Abstract

Bayesian calibration of a functional input/parameter to a time-consuming simulator based on a Gaussian process (GP) emulator involves two challenges that distinguish it from other parameter calibration problems. First, one needs to specify a flexible stochastic process prior for the input, and reduce it to a tractable number of random variables. Second, a sequential experiment design criterion that decreases the effect of emulator prediction uncertainty on calibration results is needed and the criterion should be scalable for high-dimensional input and output. In this article, we address these two issues. For the first issue, we employ a GP with a prior density for its correlation parameter as prior for the functional input, and the Karhunen-Loève (KL) expansion of this non-Gaussian stochastic process to reduce its dimension. We show that this prior gives far more robust inference results than a GP with a fixed correlation parameter. For the second issue, we propose the weighted prediction variance (WPV) criterion (with posterior density of the calibration parameter as weight) and prove the consistency of the sequence of emulator-based likelihoods given by the criterion. The proposed method is illustrated with examples on hydraulic transmissivity estimation for groundwater models.

摘要

基于高斯过程 (GP) 仿真器的耗时模拟器的功能输入/参数的贝叶斯校准涉及将其与其他参数校准问题区分开来的两个挑战。首先，需要先为输入指定一个灵活的随机过程，并将其减少为可处理的随机变量数量。其次，需要一个减少仿真器预测不确定性对校准结果的影响的顺序实验设计标准，并且该标准应该可扩展用于高维输入和输出。在本文中，我们将解决这两个问题。对于第一个问题，我们采用具有先验密度的 GP 作为其相关参数的先验作为功能输入的先验，并使用该非高斯随机过程的 Karhunen-Loève (KL) 展开来降低其维数。我们表明，与具有固定相关参数的 GP 相比，该先验给出了更稳健的推理结果。对于第二个问题，我们提出了加权预测方差（WPV）准则（以校准参数的后验密度为权重），并证明了该准则给出的基于仿真器的似然序列的一致性。所提出的方法以地下水模型的水力渗透率估计为例进行说明。