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

  • * Corresponding author: Tram Thi Ngoc Nguyen

    * Corresponding author: Tram Thi Ngoc Nguyen 
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  • 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.

    Mathematics Subject Classification: Primary: 35R30, 65M32, 65N21.

    Citation:

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  • Figure 1.  Matrix representation for a vector field in the all-at-once setting

    Figure 2.  Matrix representation for a vector field in the reduced setting

    Figure 3.  Test 1. Reconstructed $ {{\bf{m}}} $ (top) and $ {{\bf{m}}}-{{\bf{m}}}_{\rm{exact}} $ (bottom) plotted against space (x-axis) and time (y-axis)

    Figure 4.  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

    Figure 5.  Test 2. Reconstructed $ {{\bf{m}}} $ (top) and $ {{\bf{m}}}-{{\bf{m}}}_{\rm{exact}} $ (bottom). Left to right: each component

    Figure 6.  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

    Figure 7.  Test 3. Reconstructed $ {{\bf{m}}} $ (top) and $ {{\bf{m}}}-{{\bf{m}}}_{\rm{exact}} $ (bottom) plotted against space (x-axis) and time (y-axis)

    Figure 8.  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

    Figure 9.  Left: applied field $ {\textbf{h}} $. Middle: initial state $ \widetilde{{\textbf{m}}}_0 $. Right: trajectory of $ \widetilde{{\textbf{m}}}(t) $

    Figure 10.  Magnetization $ \widetilde{{\textbf{m}}} $ at different time instances

    Figure 11.  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)

    Figure 12.  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)

    Figure 13.  Test 2, all-at-once setting: reconstructed parameter over iteration index (left) and zoom of first 250 iterations (right)

    Figure 14.  Test 2, reduced setting: reconstructed parameter (left) and number of internal loops (right) in each Landweber iteration

    Figure 15.  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)

    Figure 16.  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)

    Figure 17.  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)

    Figure 18.  Test 3, 3% noise, reconstructed parameter over iteration index. Left: all-at-once setting. Right: reduced setting

    Figure 19.  Test 3, 3% noise, reduced setting: number of internal loops in each Landweber iteration

    Figure 20.  Test 3, 3% noise, residual over iteration index. Left: all-at-once setting. Right: reduced setting

    Table 1.  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
     | Show Table
    DownLoad: CSV

    Table 2.  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
     | Show Table
    DownLoad: CSV

    Table 3.  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
     | Show Table
    DownLoad: CSV
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