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doi: 10.3934/ipi.2021042

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

 1 Institute of Mathematics and Scientific Computing, University of Graz, Heinrichstraße 36, A-8010 Graz, Austria 2 Institute of Numerical and Applied Mathematics, University of Göttingen, Lotzestraße 16-18, 37073 Göttingen, Germany

* Corresponding author: Tram Thi Ngoc Nguyen

Received  January 2021 Revised  April 2021 Published  May 2021

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.

Citation: Tram Thi Ngoc Nguyen, Anne Wald. On numerical aspects of parameter identification for the Landau-Lifshitz-Gilbert equation in Magnetic Particle Imaging. Inverse Problems & Imaging, doi: 10.3934/ipi.2021042
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##### References:
Matrix representation for a vector field in the all-at-once setting
Matrix representation for a vector field in the reduced setting
Test 1. Reconstructed ${{\bf{m}}}$ (top) and ${{\bf{m}}}-{{\bf{m}}}_{\rm{exact}}$ (bottom) plotted against space (x-axis) and time (y-axis)
Test 1. Plots of step size $\mu$ (left) and relative error $\frac{\|{\textbf{m}}_k-{\textbf{m}}_\rm{ex}\|}{\|{\textbf{m}}_\rm{ex}\|}$ (right) over iteration index
Test 2. Reconstructed ${{\bf{m}}}$ (top) and ${{\bf{m}}}-{{\bf{m}}}_{\rm{exact}}$ (bottom). Left to right: each component
Test 2. Plots of step size $\mu$ (left) and relative error $\frac{\|{\textbf{m}}_k-{\textbf{m}}_\rm{ex}\|}{\|{\textbf{m}}_\rm{ex}\|}$ (right) over iteration index
Test 3. Reconstructed ${{\bf{m}}}$ (top) and ${{\bf{m}}}-{{\bf{m}}}_{\rm{exact}}$ (bottom) plotted against space (x-axis) and time (y-axis)
Test 3. Plots of step size $\mu$ (left) and relative error $\frac{\|{\textbf{m}}_k-{\textbf{m}}_\rm{ex}\|}{\|{\textbf{m}}_\rm{ex}\|}$ (right) over iteration index
Left: applied field ${\textbf{h}}$. Middle: initial state $\widetilde{{\textbf{m}}}_0$. Right: trajectory of $\widetilde{{\textbf{m}}}(t)$
Magnetization $\widetilde{{\textbf{m}}}$ at different time instances
Test 2, all-at-once setting. Reconstructed ${{\bf{m}}}$ (top) and ${{\bf{m}}}-{{\bf{m}}}_{\text{exact}}$ (bottom) plotted against space (x-axis) and time (y-axis)
Test 2, reduced setting. Reconstructed ${{\bf{m}}}$ (top) and ${{\bf{m}}}-{{\bf{m}}}_{\text{exact}}$ (bottom) plotted against space (x-axis) and time (y-axis)
Test 2, all-at-once setting: reconstructed parameter over iteration index (left) and zoom of first 250 iterations (right)
Test 2, reduced setting: reconstructed parameter (left) and number of internal loops (right) in each Landweber iteration
Test 2, residual over iteration index: reduced setting (left), first 250 iterations for the all-at-once setting (middle), and a zoom of the all-at-once residual plot (right)
Test 3, 3% noise, all-at-once setting. Reconstructed ${{\bf{m}}}$ (top) and ${{\bf{m}}}-{{\bf{m}}}_{\rm{exact}}$ (bottom) plotted against space (x-axis) and time (y-axis)
Test 3, 3% noise, reduced setting. Reconstructed ${{\bf{m}}}$ (top) and ${{\bf{m}}}-{{\bf{m}}}_{\text{exact}}$ (bottom) plotted against space (x-axis) and time (y-axis)
