Marching Over Poles: Innovative Ways to Solve Matrix Differential Riccati Equations

矩阵微分里卡提方程(MDRE)在应用数学、科学和工程中经常出现。特别是，它们在最优控制、滤波和估计、基本粒子物理和微分方程组的两点边值问题中发挥着重要作用。在过去的25年里，已经提出了许多算法来数值求解MDRE。其中包括重新调整的通用数值方法，以利用当今的计算技术来获得更好的性能。然而，这些方法由于其普遍性，本质上没有也不能充分利用MDRE的许多有价值的结构特性，其中一些仍有待发现。这个项目的目标是建立对MDRE长期解行为的更好的数学理解，特别是那些解中有奇点的解，并推导出能够优雅地处理奇点的新的数值方法，不同于通常使用的中止和重新开始过程。一个新的MDRE解算器包，作为更好的数学理解、新的算法和对当今微处理器中IEEE浮点异常处理能力的利用的结果，预计将在该项目成功完成后公开发布，供科学界使用。具有新兴的数值微分方程式专业知识的研究生将参与其中。

Matrix Differential Riccati Equations (MDREs) arise frequently throughout applied mathematics, science, and engineering. In particular, they play major roles in optimal control, filtering and estimation, elementary particle physics, and two-point boundary value problems of differential equations. A number of algorithms have been proposed over the past 25 years for solving MDREs numerically. These include general-purpose numerical methods retooled to take advantage of today's computing technology for better performance. Nevertheless, these methods, because of their generality, inherently do not and cannot fully exploit many valuable structural properties of MDREs, some of which remain to be discovered. The objectives of this project are to build a better mathematical understanding of the long-term solution behavior of MDREs, especially those that have singularities in the solutions, and to derive new numerical methods that are able to gracefully handle singularities, unlike the commonly used abort-and-restart procedures. A new MDRE solver package as the result of a better mathematical understanding, new algorithms, and the exploitation of IEEE floating-point exception handling capabilities in today's microprocessors is expected to be released publicly for the use of scientific communities upon the successful completion of the project. Graduate students with emerging expertise in numerical differential equations will be involved.