January 25 - 27, 2007

Frequency dependence of conditioning in scattering problems

Simon Chandler-Wilde (University of Reading)

In this talk we review work on conditioning for problems of time-harmonic scattering, focussing particularly on the simplest problem of scattering of time-harmonic acoustic waves by a bounded sound soft obstacle in two and three dimensions, and on the classical Brakhage-Werner formulation of this problem as a second kind boundary integral equation in which the solution is sought as a combined single- and double-layer potential. We discuss first the case of a circle/sphere for which spherical harmonics are the eigenfunctions, singular values are known explicitly, and rigorous and sharp bounds on the wave number dependence of the norm of the operator and its inverse can be established by careful estimates of the behaviour of Bessel functions, uniform in argument and order. Moreover coercivity, uniformly with respect to the wave number, has also recently been established for this special case. We then discuss what can be said for the case of general Lipschitz, starlike domains and show that Rellich-type identities yield sharp bounds on the norm of the inverse operator which are explicit in their dependence on the wave number, coupling parameter, and geometry. Crude bounds on the norm of the operator can also be given. However, obtaining bounds which are sharp in their dependence on the wave number is an open problem, as is any result on coercivity. We also briefly mention analogous results for a second formulation of the scattering problem as a standard variational/weak formulation in the part of the exterior domain contained in a large sphere, with an exact Dirichlet-to-Neumann map applied on the boundary.

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