Mathematical Sciences: Numerical Solution of Matrix Differential Equations and Approximation of Invariant Tori

小行星9306412 Luca Dieci将在计算方程领域工作。 PI和他的合作者将围绕以下两个项目开展几个项目：（1）。 矩阵微分方程的数值解法;（2）. 不变环面的数值逼近。 更具体地说，卢卡·迪耶西希望继续研究微分黎卡提方程（DREs）和与这些方程相关的应用。 数学和理论工作是专门计划的对称DREs，在工程应用中常见的数值解。 这里的主要努力将是在研究稳定的间接解决方案的战略，而不是直接离散化。 卢卡·迪耶西还打算继续研究与矩阵方程的积分有关的数值问题，其解是一个酉矩阵。 他计划投入时间和资源的近似李雅普诺夫指数，一个问题，提供了动机研究矩阵方程的一个单一的解决方案。 最后，他打算继续工作的近似不变环面动力系统。 特别是，他希望扩展和改进以前研究的PdE方法。 在这里，研究的主要方向将涉及理论和实践研究，以完成非线性情况下的近似理论，并为了从坐标系的显式知识中解放出来的数值程序参数化的环面。 主要计划包括在使用连续理论的Fenichel，并在使其成为一个数值程序，并在仔细研究相关的实施细节。 上述项目具有强大的计算组件，旨在为计算常微分方程中的重要问题开发可靠且严格合理的算法。 ***

9306412 Dieci Luca Dieci will work in the area of computational equations. The PI and his collaborators will undertake several projects, centered around the following two: (1). Numerical Solution of Matrix Differential Equations; (2). Numerical Approximation of Invariant Tori. More in particular, Luca Dieci expects to continue working on differential Riccati equations (DREs) and on applications related to these equations. Algorithmical and theoretical work is specifically planned for the numerical solution of symmetric DREs, a common occurrence in Engineering applications. The main effort here will be in studying stable indirect solution strategies for the DREs, rather than direct discretizations. Luca Dieci intends also to continue studying numerical questions connected to the integration of matrix equations whose solution is a unitary matrix. He plans to devote time and resources to the approximation of Lyapunov exponents, an issue which provided the motivation for studying matrix equations with a unitary solution. Finally, he intends to continue the work on approximation of invariant tori dynamical systems. In particular, he expects to extend and improve upon the previously studied PdE approach. Here, the main directions of research will involve theoretical and practical study in order to complete the approximation theory for the nonlinear case, and in order to free the numerical procedures from the explicit knowledge of a coordinate system in which to parametrize the tori. The main plan consists in using the continuous theory of Fenichel and in making it into a numerical procedure, and in carefully studying the associated implementation details. The above projects have a strong computational component and aim at developing reliable and rigorously justified algorithms for important problems in computational ODEs. ***