
-
Previous Article
A mathematical perspective on radar interferometry
- IPI Home
- This Issue
-
Next Article
A mathematical approach towards THz tomography for non-destructive imaging
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 |
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.
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. |




















Test | 1 | 2 | 3 |
1 | 2 | 1 | |
-1 | 0 | 0 | |
(0, 0, 0) | |||
(0, 0, 0) | (0, 0, |
(0, 0, |
|
Initial guess |
|||
Step size |
150 | 75 | 300 |
# iterations | 3050 | 5350 | 680 |
Test | 1 | 2 | 3 |
1 | 2 | 1 | |
-1 | 0 | 0 | |
(0, 0, 0) | |||
(0, 0, 0) | (0, 0, |
(0, 0, |
|
Initial guess |
|||
Step size |
150 | 75 | 300 |
# iterations | 3050 | 5350 | 680 |
Parameter | Value | Unit | |
Magnetic permeability | 4 |
H |
|
Sat. magnetization | 474 000 | J |
|
Gyromagnetic ratio | 1.75 |
rad |
|
Damping parameter | 0.1 | ||
Field of view | [-0.006, 0.006] | m | |
Max observation time | T | 0.03 |
s |
External field strength | T |
Parameter | Value | Unit | |
Magnetic permeability | 4 |
H |
|
Sat. magnetization | 474 000 | J |
|
Gyromagnetic ratio | 1.75 |
rad |
|
Damping parameter | 0.1 | ||
Field of view | [-0.006, 0.006] | m | |
Max observation time | T | 0.03 |
s |
External field strength | T |
All-at-once | Reduced | |||||||||
#it | #it | |||||||||
10% | 259 | 0.0022 | 0.0619 | 0.292 | 0.090 | 49 | 0.0703 | 0.030 | 0.034 | |
5% | 401 | 0.0011 | 0.0309 | 0.125 | 0.072 | 49 | 0.0321 | 0.040 | 0.033 | |
3% | 564 | 0.0007 | 0.0186 | 0.040 | 0.062 | 49 | 0.0200 | 0.044 | 0.033 |
All-at-once | Reduced | |||||||||
#it | #it | |||||||||
10% | 259 | 0.0022 | 0.0619 | 0.292 | 0.090 | 49 | 0.0703 | 0.030 | 0.034 | |
5% | 401 | 0.0011 | 0.0309 | 0.125 | 0.072 | 49 | 0.0321 | 0.040 | 0.033 | |
3% | 564 | 0.0007 | 0.0186 | 0.040 | 0.062 | 49 | 0.0200 | 0.044 | 0.033 |
[1] |
Catherine Choquet, Mohammed Moumni, Mouhcine Tilioua. Homogenization of the Landau-Lifshitz-Gilbert equation in a contrasted composite medium. Discrete and Continuous Dynamical Systems - S, 2018, 11 (1) : 35-57. doi: 10.3934/dcdss.2018003 |
[2] |
Mourad Bellassoued, Oumaima Ben Fraj. Stability estimates for time-dependent coefficients appearing in the magnetic Schrödinger equation from arbitrary boundary measurements. Inverse Problems and Imaging, 2020, 14 (5) : 841-865. doi: 10.3934/ipi.2020039 |
[3] |
Tetsuya Ishiwata, Kota Kumazaki. Structure preserving finite difference scheme for the Landau-Lifshitz equation with applied magnetic field. Conference Publications, 2015, 2015 (special) : 644-651. doi: 10.3934/proc.2015.0644 |
[4] |
Qinqin Chai, Ryan Loxton, Kok Lay Teo, Chunhua Yang. A unified parameter identification method for nonlinear time-delay systems. Journal of Industrial and Management Optimization, 2013, 9 (2) : 471-486. doi: 10.3934/jimo.2013.9.471 |
[5] |
Francesco Di Plinio, Gregory S. Duane, Roger Temam. Time-dependent attractor for the Oscillon equation. Discrete and Continuous Dynamical Systems, 2011, 29 (1) : 141-167. doi: 10.3934/dcds.2011.29.141 |
[6] |
Jin Takahashi, Eiji Yanagida. Time-dependent singularities in the heat equation. Communications on Pure and Applied Analysis, 2015, 14 (3) : 969-979. doi: 10.3934/cpaa.2015.14.969 |
