Quantifying uncertainties in physical or engineering systems often requires a large number of simulations of the underlying computer models that are computationally intensive. Emulators or surrogate models are often used to accelerate the computation in such problems, and in this regard the Gaussian Process (GP) emulator is a popular choice for its ability to quantify the approximation error in the emulator itself. However, a major limitation of the GP emulator is that it can not handle problems of very high dimensions, which is often addressed with dimension reduction techniques. In this work we hope to address an issue that the models of interest are so complex that they admit different low dimensional structures in different parameter regimes. Building upon the active subspace method for dimension reduction, we propose a clustered active subspace method which identifies the local low-dimensional structures as well as the parameter regimes they are in (represented as clusters), and then construct low dimensional and local GP emulators within the clusters. Specifically we design a clustering method based on the gradient information to identify these clusters, and a local GP construction procedure to construct the GP emulator within a local cluster. With numerical examples, we demonstrate that the proposed method is effective when the underlying models are of complex low-dimensional structures.
翻译:测量物理或工程系统中的不确定性往往需要大量模拟计算密集的基本计算机模型。模拟模型或代理模型往往被用来加速这些问题的计算。在这方面,高山进程模拟器是对其量化模拟器本身近似误差能力的一种受欢迎的选择。然而,GP模拟器的一个主要局限性在于它无法处理非常高的尺寸问题,而这一问题往往通过降低尺寸技术加以解决。在这项工作中,我们希望解决一个问题,即各种感兴趣的模型非常复杂,以至于它们在不同参数系统中接受不同的低维结构。在使用主动的子空间子空间方法减少尺寸的基础上,我们建议采用集束积极的子空间方法,确定当地低维结构及其在模拟器本身中的参数系统(作为集群的代表性),然后在集群内建造低维和地方的GP模拟器。具体地,我们根据梯度信息设计一种集群方法来识别这些集群,以及一个本地GP模型的构建程序,以当地多维模型为基础,我们用当地数字模型为核心。