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

这一跨学科合作研究项目的目的是通过利用上界理论计算的基函数来开发一种新的强迫耗散无限维动力系统的模型降阶技术。与流行的基于本征正交分解(POD)的方法一样，该方法将模型降阶所需的凝聚变量与相干结构联系起来，并通过Galerkin投影和有限维截断来捕捉这些线性模式之间的非线性相互作用。然而，与经验POD方法不同的是，这种新方法不需要大量的实验数据或直接对控制偏微分方程组(PDE)进行数值模拟，从而产生真正可预测的简化模型。理论和计算方法将在特定的物理系统的背景下发展，流体饱和多孔介质中的热对流具有相当大的环境和技术重要性，并且是新思想的理想试验台。这项研究将有助于发展一种通用的方法来推导在不同的科学和工程领域中出现的高度复杂的动力系统的简化数学模型。在许多有意义的应用中(例如，控制各种流体流动以实现管道中泵送的石油或流经商业喷气式飞机的空气的减阻，或者用于估计多孔岩石材料的二氧化碳封存以减少全球变暖)，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。这个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支配物理定律推导出简化的方程。

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.