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LMS Durham Symposium Computational methods for wave propagation in direct scattering Andreas Kirsch (Karlsruhe. Germany)Inverse scattering theory for time-harmonic wavesAbstractIn these talks we will give an introduction to the theory of inverse scattering theory and survey some of the more recent numerical approaches for their solution. Part I: An Introduction to Scattering Theory for Time-Harmonic WavesIn this first part we will give an introduction to the scattering of time-harmonic plane waves by bounded obstacles. We will focus on the questions of uniqueness and existence, recall the integral equation methods and will also mention the variational approach. We will also introduce the scattering amplitude as the first coefficient in the Atkinson-Wilcox expansion of the scattered field.Part II: The Scattering Amplitude and the Inverse ProblemThe second part will be devoted to the corresponding inverse scattering problem. This is the task to determine properties of the scattering medium (e.g. the shape of the obstacle or the index of refraction) from full or partial knowledge of the scattering amplitude. To study this problem, more knowledge on the scattering amplitude is required such as reciprocity principles, unitarity of the scattering matrix and normality of the far field operator. A main part of this talk will be concerned with the question of uniqueness of the inverse problem.Part III: Numerical Treatment of the Inverse Scattering ProblemIn this part we will first review some of the more recent iterative methods for solving the inverse scattering problem such as the regularized Gauss-Newton methods, the Landweber method or second-degree methods. They are all based on an efficient way to compute the domain derivatives. Then we will introduce a new approach to characterize the shape of the scattering medium (either an obstacle or an inhomogeneous medium) by the scattering amplitude which is now also known under the name ``Linear Sampling Method''. Some numerical examples show advantages and limitations of the methods. |
