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控制的反障碍问题中两种经典的直接重建方法。这两种方法的共同点是均采用定义在未知障碍物外部、并在障碍物表面发生爆炸的指示函数概念。然而，它们在表现形式上完全不同。本文以有界域内拉普拉斯方程控制的反障碍问题作为原型案例，提出了一种整合探针法与奇异源方法的统一版本，填补了两种方法指示函数之间的空白。主要结果分为三部分：首先，建立了结合探针法与卡勒曼函数概念的奇异源方法公式；其次，两种方法的指示函数可通过将第三种指示函数以两种方式进行分解而获得，第三种指示函数在外部边界和障碍物表面均发生爆炸；最后，对探针法与奇异源方法进行了重构，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也具有探针法的B面特性，即通过指示序列的爆炸性质对未知障碍物进行表征。