# American Institute of Mathematical Sciences

February  2022, 16(1): 89-117. 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  February 2022 Early access  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, 2022, 16 (1) : 89-117. doi: 10.3934/ipi.2021042
##### References:
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Prohl, Convergence of an implicit finite element method for the Landau-Lifshitz-Gilbert equation, SIAM J. Numer. Anal., 44 (2006), 1405-1419.  doi: 10.1137/050631070.  Google Scholar [6] S. Bartels and A. Prohl, Convergence of an implicit, constraint preserving finite element discretization of p-harmonic heat flow into spheres, Numerische Mathematik, 109 (2008), 489-507.  doi: 10.1007/s00211-008-0150-1.  Google Scholar [7] J. Baumeister, B. Kaltenbacher and A. Leitão, On Levenberg-Marquardt-Kaczmarz iterative methods for solving systems of nonlinear ill-posed equations, Inverse Problems and Imaging, 4 (2010), 335-350.  doi: 10.3934/ipi.2010.4.335.  Google Scholar [8] F. Binder, F. Schöpfer and T. Schuster, Defect localization in fibre-reinforced composites by computing external volume forces from surface sensor measurements, Inverse Problems, 31 (2015), 025006. doi: 10.1088/0266-5611/31/2/025006.  Google Scholar [9] S. E. Blanke, B. N. Hahn and A. 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Scherzer, Kaczmarz methods for regularizing nonlinear ill-posed equations Ⅱ: Applications, Inverse Problems and Imaging, 1 (2007), 507-523.  doi: 10.3934/ipi.2007.1.507.  Google Scholar [19] M. Haltmeier, A. Leitao and O. Scherzer, Kaczmarz methods for regularizing nonlinear ill-posed equations Ⅰ: Convergence analysis, Inverse Problems and Imaging, 1 (2007), 289-298.  doi: 10.3934/ipi.2007.1.289.  Google Scholar [20] M. Hanke, A. Neubauer and O. Scherzer, A convergence analysis of the Landweber iteration for nonlinear ill-posed problems, Numerische Mathematik, 72 (1995), 21-37.  doi: 10.1007/s002110050158.  Google Scholar [21] B. Kaltenbacher, All-at-once versus reduced iterative methods for time dependent inverse problems, Inverse Problems, 33 (2017), 064002. doi: 10.1088/1361-6420/aa6f34.  Google Scholar [22] B. Kaltenbacher, A. Neubauer and O. Scherzer, Iterative regularization methods for nonlinear ill-posed problems, in Radon Series on Computational and Applied Mathematics, Vol. 6, Walter de Gruyter GmbH & Co. KG, Berlin, 2008. doi: 10.1515/9783110208276.  Google Scholar [23] B. Kaltenbacher, T. Nguyen, A. Wald and T. Schuster, Parameter Identification for the Landau-Lifshitz-Gilbert Equation in Magnetic Particle Imaging, Parameter Identification for the Landau-Lifshitz-Gilbert Equation in Magnetic Particle Imaging, Time-Dependent Problems in Imaging and Parameter Identification doi: 10.1007/978-3-030-57784-1_13.  Google Scholar [24] A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Springer New York Dordrecht Heidelberg London, 2011. Google Scholar [25] T. Kluth, Mathematical models for magnetic particle imaging, Inverse Problems, 34 (2018), 083001. doi: 10.1088/1361-6420/aac535.  Google Scholar [26] T. Knopp and T. M. Buzug, Magnetic Particle Imaging: An Introduction to Imaging Principles and Scanner Instrumentation, Springer Berlin Heidelberg, 2012. Google Scholar [27] T. Knopp, N. Gdaniec and M. Möddel, Magnetic particle imaging: From proof of principle to preclinical applications, Physics in Medicine & Biology, 62 (2017), R124. doi: 10.1088/1361-6560/aa6c99.  Google Scholar [28] R. Kowar and O. Scherzer, Convergence analysis of a Landweber-Kaczmarz method for solving nonlinear ill-posed problems, Ill-Posed and Inverse Problems, 23 (2002), 69-90.   Google Scholar [29] M. Kružík and A. Prohl, Recent developments in the modeling, analysis and numerics of ferromagnetism, SIAM Rev., 48 (2006), 439-483.  doi: 10.1137/S0036144504446187.  Google Scholar [30] F. Natterer, The Mathematics of Computerized Tomography, B. G. Teubner, Stuttgart, John Wiley & Sons, Ltd., Chichester, 1986.  