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：我们考虑介质中含有离散点状不均匀体的时谐声波逆散射问题。在Foldy-Lax模型中，通过远场测量估计散射体位置和强度的问题可转化为从非线性观测中恢复离散测度的问题。我们提出了一种“线性化与局部优化”方法来实现该重建任务。首先在测度空间中求解一个凸规划问题（即Beurling LASSO），该问题涉及正演算子（Born近似下的远场模式）的线性化；然后利用第一步的输出作为初始值，对包含非线性正演映射的第二泛函进行局部极小化。当散射体强度较弱且间距足够大时，我们给出了第一步输出逼近待求测度的理论保证。数值实验表明，在更复杂场景下，第二步仍能实现高精度恢复。