Computation of Multiscaled and Multivalued Solutions to High Frequency Waves in Multimedia

高频波出现在各种应用中，如几何光学、地震学、水声学、量子物理和电磁波。计算上的挑战源于对大范围内短波长信号的数值分辨率的需求，即使用现代计算设备，这也是昂贵的。在过去的几年里，作者与许多合作者一起发展了几种新的计算方法，在计算线性薛定谔方程、几何光学和非均匀介质中的高频波的半经典极限时非常有效。该提议与X.Li一起，发展了线性薛定谔方程半经典极限中多值解的矩方法。他与S.Osher等人一起构造了一般线性对称双曲组高频极限多值解的水平集方法。与X.wen一起，他提出了通过势垒或材料界面的高频波的哈密顿守恒格式。在接下来的几年里，作者计划进一步发展这些方法，为这些方法建立坚实的理论基础，并探索在弹性波、曲线界面的高频波、含不连续哈密顿量的哈密顿系统和Liouville方程的蒙特卡罗方法、用于纳米结构中电子输运的多尺度计算的经典和量子力学的耦合以及在真空电子器件建模中的多值解的新应用。高频波传播是应用数学中的一个经典领域，起源于几何光学的研究。今天，它是一个丰富的领域，在电磁散射、地震学、光子学、微波、半导体、量子物理和医学成像等领域都有应用。该提出者计划开发最先进的计算方法，用于多种时间和空间尺度的高频波，并通过非均匀介质。这些方法预计将在各种现代工业应用中产生深远影响，包括纳米技术、半导体、量子点和地震学。这一研究方向还将为应用数学研究生教育提供多尺度建模和计算方面的新教材。

High frequency waves arise in a variety of applications, such as geometric optics, seismology, underwater acoustics, quantum physics, and electromagnetic waves. The computational challenges originate in the need of numerical resolution of short wave length signals over large domains, which is prohibitively expensive even by modern computational equipments. In the last few years, the proposer, with a number of collaborators, has developed several new computational methods quite effective in computing the semiclassical limit of linear Schrodinger equation, geometrical optics, and high frequency waves through inhomogeneous media.With X. Li, the proposal developed a moment method for multivalued solutions in the semiclassical limit of the linear Schrodinger equation. With S. Osher etc., he constructed level set methods for the multivalued solutions that arise in high frequency limit of general linear symmetric hyperbolic systems. With X. Wen, he introduced Hamiltonian-preserving schemes for high frequency waves through potential barriers or material interfaces. In the next few years the proposer plans to further the development of these methods, to establish a solid theoretical foundation for these methods, and to explore new applications in elastic waves, high frequency waves through curved interfaces, Monte-Carlo methods for Hamiltonian systems and Liouville equations with discontinuous Hamiltonians, coupling of classical and quantum mechanics for multiscale computation of electron transport in nanostructures, and multivalued solutions in vacuum electronics device modeling.High frequency wave propagation is a classical field in applied mathematics originated from the study of geometrical optics. Today it is a rich field with applications in electromagnetic scattering, seismology, photonics, microwaves, semiconductors, quantum physics and medical imaging. The proposer plans to develop state-of-art computational methods for high frequency waves with multiple time and space scales, and through heterogeneous media. These methods are expected to have a profound impact in a variety of modern industrial applications, including nanotechnology, semiconductors, quantum dots and seismology. This line of research will also provide new teaching materials in multiscale modeling and computation for graduate education in applied mathematics.