Google Scholar [31] T. T. N. Nguyen, Landweber-Kaczmarz for parameter identification in time-dependent inverse problems: All-at-once versus reduced version, Inverse Problems, 35 (2019), 035009. doi: 10.1088/1361-6420/aaf9ba.  Google Scholar [32] T. Roubíček, Nonlinear Partial Differential Equations with Applications, Birkhäuser/Springer Basel AG, Basel, 2013. doi: 10.1007/978-3-0348-0513-1.  Google Scholar [33] F. Tröltzsch, Optimal Control of Partial Differential Equations. Theory, Methods and Applications, Graduate Studies in Mathematics, Vol. 112, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/112.  Google Scholar

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##### References:
 [1] F. Alouges, A new finite element scheme for Landau-Lifchitz equations, Discrete Contin. Dyn. Syst. Ser. S, 1 (2008), 187-196.  doi: 10.3934/dcdss.2008.1.187.  Google Scholar [2] F. Alouges, E. Kritsikis, J. Steiner and J.-C. Toussaint, A convergent and precise finite element scheme for Landau-Lifschitz-Gilbert equation, Numerische Mathematik, 128 (2014), 407-430.  doi: 10.1007/s00211-014-0615-3.  Google Scholar [3] L. Baňas, M. Page and D. Praetorius, A convergent linear finite element scheme for the Maxwell-Landau-Lifshitz-Gilbert equations, Electronic Transactions on Numerical Analysis, 44 (2015), 250-270.   Google Scholar [4] L. Baňas, M. Page, D. Praetorius and J. Rochat, A decoupled and unconditionally convergent linear FEM integrator for the Landau-Lifshitz-Gilbert equation with magnetostriction, IMA Journal of Numerical Analysis, 34 (2014), 1361-1385.  doi: 10.1093/imanum/drt050.  Google Scholar [5] S. Bartels and A. Prohl, Convergence of an implicit finite element method for the Landau-Lifshitz-Gilbert equation, SIAM J. Numer. Anal., 44 (2006), 1405-1419.  doi: 10.1137/050631070.  Google Scholar [6] S. Bartels and A. Prohl, Convergence of an implicit, constraint preserving finite element discretization of p-harmonic heat flow into spheres, Numerische Mathematik, 109 (2008), 489-507.  doi: 10.1007/s00211-008-0150-1.  Google Scholar [7] J. Baumeister, B. Kaltenbacher and A. Leitão, On Levenberg-Marquardt-Kaczmarz iterative methods for solving systems of nonlinear ill-posed equations, Inverse Problems and Imaging, 4 (2010), 335-350.  doi: 10.3934/ipi.2010.4.335.  Google Scholar [8] F. Binder, F. Schöpfer and T. Schuster, Defect localization in fibre-reinforced composites by computing external volume forces from surface sensor measurements, Inverse Problems, 31 (2015), 025006. doi: 10.1088/0266-5611/31/2/025006.  Google Scholar [9] S. E. Blanke, B. N. Hahn and A. Wald, Inverse problems with inexact forward operator: Iterative regularization and application in dynamic imaging, Inverse Problems, 36 (2020), 124001. doi: 10.1088/1361-6420/abb5e1.  Google Scholar [10] L. Borcea, Electrical impedance tomography, Inverse Problems, 18 (2002), R99–R136. doi: 10.1088/0266-5611/18/6/201.  Google Scholar [11] J. Borgert, J. D. Schmidt, I. Schmale, J. Rahmer, C. Bontus, B. Gleich, B. David, R. Eckart, O. Woywode, J. Weizenecker, J. Schnorr, M. Taupitz, J. Haegele, F. M. Vogt and J. Barkhausen, Fundamentals and applications of magnetic particle imaging, Journal of Cardiovascular Computed Tomography, 6 (2012), 149-153.  doi: 10.1016/j.jcct.2012.04.007.  Google Scholar [12] I. Cimrák, A survey on the numerics and computations for the Landau-Lifshitz equation of micromagnetism, Archives of Computational Methods in Engineering, 15 (2008), 277-309.  doi: 10.1007/s11831-008-9021-2.  Google Scholar [13] L. R. Croft, P. W. Goodwill and S. M. Conolly, Relaxation in x-space magnetic particle imaging, IEEE Transactions on Medical Imaging, 31 (2012), 2335-2342.  doi: 10.1007/978-3-642-24133-8_24.  Google Scholar [14] P. Elbau, L. Mindrinos and O. Scherzer, Inverse problems of combined photoacoustic and optical coherence tomography, Mathematical Methods in the Applied Sciences, 40 (2017), 505-522.  