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CNA Seminar/Colloquium/Joint Pitt-CNA Colloquium

Hailiang Liu
Iowa State University
Title: Recovery of high frequency wave fields by the Gaussian beam superposition

Abstract: Gaussian beams are asymptotically valid high frequency solutions to hyperbolic partial differential equations, concentrated on a single curve through the physical domain. They can also be extended to some dispersive wave equations, such as the Schroedinger equation. Superpositions of Gaussian beams provide a powerful tool to generate more general high frequency solutions that are not necessarily concentrated on a single curve. We are concerned with the accuracy of Gaussian beam superpositions in terms of the wavelength, which was thought a rather difficult problem decades ago. We present a systematic construction of Gaussian beam superpositions for all strictly hyperbolic and Schroedinger equations subject to highly oscillatory initial data, and obtain the optimal error estimates in the appropriate norm dictated by the well-posedness estimate. The obtained results are valid for any number of spatial dimensions and are unaffected by the presence of caustics.

This talk presents key ideas and techniques involved in this newly developed recovery theory of high frequency wave fields, with materials drawn from recent works with J. Ralston (UCLA), and with N. Tanushev (UT-Austin) and O. Runborg (KTH).

Date: Thursday, November 18, 2010
Time: 1:30 pm
Location: Wean Hall 8220
Submitted by:  David Kinderlehrer