DynSyst_Special_Topics: Collaborative Research: Reduced Dynamical Descriptions of Infinite-Dimensional Nonlinear systems via a-Priori Basis Functions from Upper Bound Theories

DynSyst_Special_Topics：协作研究：通过上界理论的先验基函数简化无限维非线性系统的动态描述

• 批准号: 0927587
• 负责人: Charles Doering
• 金额: $ 24万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-01 至 2013-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0927587&HistoricalAwards=false
• 关键词: DynSyst_Special_Topics; Collaborative; Research; Reduced; Dynamical

The aim of this interdisciplinary collaborative research project is to develop a novel model reduction technique for forced dissipative infinite-dimensional dynamical systems by employing basis functions computed using upper bound theories. Like popular Proper Orthogonal Decomposition (POD) based methods, this approach associates the condensed variables needed for model reduction with coherent structures and captures nonlinear interactions between these linear modes via Galerkin projection and finite-dimensional truncation. Unlike empirical POD methods, however, this new method does not require extensive data sets from experiments or direct numerical simulations of the governing partial differential equations (PDEs) and thus yields truly predictive reduced models. The theoretical and computational methodology will be developed in the context of a particular physical system, thermal convection in fluid saturated porous media, that is of considerable environmental and technological importance and an ideal testbed for new ideas.This research will contribute to the development of a general methodology for deriving simplified mathematical models of highly complex dynamical systems arising in diverse areas of science and engineering. In many applications of interest (e.g., control of various fluid flows to achieve drag reduction for oil pumped in pipelines or for air flowing past commercial jets, or for estimation of carbon dioxide sequestration by porous rock material for reducing global warming), direct numerical simulations based on the complete governing mathematical equations are infeasible using even the world's fastest high-performance supercomputers. This project will address these challenges using novel mathematical techniques to derive simplified equations directly from the governing physical laws that are amenable to practical computation and analysis.

这个跨学科的合作研究项目的目的是开发一种新的强迫耗散无限维动力系统的模型降阶技术，采用上限理论计算的基函数。 像流行的适当的正交分解（POD）为基础的方法，这种方法相关联的凝聚变量模型简化与相干结构，并通过伽辽金投影和有限维截断捕捉这些线性模式之间的非线性相互作用。 然而，与经验POD方法不同，这种新方法不需要大量的数据集从实验或直接数值模拟的偏微分方程（PDE），从而产生真正的预测减少模型。理论和计算方法将在一个特定的物理系统，在流体饱和多孔介质中的热对流，这是相当大的环境和技术的重要性和一个理想的试验台为新的ideas.This研究将有助于开发一个通用的方法来推导出简化的数学模型的高度复杂的动力系统中出现在不同的科学和工程领域的背景下发展。 在许多感兴趣的应用中（例如，尽管这些技术（例如，控制各种流体流动以实现管道中泵送的油或流过商用喷气机的空气的减阻，或用于估计多孔岩石材料的二氧化碳封存以减少全球变暖），但基于完整的控制数学方程的直接数值模拟即使使用世界上最快的高性能超级计算机也是不可行的。该项目将使用新的数学技术来解决这些挑战，直接从适用于实际计算和分析的物理定律中推导出简化的方程。