通过测度上的稀疏优化实现点散射体定位
Localization of point scatterers via sparse optimization on measures
AI总结：
研究时谐声波逆散射问题，提出“线性化与局部优化”方法，先通过Beurling LASSO求解凸规划，再局部最小化非线性泛函，实现点散射体位置和强度的精确恢复。
摘要：
我们考虑具有点状非均匀性的介质中时谐声波的逆散射问题。在Foldy-Lax模型中，从远场测量估计散射体的位置和强度可以重新表述为从非线性观测中恢复离散测度。我们提出一种“线性化与局部优化”方法来执行此重构。我们首先在测度空间中求解一个凸规划（即Beurling LASSO），该规划涉及正向算子（Born近似下的远场模式）的线性化。然后，我们使用第一步的输出作为初始化，局部最小化一个涉及非线性正向映射的第二泛函。我们提供了保证：当散射体强度较小且充分分离时，第一步的输出接近所求测度。我们还提供了数值证据，表明在更复杂的环境中，第二步仍能实现精确恢复。
英文摘要：
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.
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