Mathematical Sciences: Towards Model Reduction of Dissipative PDE's: Theory, Computation, and Applications

9308774无限维系统的低维动力学行为是一个密集的数学研究领域，这项工作的理论部分探索并建立在许多这样的方向上。这位研究人员和他的同事们研究了惯性流形和近似惯性流形理论的几个新方面，并探索和建立了它们与经验特征函数方法之间的新联系。同时，该提议的一个目标是对新方法的计算实施和与其相关的某些数值分析方面的研究(稳定性、与对称性的相互作用和保持耗散)。耗散的含时偏微分方程解源于对各种重要的物理、化学和生物现象的描述。本项目的目的是开发、分析、计算实现和测试将此类方程简化为精确的小型动力系统的方法。这本身就是一个研究兴趣浓厚的课题；同时，它为许多复杂的物理化学和工程系统的研究提供了一座桥梁，对它们的动态行为的模拟、分析、预测和控制具有重要的意义。这项工作的一个组成部分是将这些技术应用于当前研究感兴趣的现实系统的模型，例如催化和电化学反应中的图案形成。这位研究人员和他的同事们认为，解决现实问题的尝试加强了研究的理论和计算方面，并提供了新的想法和方向。

9308774 Titi The low-dimensional dynamic behavior of infinite dimensional systems is a field of intense mathematical research, and the theoretical part of this work explores and builds on a number of such directions. The investigator and his colleagues study several new aspects of the theory of inertial and approximate inertial manifolds, and explore and establish new connections between them and the method of empirical eigenfunctions. At the same time, a goal of the proposal is the computational implementation of the new methods and the study of certain numerical analysis aspects associated with them (stability, interaction with symmetries, and preservation of dissipation). Dissipative time-dependent partial differential equations arise from descriptions of a variety of important physical, chemical, and biological phenomena. The purpose of this project is to develop, analyze, computationally implement, and test methods for the reduction of such equations to accurate, small dynamical systems. This is a subject of intense research interest in its own right; at the same time, it provides a bridge to the study of many complex physicochemical and engineering systems, with important implications for the simulation, analysis, prediction and control of their dynamic behavior. An integral part of this work is the application of the techniques to models of realistic systems of current research interest, such as pattern formation in catalytic and electrochemical reactions. The investigator and his colleagues believe that the attempt to tackle realistic problems strengthens the theoretical and computational aspects of the research, and provides new ideas and directions.