The linear sampling method is a method to reconstruct the shape of an obstacle in time-harmonic inverse scattering without a priori knowledge of either the physical properties or the number of disconnected components of the scatterer.Although it has been proven numerically to be a fast and reliable method in many situations, no mathematical argument has yet been found to prove why this is so. Using results obtained by Kirsch in deriving the related factorization method, we show in this paper that for a large class of scattering problems, linear sampling can be interpreted rigorously as a numerical method to reconstruct the shape of an obstacle; or in other words that linear sampling must work for problems in this class.

The problem of determining the shape of a rigid body from the knowledge of the far-field patterns of incident plane compressional and shear waves in two-dimensional elasticity is considered. We discuss the application of the original linear sampling method and of the related (F*F)1/4-method to tackle this problem. It is established that the far-field operator is compact and normal. A theoretical basis for both reconstruction methods is developed and numerical results are shown, illustrating the excellent quality of reconstructions attainable.

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The application of the Factorization Method, the refined version of the Linear Sampling Method, to scattering by a periodic surface is considered. Central to this method is the Near Field Operator N , mapping incident fields to the corresponding scattered fields on a horizontal line. A factorization of N forms the basis for the method. It is shown that the middle operator in this factorization, the adjoint of the single layer operator, is coercive if the wave number is small enough. Thus this operator can, for general wave number, always be written as the sum of a coecive and a compact operator. We use this property to define an auxiliary positive operator N # which can be constructed directly from N and which makes it possible to reconstruct the scattering surface directly using a simple numerical algorithm.

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