October  2021, 15(5): 865-891. doi: 10.3934/ipi.2021020

On the identification of the nonlinearity parameter in the Westervelt equation from boundary measurements

1. 

Department of Mathematics, Alpen-Adria-Universität Klagenfurt, 9020 Klagenfurt, Austria

2. 

Department of Mathematics, Texas A&M University, Texas 77843, USA

* Corresponding author: Barbara Kaltenbacher

Received  August 2020 Revised  November 2020 Published  October 2021 Early access  February 2021

Fund Project: Supported by the Austrian Science Fund fwf under grant P30054 and the National Science Foundation through award dms-1620138

We consider an undetermined coefficient inverse problem for a nonlinear partial differential equation occurring in high intensity ultrasound propagation as used in acoustic tomography. In particular, we investigate the recovery of the nonlinearity coefficient commonly labeled as $ B/A $ in the literature which is part of a space dependent coefficient $ \kappa $ in the Westervelt equation governing nonlinear acoustics. Corresponding to the typical measurement setup, the overposed data consists of time trace measurements on some zero or one dimensional set $ \Sigma $ representing the receiving transducer array. After an analysis of the map from $ \kappa $ to the overposed data, we show injectivity of its linearisation and use this as motivation for several iterative schemes to recover $ \kappa $. Numerical simulations will also be shown to illustrate the efficiency of the methods.

Citation: Barbara Kaltenbacher, William Rundell. On the identification of the nonlinearity parameter in the Westervelt equation from boundary measurements. Inverse Problems and Imaging, 2021, 15 (5) : 865-891. doi: 10.3934/ipi.2021020
References:
[1]

A. B. Bakushinskiĭ, On a convergence problem of the iterative-regularised Gauss-Newton method, Comput. Math. Math. Phys., 32 (1992), 1353-1359. 

[2]

A. B. Bakushinskii, Remarks on choosing a regularisation parameter using the quasi-optimality and ratio criterion, USSR Comput. Math. Math. Phys., 24 (1984), 181-182.  doi: 10.1016/0041-5553(84)90253-2.

[3]

L. Bjørnø, Characterization of biological media by means of their non-linearity, Ultrasonics, 24 (1986), 254-259.  doi: 10.1016/0041-624x(86)90102-2.

[4]

D. T. Blackstock, Approximate equations governing finite-amplitude sound in thermoviscous fluids, Tech Report, GD/E Report, GD-1463-52, General Dynamics Corp., Rochester, NY, 1963.

[5]

B. BlaschkeA. Neubauer and O. Scherzer, On convergence rates for the iteratively regularised Gauss-Newton method, IMA J. Numer. Anal., 17 (1997), 421-436.  doi: 10.1093/imanum/17.3.421.

[6]

J. M. Burgers, The Nonlinear Diffusion Equation, Springer, Netherlands, 1974. doi: 10.1007/978-94-010-1745-9.

[7]

V. BurovI. GurinovichO. Rudenko and E. Tagunov, Reconstruction of the spatial distribution of the nonlinearity parameter and sound velocity in acoustic nonlinear tomography, Acoustical Physics, 40 (1994), 816-823. 

[8]

C. A. Cain, Ultrasonic reflection mode imaging of the nonlinear parameter B/A: A theoretical basis, IEEE 1985 Ultrasonics Symposium, San Francisco, CA, USA, 1985. doi: 10.1109/ULTSYM.1985.198640.

[9]

C. Clason and A. Klassen, Quasi-solution of linear inverse problems in non-reflexive Banach spaces, J. Inverse Ill-Posed Probl., 26 (2018), 689-702.  doi: 10.1515/jiip-2018-0026.

[10]

C. Clason and K. Kunisch, A duality-based approach to elliptic control problems in non-reflexive Banach spaces, ESAIM Control Optim. Calc. Var., 17 (2011), 243-266.  doi: 10.1051/cocv/2010003.

[11]

D. G. Crighton, Model equations of nonlinear acoustics, Ann. Rev. Fluid Mech., 11 (1979), 11-33.  doi: 10.1146/annurev.fl.11.010179.000303.

[12]

H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Mathematics and its Applications, 375, Kluwer Academic Publishers Group, Dordrecht, 1996.

