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

高频波出现在各种应用中，例如几何光学、地震学、水下声学、量子物理学和电磁波。计算的挑战起源于需要在大的域上的短波长信号的数值分辨率，这是非常昂贵的，即使是现代计算设备。 在过去的几年里，作者与一些合作者发展了几种新的计算方法，在计算线性薛定谔方程的半经典极限、几何光学和高频波通过非均匀介质等方面都很有效。Li等人提出的方法发展了一种求解线性薛定谔方程半经典极限多值解的矩量法。 链球菌Osher等，他构造了求解一般线性对称双曲方程组高频极限多值解的水平集方法。与X。文，他介绍了保持哈密顿计划的高频波通过势垒或材料界面。在未来几年中，提议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、通过弯曲界面的高频波、哈密顿系统的蒙特-卡罗方法和具有不连续哈密顿量的刘维尔方程、纳米结构中电子输运的多尺度计算的经典力学和量子力学耦合、高频波的传播是应用数学中的一个经典领域，它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的应用。提议者计划开发具有多个时间和空间尺度的高频波的最先进的计算方法，并通过异质介质。这些方法预计将对各种现代工业应用产生深远的影响，包括纳米技术，半导体，量子点和地震学。这一研究方向也将为应用数学研究生教育提供新的多尺度建模与计算教材。

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.