论文标题
平面中的无差异度量和逆向逆向问题的逆向措施以发散形式
Divergence-free measures in the plane and inverse potential problems in divergence form
论文作者
论文摘要
我们表明，平面上的无差异度量是可整流Jordan曲线上的单位切线矢量场的连续总和。该循环分解比S.K.给出的基本电磁阀中的一般分解更为精确。 Smirnov，当应用于平面案例时。证明涉及将Fleming-Rishel公式扩展到均匀的BV函数（在任何维度上），并为此类函数建立近似于测量的Suplevel集合的理论连接组件的连续性作为水平的函数。我们将这些结果应用于源术语的逆潜在问题，其源项是某些未知（矢量值）度量的差异。一个原型的情况是当通过R3值borel测量值建模磁化时，磁化问题是逆磁化问题。我们通过惩罚其量度的理论总变异规范（TV）来研究恢复磁化μ的方法。特别是，我们证明，如果在平面上支持磁化，那么即使在存在噪声的情况下，电视调查方案也总是具有独特的最小化器。进一步显示，在以下情况下，TV-NORM最小化（在产生同一场的磁化中）独特地恢复了平面磁化：当通过足够分离的线段段和纯粹是1-不可靠的集合携带磁化时，或者是纯粹的1-无凝聚力的集合，或者当支撑的超级集是类似树状的。我们注意到，可以通过将正则化参数提高到零来通过零噪声限制的电视调查方案恢复这种磁化。这表明在当前无限维度上下文中稀疏的定义，产生的结果类似于压缩感应
We show that a divergence-free measure on the plane is a continuous sum of unit tangent vector fields on rectifiable Jordan curves. This loop decomposition is more precise than the general decomposition in elementary solenoids given by S.K. Smirnov, when applied to the planar case. The proof involves extending the Fleming-Rishel formula to homogeneous BV functions (in any dimension), and establishing for such functions approximate continuity of measure theoretic connected components of suplevel sets as functions of the level. We apply these results to inverse potential problems whose source term is the divergence of some unknown (vector-valued) measure. A prototypical case is that of inverse magnetization problems when magnetizations are modeled by R3-valued Borel measures. We investigate methods for recovering a magnetization μ by penalizing its measure theoretic total variation norm (TV). In particular, we prove that if a magnetization is supported in a plane, then TV-regularization schemes always have a unique minimizer, even in the presence of noise. It is further shown thatTV-norm minimization (among magnetizations generating the same field) uniquely recovers planar magnetizations in the following cases: when the magnetization is carried by a collection of sufficiently separated line segments and a set that is purely 1-unrectifiable, or when a superset of the support is tree-like. We note that such magnetizations can be recovered via TV-regularization schemes in the zero noise limit, by taking the regularization parameter to zero. This suggests definitions of sparsity in the present infinite dimensional context, that generate results akin to compressed sensing