[13]

H. W. EnglK. Kunisch and A. Neubauer, Convergence rates for Tikhonov regularisation of non-linear ill-posed problems, Inverse Problems, 5 (1989), 523-540.  doi: 10.1088/0266-5611/5/4/007.

[14]

L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 1998.

[15]

M. F. Hamilton and D. T. Blackstock, Nonlinear Acoustics, Vol. 1, Academic Press, San Diego, 1998.

[16]

M. Hanke, A regularizing Levenberg–Marquardt scheme, with applications to inverse groundwater filtration problems, Inverse Problems, 13 (1997), 79-95.  doi: 10.1088/0266-5611/13/1/007.

[17]

M. HankeA. Neubauer and O. Scherzer, A convergence analysis of the Landweber iteration for nonlinear ill-posed problems, Numer. Math., 72 (1995), 21-37.  doi: 10.1007/s002110050158.

[18]

F. Hettlich and W. Rundell, A second degree method for nonlinear inverse problems, SIAM J. Numer. Anal., 37 (2000), 587-620.  doi: 10.1137/S0036142998341246.

[19]

B. HofmannB. KaltenbacherC. Pöschl and O. Scherzer, A convergence rates result for Tikhonov regularisation in Banach spaces with non-smooth operators, Inverse Problems, 23 (2007), 987-1010.  doi: 10.1088/0266-5611/23/3/009.

[20]

T. Hohage, Logarithmic convergence rates of the iteratively regularised Gauß-Newton method for an inverse potential and an inverse scattering problem, Inverse Problems, 13 (1997), 1279-1299.  doi: 10.1088/0266-5611/13/5/012.

[21]

S. Hubmer and R. Ramlau, Nesterov's accelerated gradient method for nonlinear ill-posed problems with a locally convex residual functional, Inverse Problems, 34 (2018), 30pp. doi: 10.1088/1361-6420/aacebe.

[22]

N. IchidaT. Sato and M. Linzer, Imaging the nonlinear ultrasonic parameter of a medium, Ultrasonic Imaging, 5 (1983), 295-299.  doi: 10.1177/016173468300500401.

[23]

O. Y. Imanuvilov and M. Yamamoto, Carleman estimate and an inverse source problem for the Kelvin-Voigt model for viscoelasticity, Inverse Problems, 35 (2019), 45pp. doi: 10.1088/1361-6420/ab323e.

[24]

V. Isakov, Inverse Problems for Partial Differential Equations, Applied Mathematical Sciences, 127, Springer, New York, 2006. doi: 10.1007/0-387-32183-7.

[25]

V. K. Ivanov, On linear problems which are not well-posed, Dokl. Akad. Nauk SSSR, 145 (1962), 270-272. 

[26]

B. Kaltenbacher, An iteratively regularized Gauss-Newton-Halley method for solving nonlinear ill-posed problems, Numer. Math., 131 (2015), 33-57.  doi: 10.1007/s00211-014-0682-5.

[27]

B. Kaltenbacher, Mathematics of nonlinear acoustics, Evol. Equ. Control Theory, 4 (2015), 447-491.  doi: 10.3934/eect.2015.4.447.

[28]

B. Kaltenbacher, Periodic solutions and multiharmonic expansions for the Westervelt equation, to appear, Evol. Equ. Control Theory. doi: 10.3934/eect.2020063.

[29]

B. Kaltenbacher and I. Lasiecka, Global existence and exponential decay rates for the Westervelt equation, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 503-523.  doi: 10.3934/dcdss.2009.2.503.

[30]

B. Kaltenbacher and A. Klassen, On convergence and convergence rates for Ivanov and Morozov regularisation and application to some parameter identification problems in elliptic PDEs, Inverse Problems, 34 (2018), 24pp. doi: 10.1088/1361-6420/aab739.

[31]

B. Kaltenbacher, A. Neubauer and O.Scherzer, Iterative Regularization Methods for Nonlinear Ill-Posed Problems, Radon Series on Computational and Applied Mathematics, 6, Walter de Gruyter GmbH & Co. KG, Berlin, 2008. doi: 10.1515/9783110208276.

[32]

V. Kuznetsov, Equations of nonlinear acoustics, Soviet Physics - Acoustics, 16 (1971), 467-470. 

[33]

M. B. Lesser and R. Seebass, The structure of a weak shock wave undergoing reflexion from a wall, J. Fluid Mech., 31 (1968), 501-528.  doi: 10.1017/S0022112068000303.

