Stages 2009

• Scientific Context : Scattering of (acoustic or electromagnetic) time-harmonic waves by a bounded inhomogeneous medium can be formulated via partial differential equations. It is known since a long time that solutions to this system can be characterized as solutions of the Lippmann-Schwinger integral equation. Roughly speaking, this integral equation is of the form u + K(qu)= f with unknown u, right hand side f, a convolution operator K and a function q describing the medium. In 2000, Vainikko [2] used the well known fact that the Fourier transform of a convolution f _∗g is the multiplication of the Fourier transforms f^ˆ and ˆg to propose a fast numerical algorithm to solve the Lippmann-Schwinger equation numerically. This spectral method exploits the fast Fourier transform to implement the convolution operator K(v) in the Fourier domain. The numerical analysis for this method in [2] is done in the framework of so-called collocation methods. The recent work [1] suggests analysis of the method as a spectral Galerkin method.

• Project : A first part of this internship is dedicated to study Vainikko’s fast integral equation method for acoustic scattering problems. Afterwards we can start to implement this method in 2D/3D and can test the code against an existing solver. Concerning the electromagnetic scattering problem, there are several open questions concerning the numerical analysis of the method which could be tackled in this project. Of course, an implementation of the 3D electromagnetic problem would be a great success. However, depending on the preferences of the student, the emphasis can be chosen on the analytical or on the numerical side.

• Requirements : This internship is suitable for Master students. It requires a basic knowledge on (elliptic linear) PDE and Fourier theory. Since part of the project is numerical implementation, knowledge of C/C++ is helpful; candidates should at least have some experience with MATLAB.

Advisers :

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Houssem Haddar (haddar@cmapx.polytechnique.fr)

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Armin Lechleiter (alechle@cmapx.polytechnique.fr)

_• Lieu : CMAP, Ecole Polytechnique -91128 Palaiseau

References

[1] A. Kirsch and A. Lechleiter, The integral equations of Lippmann-Schwinger type for acoustic and electromagnetic scattering problems in L², 2008, submitted.

[2] G. Vainikko, Fast solvers of the Lippmann-Schwinger equation, in Direct and Inverse Problems of Mathematical Physics, D. Newark, ed., Int. Soc. Anal. Appl. Comput. 5, Dordrecht, 2000, Kluwer, p. 423.