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An Isaac Newton Institute Workshop

Effective Computational Methods for Highly Oscillatory Problems: The Interplay between Mathematical Theory and Applications

Condition Number Estimates for Oscillatory Integral Operators

2nd July 2007

Authors: Simon Chandler-Wilde (University of Reading), Ivan Graham (University of Bath), Stephen Langdon (University of Reading), Marko Lindner (University of Reading)

Abstract

We present results obtained during the programme on the conditioning of the standard combined potential boundary integral operators in time-harmonic acoustic scattering (cf. the talk by Oscar Bruno), in particular addressing behaviour as the wavenumber k tends to infinity, when the integral operator becomes increasingly oscillatory. While study of this topic goes back to Kress and Spassov (Numer. Math. 1983), the focus previously has been on the canonical case of a circle/sphere for which spherical harmonics are the eigenfunctions and the singular values are known explicitly. However, even for this case, it is only recently (Dominguez, Graham, Smyshlyaev, Preprint NI07004-HOP and Numer. Math. 2007) that rigorous upper bounds have been obtained on the operator and its inverse as a function of k. For non-spherical scatterers the only result is a recent upper bound on the inverse operator for piecewise smooth starlike domains (Chandler-Wilde & Monk, to appear SIAM J. Math. Anal.). In this talk we derive a range of lower bounds on the operator and its inverse, which show that the behaviour for large k depends subtly on the geometry. The sharpness of these lower bounds is demonstrated by numerical simulation and, in some instances, by provable upper bounds. The main computational message is that, while the condition number grows only mildly (like k^{1/3}) for a circle/sphere, behaviour can be much worse (like k^{5/4}) for non-starlike domains.