[34]

M. J. Lighthill, Viscosity effects in sound waves of finite amplitude, in Surveys in Mechanics, Cambridge, at the University Press, 1956, 250–351.

[35]

D. Lorenz and N. Worliczek, Necessary conditions for variational regularisation schemes, Inverse Problems, 29 (2013), 19pp. doi: 10.1088/0266-5611/29/7/075016.

[36]

S. Meyer and M. Wilke, Optimal regularity and long-time behavior of solutions for the Westervelt equation, Appl. Math. Optim., 64 (2011), 257-271.  doi: 10.1007/s00245-011-9138-9.

[37]

V. A. Morozov, On the solution of functional equations by the method of regularisation, Soviet Math. Dokl., 7 (1966), 414-417. 

[38]

M. MuhrV. NikolićB. Wohlmuth and L. Wunderlich, Isogeometric shape optimization for nonlinear ultrasound focusing, Evol. Equ. Control Theory, 8 (2019), 163-202.  doi: 10.3934/eect.2019010.

[39]

A. Neubauer, On Nesterov acceleration for Landweber iteration of linear ill-posed problems, J. Inverse Ill-Posed Probl., 25 (2017), 381-390.  doi: 10.1515/jiip-2016-0060.

[40]

A. Neubauer, Tikhonov-regularisation of ill-posed linear operator equations on closed convex sets, J. Approx. Theory, 53 (1988), 304-320.  doi: 10.1016/0021-9045(88)90025-1.

[41]

A. Neubauer and O. Scherzer, A convergent rate result for a steepest descent method and a minimal error method for the solution of nonlinear ill-posed problems, Z. Anal. Anwendungen, 14 (1995), 369-377.  doi: 10.4171/ZAA/679.

[42]

H. Ockendon and J. R. Ockendon, Waves and Compressible Flow, Texts in Applied Mathematics, 47, Springer-Verlag, New York, 2004. doi: 10.1007/b97537.

[43]

A. Pierce, Unique identification of eigenvalues and coefficients in a parabolic problem, SIAM J. Control Optim., 17 (1979), 494-499.  doi: 10.1137/0317035.

[44]

A. Rieder, On convergence rates of inexact Newton regularizations, Numer. Math., 88 (2001), 347-365.  doi: 10.1007/PL00005448.

[45]

W. Rundell and P. E. Sacks, Reconstruction techniques for classical inverse Sturm-Liouville problems, Math. Comp., 58 (1992), 161-183.  doi: 10.1090/S0025-5718-1992-1106979-0.

[46]

O. Scherzer, A modified Landweber iteration for solving parameter estimation problems, Appl. Math. Optim., 38 (1998), 45-68.  doi: 10.1007/s002459900081.

[47]

T. I. Seidman and C. R. Vogel, Well-posedness and convergence of some regularisation methods for non-linear ill posed problems, Inverse Problems, 5 (1989), 227-238.  doi: 10.1088/0266-5611/5/2/008.

[48]

F. VarrayO. BassetP. Tortoli and C. Cachard, Extensions of nonlinear B/A parameter imaging methods for echo mode, IEEE Trans. Ultrasonics, Ferroelectrics, and Frequency Control, 58 (2011), 1232-1244.  doi: 10.1109/TUFFC.2011.1933.

[49]

P. J. Westervelt, Parametric acoustic array, J. Acoustical Soc. Amer., 35 (1963), 535-537.  doi: 10.1121/1.1918525.

[50]

M. Yamamoto and B. Kaltenbacher, An inverse source problem related to acoustic nonlinearity parameter imaging, to appear, Time-Dependent Problems in Imaging and Parameter Identification, Springer, 2021.

[51]

E. A. Zabolotskaya and R. V. Khokhlov, Quasi-plane waves in the non-linear acoustics of confined beams, Soviet Physics - Acoustics, 15 (1969), 35-40. 

[52]

D. ZhangX. Chen and X.-F. Gong, Acoustic nonlinearity parameter tomography for biological tissues via parametric array from a circular piston source - Theoretical analysis and computer simulations, J. Acoustical Soc. Amer., 109 (2001), 1219-1225.  doi: 10.1121/1.1344160.

