A numerical method is developed for recovering both the source locations and the obstacle from the scattered Cauchy data of the time-harmonic acoustic field. First of all, the incident and scattered components are decomposed from the coupled Cauchy data by the representation of the single-layer potentials and the solution to the resulting linear integral system. As a consequence of this decomposition, the original problem of joint inversion is reformulated into two decoupled subproblems: an inverse source problem and an inverse obstacle scattering problem. Then, two sampling-type schemes are proposed to recover the shape of the obstacle and the source locations, respectively. The sampling methods rely on the specific indicator functions defined on target-oriented probing domains of circular shape. The error estimates of the decoupling procedure are established and the asymptotic behaviors of the indicator functions are analyzed. Extensive numerical experiments are also conducted to verify the performance of the sampling schemes.

翻译：针对时谐声场散射柯西数据，本文发展了一种同时恢复源位置与障碍物形状的数值方法。首先，通过单层位势表示与所得线性积分系统的解，将耦合柯西数据分解为入射场与散射场分量。基于该分解，原始联合反演问题被重构为两个解耦子问题：逆源问题与逆障碍散射问题。随后，分别提出两种采样类方案来恢复障碍物形状与源位置。这些采样方法依赖于定义在圆形目标探测量区域上的特定指示函数。本文建立了解耦过程的误差估计，并分析了指示函数的渐近行为。通过大量数值实验验证了采样方案的有效性。