Surrogate Modeling for Stochastic Simulators Using Statistical Approaches

Abstract

Nowadays, more and more complex interdependent infrastructures and networks are developed in engineering. The design and maintenance of such systems increasingly call for advanced computational models to optimize their performance and assess their reliability under various operational conditions. Unlike many conventional simulators that are deterministic, stochastic simulators feature intrinsic stochasticity. More precisely, they produce different results when run multiple times with a given set of input parameters. Due to this random nature, repeated model evaluations of the stochastic model with the same input value, called replications, are necessary to fully characterize the probability distribution of the associated model response. For the purpose of optimization or uncertainty quantification (e.g., uncertainty propagation or sensitivity analysis), computational models typically need to be evaluated a large number of times. The additional layer of randomness due to the intrinsic stochasticity of stochastic simulators makes it even more computationally demanding to perform these complex analyses. A common practice to alleviate the prohibitive cost associated with expensive simulators is to build surrogate models, which behave similarly to the original model but are much cheaper to evaluate. Contrary to the deterministic case, surrogate modeling of stochastic simulators has only emerged in the past decade. The main challenge in this field is that one model evaluation yields only a single realization of the random model response associated with the given input value. In other words, one run of a stochastic simulator provides proportionally much less information than that of a deterministic one. This thesis focuses on developing efficient and accurate surrogate models to emulate the response distribution of stochastic simulators, combining statistical methods with state-of-the-art deterministic surrogate modeling techniques. To this end, we propose two new approaches: the generalized lambda model (GLaM) and the stochastic polynomial chaos expansion (SPCE). The first one capitalizes on the use of the generalized lambda distribution to characterize the random nature of the simulator response. The distribution parameters are functions of the input variables and are represented by polynomial chaos expansions (PCEs). We explore replication-based methods to build GLaMs and improve their performance by an additional joint optimization of the overall likelihood function. We further elaborate this idea and develop a new method that does not require replications. Using this surrogate, we investigate sensitivity analysis for stochastic simulators. The second class of stochastic surrogates, SPCE, overcomes the main shortcoming of GLaM, which is unable to represent multimodal distributions. In this more versatile stochastic emulator, we extend PCE by introducing an artificial latent variable to the expansion and an additive noise variable to mimic the intrinsic stochasticity of the simulator. We also propose an adaptive algorithm to construct the surrogate model without the need for replications. For both stochastic surrogate models, we investigate basic theoretical properties of the primary estimation method. Analytical examples and engineering applications, including wind turbine design and seismic fragility analysis, are used to validate and illustrate the performance of the new approaches. Furthermore, these engineering case studies provide valuable insights into the applicability of the developed framework to real-world industrial problems.