Surrogate modelling techniques have opened up new possibilities to overcome the limitations of computationally intensive numerical models in various areas of engineering and science. However, while fundamental in many engineering applications and decision-making, the incorporation of uncertainty quantification into meta-models remains a challenging open area of research. To address this issue, this paper presents a novel stochastic simulation approach combining sparse polynomial chaos expansion (PCE) and Stochastic Kriging (SK). Specifically, the proposed approach adopts adaptive sparse PCE as the trend model in SK, achieving both global and local prediction capabilities and maximizing the role of the stochastic term to conduct uncertainty quantification. To maximize the generalization and computational efficiency of the meta-model, the Least Angle Regression (LAR) algorithm is adopted to automatically select the optimal polynomial basis in the PCE. The computational effectiveness and accuracy of the proposed approach are appraised through a comprehensive set of case studies and different quality metrics. The presented numerical results and discussion demonstrate the superior performance of the proposed approach compared to the classical ordinary SK model, offering high flexibility for the characterization of both extrinsic and intrinsic uncertainty for a wide variety of problems.
Sparse approximation of data-driven Polynomial Chaos expansions: an induced sampling approach
在前向不确定性量化（UQ）领域中的一个开放问题是形成仅具有关于随机输入的分布的不完全信息的不确定性的准确评估的能力。另一个挑战是有效地利用有限的训练数据进行复杂工程问题的UQ预测，特别是高维随机参数。我们解决这些挑战相结合的数据驱动的多项式混沌扩展与最近开发的预处理稀疏近似方法的UQ问题。在这个两步过程中的第一个任务是使用（Ahlfeld et al. 2016）中开发的程序，使用随机输入的有限数量的统计矩来构建“任意”多项式混沌扩展基。第二步是一个新的程序，通过$\ell^1$最小化，以量化的前向不确定性的影响稀疏近似。为了提高预处理的$\ell^1$最小化问题的性能，我们从所谓的诱导分布中采样，而不是从原始的未知概率测度中使用Monte Carlo（MC）采样。我们证明了测试问题，诱导抽样是一个有竞争力的，往往更好的选择相比，从渐近最优的措施（如平衡测度），当我们有不完整的信息分布的抽样。我们证明了能力的诱导抽样算法通过稀疏表示有限的数据上的检验函数，和随机杨氏模量的Kirchoff板弯曲问题。
A generalized sampling and preconditioning scheme for sparse approximation of polynomial chaos expansions
Computer Science Research Institute, Sandia National Laboratories（桑迪亚国家实验室计算机科学研究所）；Scientific Computing and Imaging (SCI) Institute and Mathematics Department, University of Utah（犹他大学科学计算与成像研究所及数学系）；Institute of Computational Mathematics and the Chinese Academy of Sciences（中国科学院计算数学研究所）
Polynomial Chaos Expansion of random coefficients and the solution of stochastic partial differential equations in the Tensor Train format
Max-Planck-Institut für Dynamik komplexer technischer Systeme（马克斯·普朗克复杂技术系统动力学研究所）；Max-Planck-Institut für Mathematik in den Naturwissenschaften（马克斯·普朗克自然科学数学研究所）；King Abdullah University of Science and Technology (KAUST)（阿卜杜拉国王科技大学）；Institute for Scientific Computing, Technische Universität Braunschweig（布伦瑞克工业大学科学计算研究所）
Discovery of interpretable structural model errors by combining Bayesian sparse regression and data assimilation: A chaotic Kuramoto-Sivashinsky test case
