Mathematical Sciences: Numerical Aspects of Riccati Transformation, Invariant Manifold Approximation, and Connected Issues

数学科学：Riccati 变换的数值方面、不变流形逼近和相关问题

• 批准号: 9104564
• 负责人: Luca Dieci
• 金额: $ 3.92万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1991
• 资助国家: 美国
• 起止时间: 1991-09-01 至 1994-02-28
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9104564&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Numerical; Aspects; Riccati

The principal investigator will conduct research on some analytical and computational problems related to the Riccati transformation, the superstability phenomenon for integration of stiff differential equations, and invariant manifold approximation. This work involves (i) further study, implementation and software production of reliable algorithms for integrating differential Riccati equations, (ii) using the differential Riccati equation to perform mesh-selection for stiff systems of two-point boundary value problems, (iii) software production of a code for nonlinear two-point boundary value problems based on the Riccati method. He will further study the so-called superstability phenomenon arising during adaptive integration of stiff differential equations, seeking efficient techniques that avoid the practical superstability occurrences and that can be incorporated into general purpose ordinary differential equation solvers. Finally, he will continue working on numerical approximation of invariant manifolds, in fact on the numerical solution of the associated systems of hyperbolic partial differential equations. This involves (i) approximation analysis for the nonlinear case, (ii) efficient solution techniques (multigrid approaches) for the resulting linear systems, (iii) exploitation of the link with the Riccati transformation. Most realistic physical phenomena are modelled by complicated nonlinear systems of differential equations. Exact solution of these systems is hopeless, and numerical simulation is a necessity. This work aims at producing reliable, and rigorously justified, algorithms to solve some of these systems. The intent is to produce software tools for the working scientist. In particular, work on the Riccati equation will be of value to people working in Games' Theory, Optimal Control and connected Engineering fields. The work on so-called stiff differential equations is useful in solving systems having different time-scales, such as in chemical kinetics and aerodynamics applications. Finally, the work on approximation of invariant manifolds is useful in understanding the long-term behavior of nonlinear dynamical systems. This has applications in many areas of applied science.

主要研究Riccati变换、刚性微分方程积分的超稳定性现象、不变流形逼近等相关的分析和计算问题。这项工作涉及(i)进一步研究、实现和软件生产可靠的积分微分Riccati方程算法，（ii）使用微分Riccati方程对两点边值问题的刚性系统进行网格选择，(3)基于Riccati方法的非线性两点边值问题的软件编写。他将进一步研究刚性微分方程自适应积分过程中出现的所谓超稳定性现象，寻求有效的技术来避免实际的超稳定性发生，并可以纳入通用的常微分方程求解器中。最后，他将继续研究不变流形的数值逼近，实际上是双曲型偏微分方程相关系统的数值解。这包括(i)非线性情况的近似分析，（ii）所得线性系统的有效求解技术（多网格方法），（iii）利用与Riccati变换的联系。大多数现实的物理现象都是用复杂的非线性微分方程组来模拟的。这些系统的精确解是不可能的，因此必须进行数值模拟。这项工作的目的是产生可靠的，严格证明，算法来解决这些系统中的一些。其目的是为工作中的科学家生产软件工具。特别是，关于Riccati方程的工作对于在博弈论、最优控制和相关工程领域工作的人来说是有价值的。所谓的刚性微分方程在求解具有不同时间尺度的系统时很有用，例如在化学动力学和空气动力学应用中。最后，研究不变流形的近似对于理解非线性动力系统的长期行为是有用的。这在应用科学的许多领域都有应用。