We develop new efficient algorithms for a class of inverse problems of gravimetry to recover an anomalous volume mass distribution (measure) in the sense that we design fast local level-set methods to simultaneously reconstruct both unknown domain and varying density of the anomalous measure from modulus of gravity force rather than from gravity force itself. The equivalent-source principle of gravitational potential forces us to consider only measures of the form $ \mu = f\,\chi_{D} $, where $ f $ is a density function and $ D $ is a domain inside a closed set in $ \bf{R}^n $. Accordingly, various constraints are imposed upon both the density function and the domain so that well-posedness theories can be developed for the corresponding inverse problems, such as the domain inverse problem, the density inverse problem, and the domain-density inverse problem. Starting from uniqueness theorems for the domain-density inverse problem, we derive a new gradient from the misfit functional to enforce the directional-independence constraint of the density function and we further introduce a new labeling function into the level-set method to enforce the geometrical constraint of the corresponding domain; consequently, we are able to recover simultaneously both unknown domain and varying density from given modulus of gravity force. Our fast level-set method is built upon localizing the level-set evolution around a narrow band near the zero level-set and upon accelerating numerical modeling by novel low-rank matrix multiplication. Numerical results demonstrate that uniqueness theorems are crucial for solving the inverse problem of gravimetry and will be impactful on gravity prospecting. To the best of our knowledge, our inversion algorithm is the first of such for the domain-density inverse problem since it is based upon the conditional well-posedness theory of the inverse problem.

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通过有效的水平集方法同时在逆重分析法中同时恢复域和变化的密度

我们针对一类重量分析的反问题开发了新的高效算法，以恢复异常的体积质量分布（度量），因为我们设计了快速的局部水平集方法，以便同时从模数重构未知域和异常度量的变化密度而不是重力本身。引力势的等效源原理迫使我们仅考虑形式为$ \ mu = f \，\ chi_ {D} $的度量，其中$ f $是密度函数，$ D $是闭集内的一个域在$ \ bf {R} ^ n $中。因此，对密度函数和域都施加了各种约束，因此可以为相应的反问题（如域反问题，密度反问题，以及域密度反问题。从域密度反问题的唯一性定理出发，我们从失配函数中推导了新的梯度以实施密度函数的方向独立约束，并将新的标记函数进一步引入到水平集方法中以实施几何相应域的约束；因此，我们能够从给定的重力模量中同时恢复未知域和变化的密度。我们的快速水平集方法是基于将水平集的演化围绕零水平集附近的窄带进行局部化，并通过新颖的低秩矩阵乘法来加速数值建模。数值结果表明，唯一性定理对于解决重力法的反问题至关重要，并将对重力勘探产生影响。据我们所知，我们的反演算法是针对域密度反演问题的第一个反演算法，因为它基于反演条件的条件适定性理论。