Sampling and Probe Methods – An Algorithmical View

Inverse scattering problems are concerned with the reconstruction of objects and parameter functions from the knowledge of scattered waves. Today, basically three different categories of methods for the treatment of the full nonlinear scattering problem are known: iterative methods, decomposition methods and sampling/probe methods. Sampling and probe methods have been proposed to detect the unknown scatterer for example in cases when the physical properties are not known or not given in a parametrized modell. A number of different approaches have been suggested over the last years. We will give a survey about these methods and explain the main ideas from an algorithmical point of view. We will group and describe the methods and discuss some relations to ideas which have been developed in the framework of impedance tomography. First, we will study the probe method of Ikehata and the singular sources method of Potthast. We show that these methods are closely related and basically form one unit with two different realizations. The second part is concerned with the range test of Potthast, Sylvester and Kusiak, the linear sampling method of Colton and Kirsch and the factorization method of Kirsch. These methods are founded on similar ideas testing the range of operators for reconstruction. In the final part we study the no response test of Luke and Potthast and the enclosure method of Ikehata. We will see that the enclosure method can be considered as a particular choice of the probing function for the no response test. These two methods are closely related and form two extremes for probing a scatterer with specially constructed waves.