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：我们考虑在具有点状非均匀性的介质中，时间谐波声波的反散射问题。在Foldy-Lax模型下，从远场测量中估计散射体的位置和强度可以归结为从非线性观测中恢复离散测度的问题。我们提出了一种“线性化加局部优化”的方法来实现这种重建。首先，我们在测度空间中求解一个凸规划（称为Beurling LASSO），该规划涉及正向算子的线性化（Born近似下的远场模式）。然后，我们使用第一步的输出作为初始化，对涉及非线性正向映射的第二个泛函进行局部最小化。我们给出了当散射体具有小强度且足够分离时，第一步输出接近所求测度的保证。我们还提供了数值证据，表明在更复杂的情形下，第二步仍能实现精确恢复。