June 2019, 13(3): 575-596. doi: 10.3934/ipi.2019027

Discrete regularization and convergence of the inverse problem for 1+1 dimensional wave equation

1. 

University of Helsinki, P.O. Box 68 FI-00014, Finland

2. 

University College London, Gower Street London WC1E 6BT, UK

Received  May 2018 Revised  October 2018 Published  March 2019

Fund Project: J. Korpela and M. Lassas were supported by the Academy of Finland, projects 263235, 273979, 284715, and 312119.
L. Oksanen was supported by EPSRC, project EP/P01593X/1

An inverse boundary value problem for the 1+1 dimensional wave equation $ (\partial_t^2 - c(x)^2 \partial_x^2)u(x,t) = 0,\quad x\in\mathbb{R}_+ $ is considered. We give a discrete regularization strategy to recover wave speed $ c(x) $ when we are given the boundary value of the wave, $ u(0,t) $, that is produced by a single pulse-like source. The regularization strategy gives an approximative wave speed $ \widetilde c $, satisfying a Hölder type estimate $ \| \widetilde c-c\|\leq C \epsilon^{\gamma} $, where $ \epsilon $ is the noise level.

Citation: Jussi Korpela, Matti Lassas, Lauri Oksanen. Discrete regularization and convergence of the inverse problem for 1+1 dimensional wave equation. Inverse Problems & Imaging, 2019, 13 (3) : 575-596. doi: 10.3934/ipi.2019027
References:
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[2]

L. BaudouinM. de Buhan and S. Ervedoza, Convergent algorithm based on carleman estimates for the recovery of a potential in the wave equation, SIAM Journal on Numerical Analysis, 55 (2017), 1578-1613. doi: 10.1137/16M1088776.

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L. Beilina and M. V. Klibanov, A globally convergent numerical method for a coefficient inverse problem, SIAM Journal on Scientific Computing, 31 (2008), 478-509. doi: 10.1137/070711414.

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M. I. Belishev, An approach to multidimensional inverse problems for the wave equation, Dokl. Akad. Nauk SSSR, 297 (1987), 524-527.

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M. I. Belishev, Boundary control in reconstruction of manifolds and metrics (the BC method), Inverse Problems, 13 (1997), R1–R45. doi: 10.1088/0266-5611/13/5/002.

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M. I. Belishev and Y. V. Kurylëv, A nonstationary inverse problem for the multidimensional wave equation "in the large", Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 165 (1987), 21–30,189. doi: 10.1007/BF01095575.

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M. Belishev and Y. Y. Gotlib, Dynamical variant of the BC-method: theory and numerical testing, Journal of Inverse and Ill-Posed Problems, 7 (1999), 221-240. doi: 10.1515/jiip.1999.7.3.221.

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M. I. Belishev and Y. V. Kurylev, To the reconstruction of a Riemannian manifold via its spectral data (BC-method), Comm. Partial Differential Equations, 17 (1992), 767-804. doi: 10.1080/03605309208820863.

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M. I. BelishevI. B. IvanovI. V. Kubyshkin and V. S. Semenov, Numerical testing in determination of sound speed from a part of boundary by the BC-method, J. Inverse Ill-Posed Probl., 24 (2016), 159-180. doi: 10.1515/jiip-2015-0052.

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M. Bellassoued and D. Dos Santos Ferreira, Stability estimates for the anisotropic wave equation from the Dirichlet-to-Neumann map, Inverse Probl. Imaging, 5 (2011), 745-773. doi: 10.3934/ipi.2011.5.745.

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K. BinghamY. KurylevM. Lassas and S. Siltanen, Iterative time-reversal control for inverse problems, Inverse Probl. Imaging, 2 (2008), 63-81. doi: 10.3934/ipi.2008.2.63.

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N. BissantzT. Hohage and A. Munk, Consistency and rates of convergence of nonlinear Tikhonov regularization with random noise, Inverse Problems, 20 (2004), 1773-1789. doi: 10.1088/0266-5611/20/6/005.

