Mathematical Sciences: Conference on Dynamical Numerical Analysis

研究者和他的同事组织了一个关于动力系统数值方法的国际会议。近年来，随着经典定量数值分析的局限性日益明显，动力系统与数值分析之间的相互作用日益增强。在实践中，计算的一个共同目标是在中等到较长的时间间隔内获得解的信息，然而经典的误差分析通常在较短的初始瞬态之后是没有意义的。因此，解决现有计算方法的有效性和设计用于长时间间隔计算的特殊方法是微分方程数值分析的基础。要涵盖的具体主题包括长时间间隔的积分算法的收敛（包括长时间间隔的误差边界和不变集的收敛），长时间间隔积分算法的稳定性（包括动态结构和伪解的保存），误差分析的新方法（包括阴影结果和向后误差分析），设计不变量集的计算技术，例如计算不变量torii、惯性流形、异斜轨道、同斜轨道和Lyapunov指数的算法。微分方程描述了一个物理系统在特定时间的空间位置如何在不久的将来影响邻近的位置。正是微分方程的这种局域性使得模拟复杂的物理情况成为可能，因为这意味着整个系统不必同时被描述。但这也意味着，获得一个系统的正确微分方程模型并不是故事的终点，因为必须解出微分方程才能获得关于原系统的信息。因此，微分方程解的研究已经发展成为数学的主要领域之一，同时在科学和工程中也变得非常重要。计算机的引入第一次使在大范围内近似求解微分方程成为可能。在过去的三十年里，数值数学家在设计近似微分方程的具体解的方法和分析近似解的准确性方面取得了很大的进展。研究微分方程数值方法的经典方法很好地描述了在短时间间隔内小空间区域内特解的近似行为。但要理解非线性方程的本质，就需要了解相对较长时间间隔内的许多解。为了满足这一需求，人们对使用数值方法研究微分方程解的结构和模式越来越感兴趣，这被称为方程的动力学行为。这次会议聚集了来自美国和世界各地的主要专家，代表了该领域的几乎所有方面，以及感兴趣的学生和其他研究人员，讨论当前的艺术状态并刺激新的发展。预计这次会议将对科学和工程领域的许多应用产生长期影响。

Dieci The investigator and his colleagues organize an international conference on numerical methods for dynamical systems. The interaction between dynamical systems and numerical analysis has grown in recent years as the limitations of classical quantitative numerical analysis have become increasingly apparent. In practice, a common goal of computation is to obtain information about solutions over moderate to long time intervals, yet classical error analysis is generally not meaningful past a short initial transient. It is thus fundamental to the numerical analysis of differential equations to address both the efficacy of existing computational methods and the design of special methods for computation over long time intervals. Specific topics to be covered include convergence of integration algorithms over long time intervals (including long-time error bounds and convergence of invariant sets), stability of integration algorithms over long time intervals (including preservation of dynamical structure and spurious solutions), new approaches to error analysis (including shadowing results and backward error analysis), and design of computational techniques for invariant sets such as algorithms to compute invariant torii, inertial manifolds, heteroclinic orbits, homoclinic orbits, and Lyapunov exponents. Differential equations describe how one position in space of a physical system at a particular time influences neighboring positions in the immediate future. It is this local nature of a differential equation that makes it possible to model complicated physical situations, because it means that the entire system does not have to be described simultaneously. But this also means that obtaining the correct differential equation model of a system is not the end of the story, because it is necessary to solve the differential equation to obtain information about the original system. Accordingly, the study of solutions of differential equations has grown int o one of the major areas of mathematics while becoming centrally important in science and engineering. The introduction of the computer made it possible to approximately solve differential equations on a wide scale for the first time. Over the last thirty years, numerical mathematicians have made great progress in devising methods to approximate a specific solution of a differential equation and in analyzing the accuracy of the approximate solution. The classical approach to study numerical methods for differential equations is very good at describing the behavior of the approximation of a particular solution in a small region of space over a short time interval. But understanding the nature of nonlinear equations requires knowledge about many solutions over relatively long time intervals. In response to this need, there is increasing interest in using numerical methods to study structures and patterns in solutions of differential equations, which is known as the dynamical behavior of the equation. This conference gathers leading experts from the United States and from around the world, representing nearly all aspects of the area, together with interested students and other researchers, to discuss the current state of the art and to stimulate new developments. The conference is expected to have a long-ranging impact on many applications in science and engineering.