Estimation method for inverse problems with linear forward operator and its application to magnetization estimation from magnetic force microscopy images using deep learning

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: It is assumed that the number of patterns that the unknown quantity can take is finite. Then, even if 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: 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.