The Landau-Lifshitz-Gilbert equation yields a mathematical model to describe the evolution of the magnetization of a magnetic material, particularly in response to an external applied magnetic field. It allows one to take into account various physical effects, such as the exchange within the magnetic material itself. In particular, the Landau-Lifshitz-Gilbert equation encodes relaxation effects, i.e., it describes the time-delayed alignment of the magnetization field with an external magnetic field. These relaxation effects are an important aspect in magnetic particle imaging, particularly in the calibration process. In this article, we address the data-driven modeling of the system function in magnetic particle imaging, where the Landau-Lifshitz-Gilbert equation serves as the basic tool to include relaxation effects in the model. We formulate the respective parameter identification problem both in the all-at-once and the reduced setting, present reconstruction algorithms that yield a regularized solution and discuss numerical experiments. Apart from that, we propose a practical numerical solver to the nonlinear Landau-Lifshitz-Gilbert equation, not via the classical finite element method, but through solving only linear PDEs in an inverse problem framework.
翻译:Landau-Lifshitz-Gilbert 方程式产生一个数学模型,用来描述磁性材料磁化的演变过程,特别是针对外部应用磁场。它允许人们考虑到各种物理效应,例如磁物质本身内部的交换。特别是Landau-Lifshitz-Gilbert 方程式编码放松效应,即,它描述磁化场与外部磁场在时间上延迟的对齐。这些放松效应是磁粒成像的一个重要方面,特别是在校准过程中。在本条中,我们处理磁质成像系统功能的数据驱动建模,其中Landau-Lifshitz-Gilbert 方程式是将放松效应纳入模型的基本工具。我们在全场和缩小的设置中分别提出参数识别问题,提出能够产生正规化解决办法的重建算法,并讨论数字实验。除此之外,我们提议为非线性Landau-Lifshitz-Gilbert 方程式提供实用的数字解算器,而不是通过古典定质元素框架,在线性定质要素中只解决问题。