ABSTRACT

This study considers an inverse problem, where the corresponding forward problem is given by a finite-dimensional linear operator T. The inverse problem has the following form: $\left(\text{data}\right)=T\left(\text{unknown}\right).$ It is assumed that the number of patterns that the unknown quantity can take is finite. Then, even if $\text{Ker }T\ne \left\{0\right\},$ the unknown quantity may be uniquely determined from the data. This case is the subject of this study. We propose a method for solving this inverse problem using numerical calculations. A famous inverse problem requires the estimation of the unknown magnetization distribution or magnetic charge distribution in an anisotropic permanent magnet sample from the magnetic force microscopy images. It is known that the solution of this problem is not unique in general. In this work, we consider the case where a magnetic sample comprises cubic cells, and the unknown magnetic moment is oriented either upward or downward in each cell. This discretized problem is an example of the above-mentioned inverse problem: $\left(\text{data}\right)=T\left(\text{unknown}\right).$ Numerical calculations were carried out to solve this model problem employing our method and deep learning. The experimental results show that the magnetization can be estimated roughly up to a certain depth.

摘要

本研究考虑了一个逆问题，其中相应的前向问题由有限维线性算子T 给出。逆问题具有以下形式：$\left(\text{数据}\right)=吨\left(\text{未知}\right).$假设未知量可以采用的模式数量是有限的。那么，即使$\text{克尔 }吨\ne \left\{0\right\},$未知量可以从数据中唯一确定。这个案例是本研究的主题。我们提出了一种使用数值计算解决这个逆问题的方法。一个著名的逆问题需要从磁力显微镜图像中估计各向异性永磁体样品中未知的磁化分布或磁荷分布。众所周知，这个问题的解一般不是唯一的。在这项工作中，我们考虑磁性样品包含立方体单元的情况，并且未知磁矩在每个单元中向上或向下取向。这个离散化问题是上述逆问题的一个例子：$\left(\text{数据}\right)=吨\left(\text{未知}\right).$使用我们的方法和深度学习进行数值计算来解决这个模型问题。实验结果表明，磁化强度可以粗略估计到一定深度。