Some Approximation Problems in Differential Equations

dieeci9973266研究者和他的合作者分析和实施技术来解决微分方程和矩阵分析中的几个问题。研究了参数相关矩阵的光滑标准正交分解及其连通应用，Lyapunov指数的计算及其应用，不变环面的稳定性和分岔，矩阵指数和矩阵其他函数的计算，Riccati方程。在所有情况下，进行理论分析和广泛的计算测试，并为概述的任务制定代码。具体目标包括实现计算基本矩阵解的QR和SVD的技术，更新不变子空间分解和参数相关矩阵的SVD的技术，以及与这些技术相关的应用；连续动力系统的李雅普诺夫指数计算方法及相关稳定性信息的实现；计算Fenichel的Lyapunov型数并利用这些量监测不变环面的连续性和分岔的技术；在矩阵为块三角形的情况下，矩阵指数的逼近，以及矩阵函数计算的其他方面。为了以紧凑的方式模拟物理现象，微分方程可能是我们拥有的最强大的工具。微分方程可以模拟复杂的化学相互作用和生物现象，制造工厂，机器人和卫星运动，材料的传输特性，以及许多应用科学中感兴趣的其他现象。然而，通常人们只能保证微分方程有解，而不能以封闭形式提供。此外，在典型情况下，所研究的现象会存在不确定性，这些不确定性会以微分方程模型中的参数的形式表现出来。粗略地说，基本问题是理解解的稳定性（随着参数的变化）。现在，假设我们已经（以很高的代价）获得了关于模型中某个参数值的模型解的详细信息。我们可能希望，如果我们稍微改变参数，我们将能够以之前支付的成本的一小部分获得新参数值的详细信息。基本上，这种期望形成了所谓的“延续技术”的原则。研究者研究求解微分方程的近似技术，目的是评估解的稳定性和开发延拓技术。目标是最终获得微分方程模型的近似值，该模型的参数在一定（物理）范围内变化时，该模型的解的全貌。这一目标在研究作为稳定性指标的量（所谓的李亚普诺夫指数）和近似这些量的计算机程序中得到具体实现。研究者还使用这些李雅普诺夫指数来预测分岔现象，并在求解微分方程时评估误差传播。最后，研究者研究了矩阵分解的延拓技术，并制作了延拓技术的计算机程序。

Dieci9973266The investigator and his collaborators analyze and implement techniques to tackle several problems in differential equations and matrix analysis. The following topics are studied: (1) smooth orthonormal factorizations of parameter dependent matrices and connected applications,(2) computation of Lyapunov exponents with applications, (3) stability and bifurcations of invariant tori, (4) computation of matrix exponential and other functions of a matrix, and Riccati equations. In all cases, theoretical analysis and extensive computational testing are undertaken, and codes for the outlined tasks are developed. Specific aims include implementation of techniques to compute QR and SVD of fundamental matrix solutions, techniques to update invariant subspaces factorizations and SVDs of parameter dependent matrices, and applications connected to such techniques; implementation of methods to compute Lyapunov exponents and related stability information of continuous dynamical systems; techniques to compute the Lyapunov type numbers of Fenichel and use these quantities to monitor continuability and bifurcations of invariant tori; approximation of the exponential of a matrix in case the matrix is block triangular, and other aspects of computation of functions of a matrix.To model physical phenomena in a compact way, differential equations are probably the most powerful tool we have. Differential equations can model complicated chemical interactions and biological phenomena, manufacturing plants, robot and satellite motions, transport properties of materials, and a host of other phenomena of interest in the applied sciences. However, typically one can only guarantee that differential equations have a solution, but it cannot be provided in closed form. Moreover, in a typical situation, there will be uncertainties in the phenomenon under study, and these will show up as parameters in the differential equation model. Loosely speaking, the fundamental question is then to understand the stability of solutions (as parameters change). Now, suppose that we have obtained (at a high price) detailed information on the solution to our model for a certain parameter value in the model. We may hope that if we change the parameter just slightly, we will be able to obtain detailed information for the new parameter value at a fraction of the cost we previously paid. Basically, this expectation forms the principle of so-called "continuation techniques." The investigator studies approximation techniques to solve differential equations which aim at assessing the stability of the solutions and at exploiting continuation techniques. The goal is to eventually obtain approximation of a differential equation model in a way which will give the complete picture of solutions of the model as its parameters vary inside a certain (physical) range. This goal finds its concrete realization in the study of quantities which serve as indicators of stability (the so-called Lyapunov exponents) and in computer programs which approximate these quantities. The investigator also uses these Lyapunov exponents for predicting bifurcation phenomena and assessing error propagation when solving differential equations. Finally, the investigator studies continuation techniques for matrix factorizations and produces computer programs for continuation techniques.