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M. F. DahlA. Kirpichnikova and M. Lassas, Focusing waves in unknown media by modified time reversal iteration, SIAM J. Control Optim., 48 (2009), 839-858. doi: 10.1137/070705192.

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M. de Hoop, P. Kepley and L. Oksanen, Recovery of a smooth metric via wave field and coordinate transformation reconstruction, SIAM J. Appl. Math., 78 (2018), 1931–1953, arXiv: 1710.02749. doi: 10.1137/17M1151481.

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T. Hohage and M. Pricop, Nonlinear Tikhonov regularization in Hilbert scales for inverse boundary value problems with random noise, Inverse Probl. Imaging, 2 (2008), 271-290. doi: 10.3934/ipi.2008.2.271.

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I. B. Ivanov, M. I. Belishev and V. S. Semenov, The reconstruction of sound speed in the marmousi model by the boundary control method, Preprint, arXiv: 1609.07586.

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L. Justen and R. Ramlau, A non-iterative regularization approach to blind deconvolution, Inverse Problems, 22 (2006), 771-800. doi: 10.1088/0266-5611/22/3/003.

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[25]

B. Kaltenbacher and A. Neubauer, Convergence of projected iterative regularization methods for nonlinear problems with smooth solutions, Inverse Problems, 22 (2006), 1105-1119. doi: 10.1088/0266-5611/22/3/023.

[26]

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

[27]

A. KatchalovY. KurylevM. Lassas and N. Mandache, Equivalence of time-domain inverse problems and boundary spectral problems, Inverse Problems, 20 (2004), 419-436. doi: 10.1088/0266-5611/20/2/007.

[28]

A. Katchalov and Y. Kurylev, Multidimensional inverse problem with incomplete boundary spectral data, Comm. Partial Differential Equations, 23 (1998), 55-95. doi: 10.1080/03605309808821338.

[29]

A. Katchalov, Y. Kurylev and M. Lassas, Inverse Boundary Spectral Problems, vol. 123 of Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2001. doi: 10.1201/9781420036220.

[30]

A. KatsudaY. Kurylev and M. Lassas, Stability of boundary distance representation and reconstruction of Riemannian manifolds, Inverse Probl. Imaging, 1 (2007), 135-157. doi: 10.3934/ipi.2007.1.135.

[31]

A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Springer-Verlag New York, Inc., New York, NY, USA, 1996. doi: 10.1007/978-1-4612-5338-9.

[32]

M. V. KlibanovA. E. KolesovL. Nguyen and A. Sullivan, Globally strictly convex cost functional for a 1-d inverse medium scattering problem with experimental data, SIAM Journal on Applied Mathematics, 77 (2017), 1733-1755. doi: 10.1137/17M1122487.

[33]

M. V. Klibanov and N. T. Thnh, Recovering dielectric constants of explosives via a globally strictly convex cost functional, SIAM Journal on Applied Mathematics, 75 (2015), 518-537. doi: 10.1137/140981198.

[34]

K. KnudsenM. LassasJ. L. Mueller and S. Siltanen, Regularized D-bar method for the inverse conductivity problem, Inverse Probl. Imaging, 3 (2009), 599-624. doi: 10.3934/ipi.2009.3.599.

[35]

J. Korpela, M. Lassas and L. Oksanen, Regularization strategy for an inverse problem for a 1 + 1 dimensional wave equation, Inverse Problems, 32 (2016), 065001, 24pp. doi: 10.1088/0266-5611/32/6/065001.

[36]

Y. Kurylev, An inverse boundary problem for the Schrödinger operator with magnetic field, J. Math. Phys., 36 (1995), 2761-2776. doi: 10.1063/1.531064.

[37]

Y. Kurylev and M. Lassas, Inverse problems and index formulae for Dirac operators, Adv. Math., 221 (2009), 170-216. doi: 10.1016/j.aim.2008.12.001.

[38]

Y. KurylevM. Lassas and E. Somersalo, Maxwell's equations with a polarization independent wave velocity: direct and inverse problems, J. Math. Pures Appl. (9), 86 (2006), 237-270. doi: 10.1016/j.matpur.2006.01.008.

