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An Isaac Newton Institute Workshop

The Theory of Highly Oscillatory Problems

Norms and condition numbers of oscillatory integral operators in acoustic scattering

28th March 2007

Author: Simon Chandler-Wilde (University of Reading)

Abstract

In this talk we discuss domain and boundary integral operators arising in the theory and numerical treatment by integral equation methods of the Helmholtz equation or time harmonic Maxwell equations. These integral operators are increasingly oscillatory as the wave number k increases (k proportional to the frequency of the time harmonic incident field). An interesting theoretical question, also of practical significance, is the dependence of the norms of these integral operators and their inverses on k. We investigate this question, in particular for classical single- and double-layer potential operators for the Helmholtz equation on the boundary of bounded Lipschitz domains. The results and techniques used depend on the domain geometry. In certain 2D cases (for example where the boundary is a starlike polygon) bounds which are sharp in their dependence on k can be obtained, but there are many open problems for more general geometries and higher dimension.

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