On numerical aspects of parameter identification for the Landau-Lifshitz-Gilbert equation in Magnetic Particle Imaging

Abstract

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.

Bi-level iterative regularization for inverse problems in nonlinear PDEs

“…where the strict inequality is achieved according to the constraint (42). This shows the claimed monotonicity of the residual sequence.…”

“…This section is dedicated to discussion regarding a real-life application: magnetic particle imaging (MPI). The material of the following discussion is extracted from the two papers [31,42]. The work in [31] puts forward the mathematical analysis of the parameter identification problem in MPI; subsequently, [42] proceeds with Landweber, Landweber-Kaczmarz algorithms and numerical experiments.…”

“…The material of the following discussion is extracted from the two papers [31,42]. The work in [31] puts forward the mathematical analysis of the parameter identification problem in MPI; subsequently, [42] proceeds with Landweber, Landweber-Kaczmarz algorithms and numerical experiments. Moreover, an ad hoc bi-level algorithm, in which lower-level iteration is used as an approximation method for the nonlinear PDE (73), was successfully implemented without full analysis.…”

Existence and uniqueness of weak solutions to the Landau–Lifshitz equation for large ferromagnetic bodies

“…It allows us to take into account various physical effects from the viewpoint of mathematical analysis. Today, the Landau–Lifshitz‐type equations have become fundamental and useful tools in the computer science and artificial intelligence, for instance, in the areas of magnetic recording 4 and magnetic particle imaging 5 . For more details, see, for example, previous studies 6–8 …”

“…Today, the Landau-Lifshitz-type equations have become fundamental and useful tools in the computer science and artificial intelligence, for instance, in the areas of magnetic recording 4 and magnetic particle imaging. 5 For more details, see, for example, previous studies. [6][7][8] Very recently, Han et al 9 considered the two-dimensional Landau-Lifshitz equation describing magnetization of ferromagnetic materials occupying a bounded domain Ω ⊂ R 2 , whose total energy is…”