Dominant Mode Extraction Based on the Four-Dimensional Variational Method

Understanding the formation mechanism of magnetic domain patterns is important to improve the performance of magnetic materials. The magnetic domain patterns depend on the parameters of the time-dependent Ginzburg-Landau (TDGL) equation and the sweep rate of the external magnetic field. Although conventional analytical approaches can predict the patterns formed in the case of a high sweep rate, more versatile methods are required to understand the formation mechanism of complicated patterns for any temporal variation in the external field. This study proposes a method that extracts dominant modes from a posterior distribution based on the four-dimensional variational method (4DVar). The method decomposes the magnetic domain patterns into eigenvectors of the Hessian matrix of the cost function, defined as the difference between the observed and simulated magnetic domain patterns. The eigenvectors are extracted using a second-order adjoint method (SOA) and power iteration. The patterns are reconstructed by superimposing the extracted eigenvectors, and their time evolution can be obtained from the weights of the eigenvectors. Experiments demonstrate that the patterns are sufficiently reconstructed using a small number of the eigenvectors. This enables us to understand the pattern evolution as the change of the dominant eigenvectors.