenomenon in the presence of discontinuities in a random space. The proposed method uses edge detection to capture the discontinuous behavior of a solution with minimal decomposition. In contrast, the existing MEgPC iterates splitting the random space into two equal parts until achieving a sufficient resolution level. We take advantage of the fact that the stochastic Galerkin (SG) methods facilitate adaptive refinement of the decomposition at every time-step during a computation for the proposed method. The numerical experiments for two-test problems demonstrate the performance of the proposed method. The results show that the proposed method is consistently more accurate than conventional methods for sufficiently high polynomial orders with minimal additional computational costs to capture discontinuities.

A Classifier-Guided Sampling Method for Computationally Expensive, Discrete-Variable, Discontinuous Design Problems

Metamodel-based design is a well-established method for providing fast and accurate approximations of expensive computer models to enable faster optimization and rapid design space exploration. Traditionally, a metamodel is developed by fitting a surface to a set of training points that are generated with an expensive computer model or simulation. A requirement of this process is that the function being approximated is continuous. However, many engineering problems have variables that are discrete and a function response that is discontinuous in nature. In this paper, a classifier-guided sampling method is presented that can be used for optimization and design space exploration of expensive computer models that have discrete variables and discontinuous responses. The method is tested on a set of example problems. Results show that the method significantly improves the rate of convergence towards known global optima, on average, when compared to random search.