[39]

Y. Kurylev, L. Oksanen and G. P. Paternain, Inverse problems for the connection Laplacian, J. Differential Geom., 110 (2018), 457–494, arXiv: 1509.02645. doi: 10.4310/jdg/1542423627.

[40]

M. Lassas and L. Oksanen, Inverse problem for the Riemannian wave equation with Dirichlet data and Neumann data on disjoint sets, Duke Math. J., 163 (2014), 1071-1103. doi: 10.1215/00127094-2649534.

[41]

M. Lassas and L. Oksanen, Local reconstruction of a Riemannian manifold from a restriction of the hyperbolic Dirichlet-to-Neumann operator, in Inverse Problems and Applications, vol. 615 of Contemp. Math., Amer. Math. Soc., Providence, RI, 2014,223–231. doi: 10.1090/conm/615/12278.

[42]

S. Liu and L. Oksanen, A Lipschitz stable reconstruction formula for the inverse problem for the wave equation, Trans. Amer. Math. Soc., 368 (2016), 319-335. doi: 10.1090/tran/6332.

[43]

S. LuS. V. Pereverzev and R. Ramlau, An analysis of Tikhonov regularization for nonlinear ill-posed problems under a general smoothness assumption, Inverse Problems, 23 (2007), 217-230. doi: 10.1088/0266-5611/23/1/011.

[44]

P. Mathé and B. Hofmann, How general are general source conditions?, Inverse Problems, 24 (2008), 015009, 5pp. doi: 10.1088/0266-5611/24/1/015009.

[45]

A. Nachman, J. Sylvester and G. Uhlmann, An $n$-dimensional Borg-Levinson theorem, Comm. Math. Phys., 115 (1988), 595–605, URL http://projecteuclid.org/getRecord?id=euclid.cmp/1104161086. doi: 10.1007/BF01224129.

[46]

L. Oksanen, Inverse obstacle problem for the non-stationary wave equation with an unknown background, Comm. Partial Differential Equations, 38 (2013), 1492-1518. doi: 10.1080/03605302.2013.804550.

[47]

L. PestovV. Bolgova and O. Kazarina, Numerical recovering of a density by the BC-method, Inverse Probl. Imaging, 4 (2010), 703-712. doi: 10.3934/ipi.2010.4.703.

[48]

R. Ramlau, Regularization properties of Tikhonov regularization with sparsity constraints, Electron. Trans. Numer. Anal., 30 (2008), 54-74.

[49]

R. Ramlau and G. Teschke, A Tikhonov-based projection iteration for nonlinear ill-posed problems with sparsity constraints, Numer. Math., 104 (2006), 177-203. doi: 10.1007/s00211-006-0016-3.

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E. Resmerita, Regularization of ill-posed problems in Banach spaces: convergence rates, Inverse Problems, 21 (2005), 1303-1314. doi: 10.1088/0266-5611/21/4/007.

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O. Scherzer, The use of morozov's discrepancy principle for tikhonov regularization for solving nonlinear ill-posed problems, Computing, 51 (1993), 45-60. doi: 10.1007/BF02243828.

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P. Stefanov and G. Uhlmann, Stability estimates for the hyperbolic Dirichlet to Neumann map in anisotropic media, J. Funct. Anal., 154 (1998), 330-358. doi: 10.1006/jfan.1997.3188.

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P. Stefanov and G. Uhlmann, Recovery of a source term or a speed with one measurement and applications, Trans. Amer. Math. Soc., 365 (2013), 5737-5758. doi: 10.1090/S0002-9947-2013-05703-0.

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show all references

References:
[1]

M. AndersonA. KatsudaY. KurylevM. Lassas and M. Taylor, Boundary regularity for the Ricci equation, geometric convergence, and Gel'fand's inverse boundary problem, Invent. Math., 158 (2004), 261-321. doi: 10.1007/s00222-004-0371-6.

