Nonlinear Waves

非线性波

• 批准号: 9705380
• 负责人: Jonathan Goodman
• 金额: $ 21.27万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-08-01 至 2001-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9705380&HistoricalAwards=false
• 关键词: Nonlinear; Waves

9705380 Jonathan Goodman ABSTRACT OF PROPOSAL BY JONATHAN GOODMAN APPLIED ANALYSIS AND COMPUTATION This proposal covers several areas of applied mathematics, applied analysis, and computational science. One area is the mathematical analysis of nonlinear waves and fronts, particularly multidimensional shocks. Another area is control of systems governed by hyperbolic partial differential equations (wave propagation equations) and discrete ``lumped parameter'' approximations to them that would be used in numerical computation of optimal controls. Two main computational areas are: 1: anisotropic adaptive refinement methods for multidimensional approximation and finite element computation, and 2: Monte Carlo methods for computing quantum mechanical properties of systems of interacting electrons. Several other areas, including the research of graduate students under my supervision are discussed. There are several projects described here, most involving collaborations with colleagues, postdoctoral trainees, or graduate students. The project on control of systems governed by hyperbolic differential equations is about methods for removing acoustic noise from structures. This has applications in aircraft and submarine technology and in other places. It fits in with a larger effort to design "smart materials" and "smart structures". Modern sensors, actuators, and computers are fast enough to react to individual sound waves. The mathematical problem is to design good computational "control strategies" that use this ability effectively. The existing mathematical theory of control was developed with smaller, simpler systems in mind and does not apply directly to control of systems where acoustic waves (sound waves) are propagating. We hope that our theory of control will apply to such problems. A more computational project is the attempt to compute the "electronic structure" of molecules from quantum mechanics. The equation to be solved (the Schrodin ger equation) has been known since 1926. Still in 1997, there is no reliable way to solve the Schrodinger equation for systems involving more than one electron (an oxygen atom has 8). The ability to do this would have enormous scientific and technological impact, with applications ranging from superconductivity to drug design.

应用分析与计算这个建议涵盖了应用数学、应用分析和计算科学的几个领域。一个领域是非线性波和锋面的数学分析，特别是多维冲击。另一个领域是由双曲偏微分方程（波传播方程）和离散的“集中参数”近似控制的系统控制，这些近似将用于最优控制的数值计算。两个主要的计算领域是：1：用于多维逼近和有限元计算的各向异性自适应细化方法；2：用于计算相互作用电子系统量子力学特性的蒙特卡罗方法。讨论了其他几个领域，包括我指导的研究生的研究。这里描述了几个项目，大多数涉及与同事、博士后实习生或研究生的合作。双曲型微分方程控制系统的课题是关于从结构中去除噪声的方法。这在飞机和潜艇技术以及其他地方都有应用。它符合设计“智能材料”和“智能结构”的更大努力。现代传感器、执行器和计算机对单个声波的反应速度足够快。数学问题是设计良好的计算“控制策略”，有效地利用这种能力。现有的控制数学理论是在考虑更小、更简单的系统的情况下发展起来的，不能直接应用于声波传播的系统的控制。我们希望我们的控制理论能适用于这类问题。一个更具计算性的项目是尝试从量子力学中计算分子的“电子结构”。要解的方程（薛定谔方程）早在1926年就为人所知。在1997年，还没有可靠的方法来解决涉及多个电子（氧原子有8个）的系统的薛定谔方程。这样做的能力将产生巨大的科学和技术影响，应用范围从超导到药物设计。