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.

翻译：探针法与奇异源方法是偏微分方程支配的反障碍问题中两种经典且广为人知的直接重构方法。二者的共同之处在于均定义了未知障碍物外部的指示函数，且该函数在障碍物表面发生爆破。然而，它们的外在表现形式截然不同。本文以有界域内拉普拉斯方程支配的反障碍问题为原型案例，提出了一种整合版本的探针法与奇异源方法，填补了二者指示函数之间的鸿沟。主要结论分为三个部分：首先，将奇异源方法与探针法及Carleman函数概念相结合进行公式化；其次，通过对第三个指示函数进行两种方式分解，可分别得到两种方法的指示函数，而这第三个指示函数在外部边界与障碍物表面均发生爆破；最后，对探针法与奇异源方法进行重构，并证明这两种重构方法所基于的指示函数完全一致。作为副产品，重构后的奇异源方法还具备了探针法的Side B特性——即通过指示序列的爆破性质对未知障碍物进行刻画。