论文标题
有界3D样品的磁化矩：从大磁盘上的平面测量结果渐近恢复
Magnetisation moment of a bounded 3D sample: asymptotic recovery from planar measurements on a large disk
论文作者
论文摘要
我们考虑了从磁场的部分数据中重建样品的整体磁化载体（净力矩）的问题。也就是说，是出于具体的实验设置的动机，我们处理在样品附近一部分在平面上测量磁场的情况，并且只有一个（与平面）分量可用。在假设测量区域是一个足够大的磁盘（位于样品上方的水平面）中，我们获得了净力矩矢量组件的一组估计值，其准确性随着测量磁盘半径的增加而渐近。与我们先前的初步结果相比，渐近公式现在是严格的合理性，并且得出了高阶估计。此外，根据在傅立叶域中的适当分裂和振荡积分的估计值（涉及小参数和大参数），阐明了所提出的方法，阐明了任意顺序的渐近估计的推导，这是以前不清楚的可能性。获得的结果是数值说明的，并且讨论了它们相对于噪声的鲁棒性。所提出的方法应适用于其他平面测量的其他磁性和重量问题。
We consider the problem of reconstruction of the overall magnetisation vector (net moment) of a sample from partial data of the magnetic field. Namely, motivated by a concrete experimental set-up, we deal with a situation when the magnetic field is measured on a portion of the plane in vicinity of the sample and only one (normal to the plane) component of the field is available. Under assumption that the measurement area is a sufficiently large disk (lying in a horizontal plane above the sample), we obtain a set of estimates for the components of the net moment vector with the accuracy which improves asymptotically with the increase of the measurement disk radius. Compared to our previous preliminary results, the asymptotic formulas are now rigorously justified and higher-order estimates are derived. Moreover, the presented approach, based on an appropriate splitting in the Fourier domain and estimates of oscillatory integrals (involving both small and large parameters), elucidates the derivation of asymptotic estimates of an arbitrary order, a possibility that was previously unclear. The obtained results are illustrated numerically and their robustness with respect to noise is discussed. The proposed methodology should be applicable to other magnetic and gravimetric problems with planar measurements.
