A data-driven Bayesian framework for the solution of SPDEs on random heterogeneous media

PI和他的同事们研究随机偏微分方程（SPDEs）的解，重点是在随机异质介质中建模过程。采用了一个整体的数学观点来看待这个问题，其中随机输入是根据实验数据明确建模的，不确定性是通过确定性求解器传播的。建议的工作重点如下：(A)数据驱动的随机输入建模。这包括开发实验观察到的输入场的降阶表示，构建从降阶输入空间到高维输入空间的映射，以及估计由观测引起的降阶输入空间的概率密度。(B)不确定性传播。这包括构建一个层次的、局部的、贝叶斯代理的确定性代码，通过使用主动学习的确定性运行的信息选择来提高效率，求解器离散化的顺序精炼，输出相关性的明确处理和/或输出降维，以及代理的解析/数值积分，用于计算响应统计数据和误差条。在这个项目中执行的工作允许更好地理解在工程和自然系统的分析和设计中考虑可变性的需要。正是这样的思考，将有助于在一个容易受到自然和制造不确定性影响的环境中实现组件和系统的可重用多功能设计。特别是，本研究系统地解决了随机异构介质建模中出现的关键问题（现有的，非常昂贵的确定性多尺度求解器，很少的实验输入实现，响应的非各向同性，输入/输出的高维性），并为其解决提出了具体的数学问题。它作为一个范例，成功地结合了应用数学和计算统计学的看似不同的想法，以回答现实的工程问题。信息性（数据驱动）随机模拟将提高安全性，并极大地有利于众多行业的设计和优化过程，同时降低成本。参与这项研究的学生在计算数学和贝叶斯统计的接口训练。组织的研究活动和课程使许多对不确定性存在下的预测科学和设计感兴趣的不同社区受益。开发的算法、数据和软件通过网络向社区传播。

The PI and his colleagues study the solution of Stochastic Partial Differential Equations (SPDEs) with emphasis on modeling processes in random heterogeneous media. A holistic mathematical view of the problem is adopted in which the stochastic input is explicitly modeled from experimental data and uncertainty is propagated via a deterministic solver. The key aspects of the proposed work are as follows: (A) Data-driven stochastic input modeling. This includes the development of reduced-order representations of experimentally observed input fields, construction of the mapping from the reduced input space to the high dimensional input space and estimating the probability density of the reduced input space as induced by the observations. (B) Uncertainty propagation. This includes the construction of a hierarchical, local, Bayesian surrogate of the deterministic code made efficient by the informative selection of deterministic runs using active learning, the sequential refining of the discretization of the solver, the explicit treatment of output correlations and/or output dimensionality reduction and analytical/ numerical integration of the surrogate for the calculation of the response statistics as well as error bars. The work performed in this project allows better appreciation for the need to account for variabilities in the analysis and design of engineered and natural systems. It is such thinking that will contribute towards re-usable multifunctional design of components and systems in an environment prone to natural and manufacturing uncertainties. In particular, this research systematically addresses key problems that arise in modeling random heterogeneous media (existing, very expensive deterministic multiscale solvers, few experimental input realizations, non-isotropy of the response, high-dimensionality of input/output) and poses concrete mathematical questions for their resolution. It serves as a paradigm of a successful combination of seemingly diverse ideas of applied mathematics and computational statistics in order to answer realistic engineering questions. Informative (data-driven) stochastic simulations would improve safety and greatly benefit the design and optimization process for a wide array of industries while at the same time reducing costs. The student that participates in this research is trained in the interface of computational mathematics and Bayesian statistics. The research activity and courses organized benefit many diverse communities that have interests in predictive science and design in the presence of uncertainties. The algorithms, data and software developed are disseminated to the community through the web.