Test 3, 3% noise, reconstructed parameter over iteration index. Left: all-at-once setting. Right: reduced setting
Test 3, 3% noise, reduced setting: number of internal loops in each Landweber iteration
Test 3, 3% noise, residual over iteration index. Left: all-at-once setting. Right: reduced setting
Test cases and run parameters
 Test 1 2 3 ${\hat{\alpha}}_1$ 1 2 1 ${\hat{\alpha}}_2$ -1 0 0 ${{\bf{h}}}$ $\cfrac{2}{5}(0,3,4)$ $-(\cos(x),\cos(x),0)$ (0, 0, 0) ${{\bf{m}}}_{\rm{exact}}$ $\cfrac{1}{5}(0,3,4)$ $(\cos(x),\cos(x),e^t)$ $(\sin(x),\cos(x),e^t)$ ${\textbf{m}}_0$ $\cfrac{1}{5}(0,3,4)$ $(\cos(x),\cos(x),1)$ $(\sin(x),\cos(x),1)$ $\hat{{{\bf{m}}}}_{exact}$ (0, 0, 0) (0, 0, $e^t-1$) (0, 0, $e^t-1$) Initial guess ${\hat{{\bf{m}}}}$ $-5t(1,1,1)$ $-5t\cos(x)(1,1,1)$ $-\cfrac{\sin(30t)}{5}(1,1,1)$ Step size $\mu$ 150 75 300 # iterations 3050 5350 680
 Test 1 2 3 ${\hat{\alpha}}_1$ 1 2 1 ${\hat{\alpha}}_2$ -1 0 0 ${{\bf{h}}}$ $\cfrac{2}{5}(0,3,4)$ $-(\cos(x),\cos(x),0)$ (0, 0, 0) ${{\bf{m}}}_{\rm{exact}}$ $\cfrac{1}{5}(0,3,4)$ $(\cos(x),\cos(x),e^t)$ $(\sin(x),\cos(x),e^t)$ ${\textbf{m}}_0$ $\cfrac{1}{5}(0,3,4)$ $(\cos(x),\cos(x),1)$ $(\sin(x),\cos(x),1)$ $\hat{{{\bf{m}}}}_{exact}$ (0, 0, 0) (0, 0, $e^t-1$) (0, 0, $e^t-1$) Initial guess ${\hat{{\bf{m}}}}$ $-5t(1,1,1)$ $-5t\cos(x)(1,1,1)$ $-\cfrac{\sin(30t)}{5}(1,1,1)$ Step size $\mu$ 150 75 300 # iterations 3050 5350 680
Common physical parameters
 Parameter Value Unit Magnetic permeability $\mu_0$ 4$\pi\times 10^{-7}$ H $\rm{m}^{-1}$ Sat. magnetization $m_{\mathrm{S}}$ 474 000 J $\rm{m}^{-3} \rm{T}^{-1}$ Gyromagnetic ratio $\gamma$ 1.75$\times 10^{11}$ rad $\rm{s}^{-1}$ Damping parameter $\alpha_\rm{D}$ 0.1 Field of view $\Omega$ [-0.006, 0.006] m Max observation time T 0.03$\times 10^{-3}$ s External field strength $|{\textbf{h}}|$ $10^{-4}$ T
 Parameter Value Unit Magnetic permeability $\mu_0$ 4$\pi\times 10^{-7}$ H $\rm{m}^{-1}$ Sat. magnetization $m_{\mathrm{S}}$ 474 000 J $\rm{m}^{-3} \rm{T}^{-1}$ Gyromagnetic ratio $\gamma$ 1.75$\times 10^{11}$ rad $\rm{s}^{-1}$ Damping parameter $\alpha_\rm{D}$ 0.1 Field of view $\Omega$ [-0.006, 0.006] m Max observation time T 0.03$\times 10^{-3}$ s External field strength $|{\textbf{h}}|$ $10^{-4}$ T
Reconstruction with noisy data
 All-at-once Reduced $\delta$ #it $r_{llg}$ $r_{obs}$ $e_{\alpha_1}$ $e_{\alpha_2}$ #it $r_{llg}$ $r_{obs}$ $e_{\alpha_1}$ $e_{\alpha_2}$ 10% 259 0.0022 0.0619 0.292 0.090 49 $3\times10^{-6}$ 0.0703 0.030 0.034 5% 401 0.0011 0.0309 0.125 0.072 49 $3\times10^{-6}$ 0.0321 0.040 0.033 3% 564 0.0007 0.0186 0.040 0.062 49 $3\times10^{-6}$ 0.0200 0.044 0.033
 All-at-once Reduced $\delta$ #it $r_{llg}$ $r_{obs}$ $e_{\alpha_1}$ $e_{\alpha_2}$ #it $r_{llg}$ $r_{obs}$ $e_{\alpha_1}$ $e_{\alpha_2}$ 10% 259 0.0022 0.0619 0.292 0.090 49 $3\times10^{-6}$ 0.0703 0.030 0.034 5% 401 0.0011 0.0309 0.125 0.072 49 $3\times10^{-6}$ 0.0321 0.040 0.033 3% 564 0.0007 0.0186 0.040 0.062 49 $3\times10^{-6}$ 0.0200 0.044 0.033
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