doi: 10.1002/mma.3915.  Google Scholar [15] L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, Vol. 19, AMS, Providence, RI, 1998.  Google Scholar [16] B. Gleich and J. Weizenecker, Tomographic imaging using the nonlinear response of magnetic particles, Nature, 435 (2005), 1214-1217.  doi: 10.1038/nature03808.  Google Scholar [17] B. Guo and M.-C. Hong, The Landau-Lifshitz equation of the ferromagnetic spin chain and harmonic maps, Calc. Var. Partial Differential Equations, 1 (1993), 311-334.  doi: 10.1007/BF01191298.  Google Scholar [18] M. Haltmeier, R. Kowar, A. Leitao and O. Scherzer, Kaczmarz methods for regularizing nonlinear ill-posed equations Ⅱ: Applications, Inverse Problems and Imaging, 1 (2007), 507-523.  doi: 10.3934/ipi.2007.1.507.  Google Scholar [19] M. Haltmeier, A. Leitao and O. Scherzer, Kaczmarz methods for regularizing nonlinear ill-posed equations Ⅰ: Convergence analysis, Inverse Problems and Imaging, 1 (2007), 289-298.  doi: 10.3934/ipi.2007.1.289.  Google Scholar [20] M. Hanke, A. Neubauer and O. Scherzer, A convergence analysis of the Landweber iteration for nonlinear ill-posed problems, Numerische Mathematik, 72 (1995), 21-37.  doi: 10.1007/s002110050158.  Google Scholar [21] B. Kaltenbacher, All-at-once versus reduced iterative methods for time dependent inverse problems, Inverse Problems, 33 (2017), 064002. doi: 10.1088/1361-6420/aa6f34.  Google Scholar [22] B. Kaltenbacher, A. Neubauer and O. Scherzer, Iterative regularization methods for nonlinear ill-posed problems, in Radon Series on Computational and Applied Mathematics, Vol. 6, Walter de Gruyter GmbH & Co. KG, Berlin, 2008. doi: 10.1515/9783110208276.  Google Scholar [23] B. Kaltenbacher, T. Nguyen, A. Wald and T. Schuster, Parameter Identification for the Landau-Lifshitz-Gilbert Equation in Magnetic Particle Imaging, Parameter Identification for the Landau-Lifshitz-Gilbert Equation in Magnetic Particle Imaging, Time-Dependent Problems in Imaging and Parameter Identification doi: 10.1007/978-3-030-57784-1_13.  Google Scholar [24] A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Springer New York Dordrecht Heidelberg London, 2011. Google Scholar [25] T. Kluth, Mathematical models for magnetic particle imaging, Inverse Problems, 34 (2018), 083001. doi: 10.1088/1361-6420/aac535.  Google Scholar [26] T. Knopp and T. M. Buzug, Magnetic Particle Imaging: An Introduction to Imaging Principles and Scanner Instrumentation, Springer Berlin Heidelberg, 2012. Google Scholar [27] T. Knopp, N. Gdaniec and M. Möddel, Magnetic particle imaging: From proof of principle to preclinical applications, Physics in Medicine & Biology, 62 (2017), R124. doi: 10.1088/1361-6560/aa6c99.  Google Scholar [28] R. Kowar and O. Scherzer, Convergence analysis of a Landweber-Kaczmarz method for solving nonlinear ill-posed problems, Ill-Posed and Inverse Problems, 23 (2002), 69-90.   Google Scholar [29] M. Kružík and A. Prohl, Recent developments in the modeling, analysis and numerics of ferromagnetism, SIAM Rev., 48 (2006), 439-483.  doi: 10.1137/S0036144504446187.  Google Scholar [30] F. Natterer, The Mathematics of Computerized Tomography, B. G. Teubner, Stuttgart, John Wiley & Sons, Ltd., Chichester, 1986.  Google Scholar [31] T. T. N. Nguyen, Landweber-Kaczmarz for parameter identification in time-dependent inverse problems: All-at-once versus reduced version, Inverse Problems, 35 (2019), 035009. doi: 10.1088/1361-6420/aaf9ba.  Google Scholar [32] T. Roubíček, Nonlinear Partial Differential Equations with Applications, Birkhäuser/Springer Basel AG, Basel, 2013. doi: 10.1007/978-3-0348-0513-1.  Google Scholar [33] F. Tröltzsch, Optimal Control of Partial Differential Equations. Theory, Methods and Applications, Graduate Studies in Mathematics, Vol. 112, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/112.  Google Scholar
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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