Integrating the probe and singular sources methods

The probe and singular sources methods are well-known two classical direct reconstruction methods in inverse obstacle problems governed by partial differential equations. The common part of both methods is the notion of the indicator functions which are defined outside an unknown obstacle and blow up on the surface of the obstacle. However, their appearance is completely different. In this paper, by considering an inverse obstacle problem governed by the Laplace equation in a bounded domain as a prototype case, an integrated version of the probe and singular sources methods which fills the gap between their indicator functions is introduced. The main result is decomposed into three parts. First, the singular sources method combined with the probe method and notion of the Carleman function is formulated. Second, the indicator functions of both methods can be obtained as a result of decomposing a third indicator function into two ways. The third indicator function blows up on both the outer and obstacle surfaces. Third, the probe and singular sources methods are reformulated and it is shown that the indicator functions on which both reformulated methods based, completely coincide with each other. As a byproduct, it turns out that the reformulated singular sources method has also the Side B of the probe method, which is a characterization of the unknown obstacle by means of the blowing up property of an indicator sequence.