Data-driven stochastic analysis of flow in random heterogeneous media

这一建议涉及分析非均匀随机介质中的输运现象，重点是这些现象表现出的多长度尺度的特性变化，以及可用于量化这些特性变化的内在有限的信息，这些信息需要将这些现象作为随机过程来假设。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)将关于(多尺度)材料性质(渗透率)变异性的有限信息编码到降阶随机输入模型中的计算框架，(B)用于求解所涉及的随机偏微分方程组的自适应稀疏网格配置框架，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。数据驱动降阶随机输入模型构建的关键概念是嵌入高维空间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。该框架基于多维层次基函数。通过利用本地支持来确保适应性和收敛，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算法和多尺度算法的解耦结构。这项研究的结果将影响对随机介质中流动过程的理解。随机非均匀介质中的热和水动力输运是从大尺度(如地热能系统、石油开采、地球-S地壳的地质加热)到小尺度(如复合材料、多晶、孔隙流动、凝固中的枝晶间流动、沸腾床中的换热)的无处不在的过程。对这类介质的热力和水动力行为的预测建模已经引起了越来越多的科学、技术和经济兴趣。此外，这项工作对于理解其他系统可能很有价值，这些系统由于可用于描述它们的不准确和不准确的数据而很难被理解和/或控制。所解决的问题为学生提供了一个独特而宝贵的培训机会，让他们学习、开发并将尖端计算数学技术应用于各种复杂系统。

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.