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.

中文翻译：

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磁粒子成像中Landau-Lifshitz-Gilbert方程参数识别的数值方面

Landau-Lifshitz-Gilbert 方程产生了一个数学模型来描述磁性材料的磁化强度的演变，特别是对外部施加的磁场的响应。它允许人们考虑各种物理效应，例如磁性材料本身内部的交换。特别是，Landau-Lifshitz-Gilbert 方程对弛豫效应进行编码，即它描述了磁化场与外部磁场的时间延迟对齐。这些弛豫效应是磁性粒子成像的一个重要方面，尤其是在校准过程中。在本文中，我们讨论了磁粒子成像中系统函数的数据驱动建模，其中 Landau-Lifshitz-Gilbert 方程用作在模型中包含弛豫效应的基本工具。我们在一次性和简化设置中制定了各自的参数识别问题，提出了产生正则化解决方案的重建算法并讨论了数值实验。除此之外，我们提出了一种实用的非线性 Landau-Lifshitz-Gilbert 方程数值求解器，不是通过经典的有限元方法，而是通过在逆问题框架中仅求解线性偏微分方程。