The idea that a solution to a nonlinear inverse scattering problem (ISP) can contain information about the target on a subwavelength scale and thus allow one to achieve super-resolution (spatial resolution beyond the diffraction limit) has been around since the 1990s. However, a solid mathematical theory of super-resolution in nonlinear image reconstruction is still lacking. In this paper, we investigate the effect of super-resolution in nonlinear ISPs (both analytically and numerically) by analyzing several inverse problems in which the limit of spatial resolution can be defined precisely. The conclusions we obtain are not optimistic. Although it is possible to create examples of exactly solvable models in which account of nonlinearity in the ISP results in additional mathematically independent equations (one such example is shown herein), our results indicate that super-resolution is not achievable in any practical sense. Rather, we find that the linear subspace of possible solutions to a band-limited linearized ISP is transformed into a more general curved manifold due to the effects of nonlinearity. In the one-dimensional problem with realistic interaction that we have considered, the manifold can have a slightly smaller dimensionality that the subspace of solutions to the linearized problem but it does not contract to a point and the effect is practically insignificant.

中文翻译：

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研究超分辨率在非线性逆散射中的作用

自1990年代以来，一直存在这样一种想法：一种解决非线性逆散射问题（ISP）的解决方案可以在亚波长范围内包含有关目标的信息，从而可以实现超分辨率（超出衍射极限的空间分辨率）。但是，仍然缺乏扎实的数学理论来解决非线性图像重建问题。在本文中，我们将通过分析几个可以精确定义空间分辨率极限的反问题，研究非线性解析度中超分辨率的影响（无论是解析还是数值）。我们得出的结论并不乐观。尽管可以创建精确可解模型的示例，其中ISP中的非线性因素会导致其他数学上独立的方程式（此处显示了一个这样的示例），我们的结果表明，从任何实际意义上讲都无法实现超分辨率。相反，我们发现由于非线性的影响，带限线性化ISP的可能解的线性子空间被转换为更通用的弯曲流形。在我们考虑过的具有实际交互作用的一维问题中，流形可以具有稍小的维数，比线性化问题的解的子空间要小，但它不会收缩到一点，效果实际上是微不足道的。