[7] |
Alexander Zlotnik, Ilya Zlotnik. Finite element method with discrete transparent boundary conditions for the time-dependent 1D Schrödinger equation. Kinetic and Related Models, 2012, 5 (3) : 639-667. doi: 10.3934/krm.2012.5.639 |
[8] |
Simon Hubmer, Andreas Neubauer, Ronny Ramlau, Henning U. Voss. On the parameter estimation problem of magnetic resonance advection imaging. Inverse Problems and Imaging, 2018, 12 (1) : 175-204. doi: 10.3934/ipi.2018007 |
[9] |
Deyue Zhang, Yue Wu, Yinglin Wang, Yukun Guo. A direct imaging method for the exterior and interior inverse scattering problems. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022025 |
[10] |
Gaël Bonithon. Landau-Lifschitz-Gilbert equation with applied eletric current. Conference Publications, 2007, 2007 (Special) : 138-144. doi: 10.3934/proc.2007.2007.138 |
[11] |
Zhidong Zhang. An undetermined time-dependent coefficient in a fractional diffusion equation. Inverse Problems and Imaging, 2017, 11 (5) : 875-900. doi: 10.3934/ipi.2017041 |
[12] |
Chan Liu, Jin Wen, Zhidong Zhang. Reconstruction of the time-dependent source term in a stochastic fractional diffusion equation. Inverse Problems and Imaging, 2020, 14 (6) : 1001-1024. doi: 10.3934/ipi.2020053 |
[13] |
Jiayun Lin, Kenji Nishihara, Jian Zhai. Critical exponent for the semilinear wave equation with time-dependent damping. Discrete and Continuous Dynamical Systems, 2012, 32 (12) : 4307-4320. doi: 10.3934/dcds.2012.32.4307 |
[14] |
Jungkwon Kim, Hyeongjin Lee, Ihyeok Seo, Jihyeon Seok. On Morawetz estimates with time-dependent weights for the Klein-Gordon equation. Discrete and Continuous Dynamical Systems, 2020, 40 (11) : 6275-6288. doi: 10.3934/dcds.2020279 |
[15] |
Holger Teismann. The Schrödinger equation with singular time-dependent potentials. Discrete and Continuous Dynamical Systems, 2000, 6 (3) : 705-722. doi: 10.3934/dcds.2000.6.705 |
[16] |
Haixia Li. Lifespan of solutions to a parabolic type Kirchhoff equation with time-dependent nonlinearity. Evolution Equations and Control Theory, 2021, 10 (4) : 723-732. doi: 10.3934/eect.2020088 |
[17] |
Aowen Kong, Carlos Nonato, Wenjun Liu, Manoel Jeremias dos Santos, Carlos Raposo. Equivalence between exponential stabilization and observability inequality for magnetic effected piezoelectric beams with time-varying delay and time-dependent weights. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 2959-2978. doi: 10.3934/dcdsb.2021168 |
[18] |
Michiyuki Watanabe. Inverse $N$-body scattering with the time-dependent hartree-fock approximation. Inverse Problems and Imaging, 2021, 15 (3) : 499-517. doi: 10.3934/ipi.2021002 |
[19] |
Plamen Stefanov, Yang Yang. Multiwave tomography with reflectors: Landweber's iteration. Inverse Problems and Imaging, 2017, 11 (2) : 373-401. doi: 10.3934/ipi.2017018 |
[20] |
Mohammed Al Horani, Angelo Favini, Hiroki Tanabe. Inverse problems for evolution equations with time dependent operator-coefficients. Discrete and Continuous Dynamical Systems - S, 2016, 9 (3) : 737-744. doi: 10.3934/dcdss.2016025 |
2021 Impact Factor: 1.483
Tools
Metrics
Other articles
by authors
[Back to Top]