[2]

L. BaudouinM. de Buhan and S. Ervedoza, Convergent algorithm based on carleman estimates for the recovery of a potential in the wave equation, SIAM Journal on Numerical Analysis, 55 (2017), 1578-1613. doi: 10.1137/16M1088776.

[3]

L. Beilina and M. Klibanov, Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems, Springer New York, 2012. URL https://books.google.fi/books?id=ldlFcf8RqBYC. doi: 10.1007/978-1-4419-7805-9.

[4]

L. Beilina and M. V. Klibanov, A globally convergent numerical method for a coefficient inverse problem, SIAM Journal on Scientific Computing, 31 (2008), 478-509. doi: 10.1137/070711414.

[5]

M. I. Belishev, An approach to multidimensional inverse problems for the wave equation, Dokl. Akad. Nauk SSSR, 297 (1987), 524-527.

[6]

M. I. Belishev, Boundary control in reconstruction of manifolds and metrics (the BC method), Inverse Problems, 13 (1997), R1–R45. doi: 10.1088/0266-5611/13/5/002.

[7]

M. I. Belishev and Y. V. Kurylëv, A nonstationary inverse problem for the multidimensional wave equation "in the large", Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 165 (1987), 21–30,189. doi: 10.1007/BF01095575.

[8]

M. Belishev and Y. Y. Gotlib, Dynamical variant of the BC-method: theory and numerical testing, Journal of Inverse and Ill-Posed Problems, 7 (1999), 221-240. doi: 10.1515/jiip.1999.7.3.221.

[9]

M. I. Belishev and Y. V. Kurylev, To the reconstruction of a Riemannian manifold via its spectral data (BC-method), Comm. Partial Differential Equations, 17 (1992), 767-804. doi: 10.1080/03605309208820863.

[10]

M. I. BelishevI. B. IvanovI. V. Kubyshkin and V. S. Semenov, Numerical testing in determination of sound speed from a part of boundary by the BC-method, J. Inverse Ill-Posed Probl., 24 (2016), 159-180. doi: 10.1515/jiip-2015-0052.

[11]

M. Bellassoued and D. Dos Santos Ferreira, Stability estimates for the anisotropic wave equation from the Dirichlet-to-Neumann map, Inverse Probl. Imaging, 5 (2011), 745-773. doi: 10.3934/ipi.2011.5.745.

[12]

K. BinghamY. KurylevM. Lassas and S. Siltanen, Iterative time-reversal control for inverse problems, Inverse Probl. Imaging, 2 (2008), 63-81. doi: 10.3934/ipi.2008.2.63.

[13]

N. BissantzT. Hohage and A. Munk, Consistency and rates of convergence of nonlinear Tikhonov regularization with random noise, Inverse Problems, 20 (2004), 1773-1789. doi: 10.1088/0266-5611/20/6/005.

[14]

A. S. Blagoveščenskiĭ, A one-dimensional inverse boundary value problem for a second order hyperbolic equation, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 15 (1969), 85-90.

[15]

A. L. Bukhgeĭm and M. V. Klibanov, Uniqueness in the large of a class of multidimensional inverse problems, Dokl. Akad. Nauk SSSR, 260 (1981), 269-272.

[16]

M. F. DahlA. Kirpichnikova and M. Lassas, Focusing waves in unknown media by modified time reversal iteration, SIAM J. Control Optim., 48 (2009), 839-858. doi: 10.1137/070705192.

[17]

M. de Hoop, P. Kepley and L. Oksanen, Recovery of a smooth metric via wave field and coordinate transformation reconstruction, SIAM J. Appl. Math., 78 (2018), 1931–1953, arXiv: 1710.02749. doi: 10.1137/17M1151481.

[18]

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

[19]

M. Hanke, Regularizing properties of a truncated Newton-CG algorithm for nonlinear inverse problems, Numer. Funct. Anal. Optim., 18 (1997), 971-993. doi: 10.1080/01630569708816804.