[53]

D. ZhangX. Gong and S. Ye, Acoustic nonlinearity parameter tomography for biological specimens via measurements of the second harmonics, J. Acoustical Soc. Amer., 99 (1996), 2397-2402.  doi: 10.1121/1.415427.

show all references

References:
[1]

A. B. Bakushinskiĭ, On a convergence problem of the iterative-regularised Gauss-Newton method, Comput. Math. Math. Phys., 32 (1992), 1353-1359. 

[2]

A. B. Bakushinskii, Remarks on choosing a regularisation parameter using the quasi-optimality and ratio criterion, USSR Comput. Math. Math. Phys., 24 (1984), 181-182.  doi: 10.1016/0041-5553(84)90253-2.

[3]

L. Bjørnø, Characterization of biological media by means of their non-linearity, Ultrasonics, 24 (1986), 254-259.  doi: 10.1016/0041-624x(86)90102-2.

[4]

D. T. Blackstock, Approximate equations governing finite-amplitude sound in thermoviscous fluids, Tech Report, GD/E Report, GD-1463-52, General Dynamics Corp., Rochester, NY, 1963.

[5]

B. BlaschkeA. Neubauer and O. Scherzer, On convergence rates for the iteratively regularised Gauss-Newton method, IMA J. Numer. Anal., 17 (1997), 421-436.  doi: 10.1093/imanum/17.3.421.

[6]

J. M. Burgers, The Nonlinear Diffusion Equation, Springer, Netherlands, 1974. doi: 10.1007/978-94-010-1745-9.

[7]

V. BurovI. GurinovichO. Rudenko and E. Tagunov, Reconstruction of the spatial distribution of the nonlinearity parameter and sound velocity in acoustic nonlinear tomography, Acoustical Physics, 40 (1994), 816-823. 

[8]

C. A. Cain, Ultrasonic reflection mode imaging of the nonlinear parameter B/A: A theoretical basis, IEEE 1985 Ultrasonics Symposium, San Francisco, CA, USA, 1985. doi: 10.1109/ULTSYM.1985.198640.

[9]

C. Clason and A. Klassen, Quasi-solution of linear inverse problems in non-reflexive Banach spaces, J. Inverse Ill-Posed Probl., 26 (2018), 689-702.  doi: 10.1515/jiip-2018-0026.

[10]

C. Clason and K. Kunisch, A duality-based approach to elliptic control problems in non-reflexive Banach spaces, ESAIM Control Optim. Calc. Var., 17 (2011), 243-266.  doi: 10.1051/cocv/2010003.

[11]

D. G. Crighton, Model equations of nonlinear acoustics, Ann. Rev. Fluid Mech., 11 (1979), 11-33.  doi: 10.1146/annurev.fl.11.010179.000303.

[12]

H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Mathematics and its Applications, 375, Kluwer Academic Publishers Group, Dordrecht, 1996.

[13]

H. W. EnglK. Kunisch and A. Neubauer, Convergence rates for Tikhonov regularisation of non-linear ill-posed problems, Inverse Problems, 5 (1989), 523-540.  doi: 10.1088/0266-5611/5/4/007.

[14]

L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 1998.

[15]

M. F. Hamilton and D. T. Blackstock, Nonlinear Acoustics, Vol. 1, Academic Press, San Diego, 1998.

[16]

M. Hanke, A regularizing Levenberg–Marquardt scheme, with applications to inverse groundwater filtration problems, Inverse Problems, 13 (1997), 79-95.  doi: 10.1088/0266-5611/13/1/007.

[17]

M. HankeA. Neubauer and O. Scherzer, A convergence analysis of the Landweber iteration for nonlinear ill-posed problems, Numer. Math., 72 (1995), 21-37.  doi: 10.1007/s002110050158.

[18]

F. Hettlich and W. Rundell, A second degree method for nonlinear inverse problems, SIAM J. Numer. Anal., 37 (2000), 587-620.  doi: 10.1137/S0036142998341246.

[19]

B. HofmannB. KaltenbacherC. Pöschl and O. Scherzer, A convergence rates result for Tikhonov regularisation in Banach spaces with non-smooth operators, Inverse Problems, 23 (2007), 987-1010.  doi: 10.1088/0266-5611/23/3/009.

[20]

T. Hohage, Logarithmic convergence rates of the iteratively regularised Gauß-Newton method for an inverse potential and an inverse scattering problem, Inverse Problems, 13 (1997), 1279-1299.  doi: 10.1088/0266-5611/13/5/012.

