January 25 - 27, 2007

High-frequency scattering by a collection of convex bodies

Fatih Ecevit (MPI Leipzig)

In the last twenty years, very efficient methodologies have been devised to simulate the propagation of acoustic and electromagnetic waves in rather complicated settings. Yet, these algorithms are restricted in applicability to moderately low frequencies due to need for wave-length dependent resolution.

In this talk, we present an analysis of a recently proposed integral equation method for the solution of electromagnetic-acoustic scattering problems that bypasses the need for discretization on the order of the wave-length, and that thereby delivers solutions in frequency-independent computational times. Focusing on a collection of convex bodies, we first show that the (now) classical Neumann series solution (that accounts rigorously for multiple scattering) of the underlying integral equations can be re-arranged into a sum over primitive periodic orbits that constitute a fundamental building block for the multiple-scattering effects, and we proceed to analyze their properties in the high-frequency regime. Our approach is based on a derivation of precise aymptotic expansions for the "currents" (i.e. the normal velocity of the total fields) that they induce on the surface of the obstacles. As we demonstrate, the ratios (or "modulations") of these later expansions on a periodic orbit converge to an explicitly computable complex number Rk in the form of a wave-number dependent phase term modulated by a (real) amplitude. Moreover, as we show, this latter convergence is exponential in the number of reflections, uniform over the entire boundaries, and that the analysis is optimal with regards to the length of the periodic orbits. Moreover, we show that this detailed asymptotic analysis can be used to accelerate the convergence of multiple scattering iterates at no additional computational cost and, thus, to reduce the number of iterations necessary to attain a prescribed accuracy. Finally, we complement our theoretical developments by a variety of numerical results that confirm the accuracy of high-frequency expansions as well as the benefits of the proposed acceleration strategies.

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