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.
High frequency wave propagation is a classical example in applied mathematics dating back to the development of geometrical optics. Today it is a rich field with a variety of applications in electromagnetic scattering, seismology, photonics, microwaves, semiconductors, quantum physics and medical imaging. The computational challenges originate in the need of resolving short wave length signals over large domains, which is usually prohibitively expensive. In the duration of this NSF support, the proposer, with a number of collaborators, has developed several new computational methods quite effective in computing multiscale, high frequency wave problems. The new methods introduced include the Eulerian Gaussian beam methods for high frequency waves, Bloch-decomposition based on Eulerian Gaussian beam method for quantum dyanmics in periodic media, and Eulerian surface hopping methods in quantum dynamics, which significantly reduced the computational cost of the previous methods in the relevant fields; the BGK-penalization method for Boltzmann and other kinetic equations, allowing the methods to deal with different scales--both kinetic and hydrodynamic--with ease, thus significantly extended the applicability of tradiational numerial methods for kinetic equations. The intellectual merits of these new methods are efficient and simple computational methods for a wide variation of problems, from quantum mechanics, to solid state physics, and to rarified gas and semiconductors. The broad impact of these works are two-fold. First it helped to train many excellent Ph.D. students. Over the previous 5 years, 18 Ph.D. students were involved and partially supported by this award, and 12 have graduated and obtained postdoc jobs from Princeton, Harvard, UCLA, Texas-Austin, Michigan, etc. Many of the research results have been integrated into graduate course materials taught at UW-Madison and summer schools in US, Europe and China. Second, The materials studied here cover some fundamental issues in scientific computation in the modern age--the multiscale modeling and simulation, which plays essential roles in nanotechnology, material sciences and other areas of modern science and technology.
