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

TL;DR

This paper investigates the numerical challenges of parameter identification in the Landau-Lifshitz-Gilbert equation for magnetic particle imaging, proposing new algorithms and a linear PDE-based solver for improved modeling of relaxation effects.

Contribution

It introduces a novel data-driven approach for system function modeling in magnetic particle imaging using the Landau-Lifshitz-Gilbert equation, with new algorithms and a linear PDE solver.

Findings

Reconstruction algorithms for regularized solutions

Numerical experiments demonstrating effectiveness

A practical linear PDE solver for the nonlinear equation

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.