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

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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

Tram Thi Ngoc Nguyen ^1,, and Anne Wald ^2,

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.

Keywords: Magnetic particle imaging, Landau-Lifshitz-Gilbert equation, time-dependent inverse problems, parameter identification, all-at-once formulation, Landweber iteration, Kaczmarz method.

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

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 and Imaging, 2022, 16 (1) : 89-117. doi: 10.3934/ipi.2021042

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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[10]	L. Borcea, Electrical impedance tomography, Inverse Problems, 18 (2002), R99–R136. doi: 10.1088/0266-5611/18/6/201.
[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.
[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.
[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.
[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.
[15]	L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, Vol. 19, AMS, Providence, RI, 1998.
[16]	B. Gleich and J. Weizenecker, Tomographic imaging using the nonlinear response of magnetic particles, Nature, 435 (2005), 1214-1217. doi: 10.1038/nature03808.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[24]	A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Springer New York Dordrecht Heidelberg London, 2011.
[25]	T. Kluth, Mathematical models for magnetic particle imaging, Inverse Problems, 34 (2018), 083001. doi: 10.1088/1361-6420/aac535.
[26]	T. Knopp and T. M. Buzug, Magnetic Particle Imaging: An Introduction to Imaging Principles and Scanner Instrumentation, Springer Berlin Heidelberg, 2012.
[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.
[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.
[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.
[30]	F. Natterer, The Mathematics of Computerized Tomography, B. G. Teubner, Stuttgart, John Wiley & Sons, Ltd., Chichester, 1986.
[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.
[32]	T. Roubíček, Nonlinear Partial Differential Equations with Applications, Birkhäuser/Springer Basel AG, Basel, 2013. doi: 10.1007/978-3-0348-0513-1.
[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.

show all references

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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[10]	L. Borcea, Electrical impedance tomography, Inverse Problems, 18 (2002), R99–R136. doi: 10.1088/0266-5611/18/6/201.
[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.
[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.
[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.
[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.
[15]	L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, Vol. 19, AMS, Providence, RI, 1998.
[16]	B. Gleich and J. Weizenecker, Tomographic imaging using the nonlinear response of magnetic particles, Nature, 435 (2005), 1214-1217. doi: 10.1038/nature03808.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[24]	A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Springer New York Dordrecht Heidelberg London, 2011.
[25]	T. Kluth, Mathematical models for magnetic particle imaging, Inverse Problems, 34 (2018), 083001. doi: 10.1088/1361-6420/aac535.
[26]	T. Knopp and T. M. Buzug, Magnetic Particle Imaging: An Introduction to Imaging Principles and Scanner Instrumentation, Springer Berlin Heidelberg, 2012.
[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.
[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.
[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.
[30]	F. Natterer, The Mathematics of Computerized Tomography, B. G. Teubner, Stuttgart, John Wiley & Sons, Ltd., Chichester, 1986.
[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.
[32]	T. Roubíček, Nonlinear Partial Differential Equations with Applications, Birkhäuser/Springer Basel AG, Basel, 2013. doi: 10.1007/978-3-0348-0513-1.
[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.

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

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

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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)

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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

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

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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

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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)

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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

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

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

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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)

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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)

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

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

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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)

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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)

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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)

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

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

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

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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

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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

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

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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

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

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	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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