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

University Of Wisconsin-Madison, Madison WI

Investigators

Abstract

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.