[21]

S. Hubmer and R. Ramlau, Nesterov's accelerated gradient method for nonlinear ill-posed problems with a locally convex residual functional, Inverse Problems, 34 (2018), 30pp. doi: 10.1088/1361-6420/aacebe.

[22]

N. IchidaT. Sato and M. Linzer, Imaging the nonlinear ultrasonic parameter of a medium, Ultrasonic Imaging, 5 (1983), 295-299.  doi: 10.1177/016173468300500401.

[23]

O. Y. Imanuvilov and M. Yamamoto, Carleman estimate and an inverse source problem for the Kelvin-Voigt model for viscoelasticity, Inverse Problems, 35 (2019), 45pp. doi: 10.1088/1361-6420/ab323e.

[24]

V. Isakov, Inverse Problems for Partial Differential Equations, Applied Mathematical Sciences, 127, Springer, New York, 2006. doi: 10.1007/0-387-32183-7.

[25]

V. K. Ivanov, On linear problems which are not well-posed, Dokl. Akad. Nauk SSSR, 145 (1962), 270-272. 

[26]

B. Kaltenbacher, An iteratively regularized Gauss-Newton-Halley method for solving nonlinear ill-posed problems, Numer. Math., 131 (2015), 33-57.  doi: 10.1007/s00211-014-0682-5.

[27]

B. Kaltenbacher, Mathematics of nonlinear acoustics, Evol. Equ. Control Theory, 4 (2015), 447-491.  doi: 10.3934/eect.2015.4.447.

[28]

B. Kaltenbacher, Periodic solutions and multiharmonic expansions for the Westervelt equation, to appear, Evol. Equ. Control Theory. doi: 10.3934/eect.2020063.

[29]

B. Kaltenbacher and I. Lasiecka, Global existence and exponential decay rates for the Westervelt equation, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 503-523.  doi: 10.3934/dcdss.2009.2.503.

[30]

B. Kaltenbacher and A. Klassen, On convergence and convergence rates for Ivanov and Morozov regularisation and application to some parameter identification problems in elliptic PDEs, Inverse Problems, 34 (2018), 24pp. doi: 10.1088/1361-6420/aab739.

[31]

B. Kaltenbacher, A. Neubauer and O.Scherzer, Iterative Regularization Methods for Nonlinear Ill-Posed Problems, Radon Series on Computational and Applied Mathematics, 6, Walter de Gruyter GmbH & Co. KG, Berlin, 2008. doi: 10.1515/9783110208276.

[32]

V. Kuznetsov, Equations of nonlinear acoustics, Soviet Physics - Acoustics, 16 (1971), 467-470. 

[33]

M. B. Lesser and R. Seebass, The structure of a weak shock wave undergoing reflexion from a wall, J. Fluid Mech., 31 (1968), 501-528.  doi: 10.1017/S0022112068000303.

[34]

M. J. Lighthill, Viscosity effects in sound waves of finite amplitude, in Surveys in Mechanics, Cambridge, at the University Press, 1956, 250–351.

[35]

D. Lorenz and N. Worliczek, Necessary conditions for variational regularisation schemes, Inverse Problems, 29 (2013), 19pp. doi: 10.1088/0266-5611/29/7/075016.

[36]

S. Meyer and M. Wilke, Optimal regularity and long-time behavior of solutions for the Westervelt equation, Appl. Math. Optim., 64 (2011), 257-271.  doi: 10.1007/s00245-011-9138-9.

[37]

V. A. Morozov, On the solution of functional equations by the method of regularisation, Soviet Math. Dokl., 7 (1966), 414-417. 

[38]

M. MuhrV. NikolićB. Wohlmuth and L. Wunderlich, Isogeometric shape optimization for nonlinear ultrasound focusing, Evol. Equ. Control Theory, 8 (2019), 163-202.  doi: 10.3934/eect.2019010.

[39]

A. Neubauer, On Nesterov acceleration for Landweber iteration of linear ill-posed problems, J. Inverse Ill-Posed Probl., 25 (2017), 381-390.  doi: 10.1515/jiip-2016-0060.

[40]

A. Neubauer, Tikhonov-regularisation of ill-posed linear operator equations on closed convex sets, J. Approx. Theory, 53 (1988), 304-320.  doi: 10.1016/0021-9045(88)90025-1.

