We consider the inverse scattering problem for time-harmonic acoustic waves in a medium with pointwise inhomogeneities. In the Foldy-Lax model, the estimation of the scatterers' locations and intensities from far field measurements can be recast as the recovery of a discrete measure from nonlinear observations. We propose a "linearize and locally optimize" approach to perform this reconstruction. We first solve a convex program in the space of measures (known as the Beurling LASSO), which involves a linearization of the forward operator (the far field pattern in the Born approximation). Then, we locally minimize a second functional involving the nonlinear forward map, using the output of the first step as initialization. We provide guarantees that the output of the first step is close to the sought-after measure when the scatterers have small intensities and are sufficiently separated. We also provide numerical evidence that the second step still allows for accurate recovery in settings that are more involved.

翻译：本文研究含点状非均匀介质中时谐声波的逆散射问题。在Foldy-Lax模型中，根据远场测量数据估计散射体位置与强度可转化为从非线性观测中恢复离散测度的问题。我们提出"线性化后局部优化"的重建方法：首先在测度空间求解凸优化问题（即Beurling LASSO），该过程涉及前向算子（Born近似下的远场模式）的线性化；随后以第一步输出为初值，通过局部最小化包含非线性前向映射的第二泛函完成重建。我们证明当散射体强度较小且间距足够大时，第一步输出能逼近目标测度。数值实验表明，第二步在更复杂场景下仍能实现精确重建。