Nonlinear Waves

9705380乔纳森·古德曼提案摘要应用分析和计算这个提案涵盖了应用数学、应用分析和计算科学的几个领域。一个领域是对非线性波和锋面的数学分析，特别是多维冲击。另一个领域是对由双曲型偏微分方程组(波传播方程)及其离散的“集中参数”近似所支配的系统的控制，这将用于最优控制的数值计算。两个主要的计算领域是：1：用于多维近似和有限元计算的各向异性自适应精化方法；2：用于计算相互作用电子系统量子力学性质的蒙特卡罗方法。还讨论了其他几个领域，包括在我指导下的研究生的研究。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。关于双曲型微分方程组控制系统的项目是关于消除结构中的声学噪声的方法。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、执行器和计算机的速度足够快，可以对单个声波做出反应。数学问题是如何设计出有效利用这一能力的好的计算“控制策略”。现有的控制数学理论是为更小、更简单的系统而开发的，不能直接应用于声波(声波)传播的系统的控制。我们希望我们的控制理论将适用于此类问题。一个更具计算性的项目是尝试从量子力学计算分子的“电子结构”。要解的方程(施罗丁-格尔方程)从1926年就知道了。仍然是在1997年，对于涉及一个以上电子(一个氧原子有8个)的系统，还没有可靠的方法来求解薛定谔方程。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。

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.