[41]

A. Neubauer and O. Scherzer, A convergent rate result for a steepest descent method and a minimal error method for the solution of nonlinear ill-posed problems, Z. Anal. Anwendungen, 14 (1995), 369-377.  doi: 10.4171/ZAA/679.

[42]

H. Ockendon and J. R. Ockendon, Waves and Compressible Flow, Texts in Applied Mathematics, 47, Springer-Verlag, New York, 2004. doi: 10.1007/b97537.

[43]

A. Pierce, Unique identification of eigenvalues and coefficients in a parabolic problem, SIAM J. Control Optim., 17 (1979), 494-499.  doi: 10.1137/0317035.

[44]

A. Rieder, On convergence rates of inexact Newton regularizations, Numer. Math., 88 (2001), 347-365.  doi: 10.1007/PL00005448.

[45]

W. Rundell and P. E. Sacks, Reconstruction techniques for classical inverse Sturm-Liouville problems, Math. Comp., 58 (1992), 161-183.  doi: 10.1090/S0025-5718-1992-1106979-0.

[46]

O. Scherzer, A modified Landweber iteration for solving parameter estimation problems, Appl. Math. Optim., 38 (1998), 45-68.  doi: 10.1007/s002459900081.

[47]

T. I. Seidman and C. R. Vogel, Well-posedness and convergence of some regularisation methods for non-linear ill posed problems, Inverse Problems, 5 (1989), 227-238.  doi: 10.1088/0266-5611/5/2/008.

[48]

F. VarrayO. BassetP. Tortoli and C. Cachard, Extensions of nonlinear B/A parameter imaging methods for echo mode, IEEE Trans. Ultrasonics, Ferroelectrics, and Frequency Control, 58 (2011), 1232-1244.  doi: 10.1109/TUFFC.2011.1933.

[49]

P. J. Westervelt, Parametric acoustic array, J. Acoustical Soc. Amer., 35 (1963), 535-537.  doi: 10.1121/1.1918525.

[50]

M. Yamamoto and B. Kaltenbacher, An inverse source problem related to acoustic nonlinearity parameter imaging, to appear, Time-Dependent Problems in Imaging and Parameter Identification, Springer, 2021.

[51]

E. A. Zabolotskaya and R. V. Khokhlov, Quasi-plane waves in the non-linear acoustics of confined beams, Soviet Physics - Acoustics, 15 (1969), 35-40. 

[52]

D. ZhangX. Chen and X.-F. Gong, Acoustic nonlinearity parameter tomography for biological tissues via parametric array from a circular piston source - Theoretical analysis and computer simulations, J. Acoustical Soc. Amer., 109 (2001), 1219-1225.  doi: 10.1121/1.1344160.

[53]

D. ZhangX. Gong and S. Ye, Acoustic nonlinearity parameter tomography for biological specimens via measurements of the second harmonics, J. Acoustical Soc. Amer., 99 (1996), 2397-2402.  doi: 10.1121/1.415427.

Figure 1.  The surface Σ
Figure 2.  Reconstructions of a smooth $ \kappa(x) $ from time trace data at $ \,x = 1\, $ under 0.1% (left) and 1% (right) noise using Newton's method
Figure 3.  Reconstructions of piecewise linear $ \kappa(x) $ from time trace data at $ \,x = 1 $ under 0.1% (left) and 1% (right) noise using Newton's method
Figure 4.  Reconstructions of a piecewise constant $ \kappa(x) $ from time trace data at $ x = 1 $ under $ 0.1\% $ noise using Newton iteration
Figure 5.  Comparison of Newton (in red) and Halley (in blue) final reconstructions under $ 0.1\% $ noise
Figure 6.  Comparison of Newton (in red) and Halley (in blue) final reconstructions and norm differences of the $ n^{\rm th} $ iterate $ \kappa_n $ and the actual $ \kappa $. Noise level was $ 1\% $
Figure 7.  Reconstructions of a piecewise linear $ \kappa(x) $ from time trace data at $ x = 1 $ under $ 1\% $ noise using Landweber iteration
Figure 8.  The leftmost figure shows reconstructions of $ \kappa(x) $ under $ 0.1\% $ noise using Landweber iteration. The rightmost figure shows the decay of the norm $ \kappa_n(x)-\kappa_{\rm act}(x) $
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