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

无限维系统的低维动态行为是一个激烈的数学研究领域，而这项工作的理论部分探索和建立在许多这样的方向上。研究者和他的同事研究了惯性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，该提案的目标是新方法的计算实现以及与之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耗散的保持）。耗散时变偏微分方程产生于对各种重要的物理、化学和生物现象的描述。该项目的目的是开发、分析、计算实现和测试方法，将这些方程简化为精确的小动力系统。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，对其动态行为的模拟、分析、预测和控制具有重要意义。这项工作的一个组成部分是将这些技术应用于当前研究兴趣的现实系统模型，例如催化和电化学反应中的模式形成。研究者和他的同事们认为，解决现实问题的尝试加强了研究的理论和计算方面，并提供了新的思路和方向。

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.