Data-driven stochastic analysis of flow in random heterogeneous media

本提案涉及异质随机介质中输运现象的分析，重点是这些现象所表现出的性质的多长度尺度变化，以及固有的有限信息，可用于量化这些性质变化，因此有必要将这些现象作为随机过程。正在开发的非侵入性随机多尺度框架有三个关键组成部分：(a)将（多尺度）材料特性（渗透率）可变性的有限可用信息编码为降阶随机输入模型的计算框架，(b)用于解决所涉及的随机偏微分方程的自适应稀疏网格搭配框架，以及(c)用于解决随机多尺度问题的跨长度尺度交换信息的数学一致策略。数据驱动的降阶随机输入模型构建的关键概念是嵌入在高维空间中的流形的低维参数化。稀疏网格配置方法仅基于对相应确定性物理模拟器的函数调用来构造随机解。该框架是基于多维层次基函数的。通过使用本地支持来确保适应性和收敛性，而通过仔细选择适当的数据结构来保证可伸缩性。信息传递策略基于随机和多尺度算法的解耦结构。这一研究结果将影响对随机介质中流动过程的理解。随机非均质介质中的热和水动力输运是普遍存在的过程，发生在各种尺度上，从大尺度(例如地热能系统、石油开采、地质加热地球？（如复合材料传热、多晶传热、孔隙流动、凝固过程中的枝晶间流动、流化床传热等）。对这类介质的热和流体力学行为进行预测建模的科学、技术和经济兴趣日益浓厚。此外，这项工作对于理解由于可用的缺口和不准确的描述数据而难以理解和/或控制的其他系统是有价值的。解决的问题提供了一个独特的和有价值的训练机会，让学生学习，发展和应用尖端的计算数学技术，以各种复杂的系统。

This proposal concerns the analysis of transport phenomena in heterogeneous random media with emphasis on the multi-length scale variations in properties that these phenomena exhibit, and the inherently limited information available to quantify these property variations that necessitates posing these phenomena as stochastic processes. The non-intrusive stochastic multiscale framework being developed has three key components: (a) A computational framework that encodes the limited information available about the variability of the (multiscale) material properties (permeability) into a reduced-order stochastic input model, (b) An adaptive sparse grid collocation framework for solving the stochastic PDEs involved and (c) A mathematically consistent strategy to exchange information across length scales for the solution of stochastic multiscale problems. The key concept explored in the data-driven reduced-order stochastic input model construction is the low-dimensional parametrization of manifolds embedded in high-dimensional spaces. The sparse grid collocation approach constructs the stochastic solution solely based on function calls to the corresponding deterministic physical simulator. The framework is based on hierarchical basis functions in multiple dimensions. Adaptivity and convergence are ensured by utilizing a local support while scalability is guaranteed by the careful choice of appropriate data structures. The information transfer strategies are based on the decoupled structure of the stochastic and multiscale algorithms. The results of this research will impact the understanding of flow processes in random media. Thermal and hydrodynamic transport in random heterogeneous media are ubiquitous processes occurring in various scales ranging from the large scale (e.g. geothermal energy systems, oil recovery, geological heating of the earth?s crust) to smaller scales (e.g. heat transfer through composites, polycrystals, flow through pores, inter-dendritic flow in solidification, heat transfer through fluidized beds). There has been increasing scientific, technological and economic interests in predictive modeling of the thermal and hydrodynamic behavior of such media. In addition, this work can be valuable in understanding other systems that are poorly understood and/or controlled due to the gappy and inaccurate data available for their description. The problems addressed provide a unique and valuable training opportunity for students to learn, develop and apply cutting edge computational mathematics techniques to a variety of complex systems.