[20]

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

[21]

T. Hohage and M. Pricop, Nonlinear Tikhonov regularization in Hilbert scales for inverse boundary value problems with random noise, Inverse Probl. Imaging, 2 (2008), 271-290. doi: 10.3934/ipi.2008.2.271.

[22]

I. B. Ivanov, M. I. Belishev and V. S. Semenov, The reconstruction of sound speed in the marmousi model by the boundary control method, Preprint, arXiv: 1609.07586.

[23]

L. Justen and R. Ramlau, A non-iterative regularization approach to blind deconvolution, Inverse Problems, 22 (2006), 771-800. doi: 10.1088/0266-5611/22/3/003.

[24]

S. I. Kabanikhin, A. D. Satybaev and M. A. Shishlenin, Direct Methods of Solving Multidimensional Inverse Hyperbolic Problems, Inverse and Ill-posed Problems Series, VSP, Utrecht, 2005.

[25]

B. Kaltenbacher and A. Neubauer, Convergence of projected iterative regularization methods for nonlinear problems with smooth solutions, Inverse Problems, 22 (2006), 1105-1119. doi: 10.1088/0266-5611/22/3/023.

[26]

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

[27]

A. KatchalovY. KurylevM. Lassas and N. Mandache, Equivalence of time-domain inverse problems and boundary spectral problems, Inverse Problems, 20 (2004), 419-436. doi: 10.1088/0266-5611/20/2/007.

[28]

A. Katchalov and Y. Kurylev, Multidimensional inverse problem with incomplete boundary spectral data, Comm. Partial Differential Equations, 23 (1998), 55-95. doi: 10.1080/03605309808821338.

[29]

A. Katchalov, Y. Kurylev and M. Lassas, Inverse Boundary Spectral Problems, vol. 123 of Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2001. doi: 10.1201/9781420036220.

[30]

A. KatsudaY. Kurylev and M. Lassas, Stability of boundary distance representation and reconstruction of Riemannian manifolds, Inverse Probl. Imaging, 1 (2007), 135-157. doi: 10.3934/ipi.2007.1.135.

[31]

A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Springer-Verlag New York, Inc., New York, NY, USA, 1996. doi: 10.1007/978-1-4612-5338-9.

[32]

M. V. KlibanovA. E. KolesovL. Nguyen and A. Sullivan, Globally strictly convex cost functional for a 1-d inverse medium scattering problem with experimental data, SIAM Journal on Applied Mathematics, 77 (2017), 1733-1755. doi: 10.1137/17M1122487.

[33]

M. V. Klibanov and N. T. Thnh, Recovering dielectric constants of explosives via a globally strictly convex cost functional, SIAM Journal on Applied Mathematics, 75 (2015), 518-537. doi: 10.1137/140981198.

[34]

K. KnudsenM. LassasJ. L. Mueller and S. Siltanen, Regularized D-bar method for the inverse conductivity problem, Inverse Probl. Imaging, 3 (2009), 599-624. doi: 10.3934/ipi.2009.3.599.

[35]

J. Korpela, M. Lassas and L. Oksanen, Regularization strategy for an inverse problem for a 1 + 1 dimensional wave equation, Inverse Problems, 32 (2016), 065001, 24pp. doi: 10.1088/0266-5611/32/6/065001.

[36]

Y. Kurylev, An inverse boundary problem for the Schrödinger operator with magnetic field, J. Math. Phys., 36 (1995), 2761-2776. doi: 10.1063/1.531064.

[37]

Y. Kurylev and M. Lassas, Inverse problems and index formulae for Dirac operators, Adv. Math., 221 (2009), 170-216. doi: 10.1016/j.aim.2008.12.001.

[38]

Y. KurylevM. Lassas and E. Somersalo, Maxwell's equations with a polarization independent wave velocity: direct and inverse problems, J. Math. Pures Appl. (9), 86 (2006), 237-270. doi: 10.1016/j.matpur.2006.01.008.

[39]

Y. Kurylev, L. Oksanen and G. P. Paternain, Inverse problems for the connection Laplacian, J. Differential Geom., 110 (2018), 457–494, arXiv: 1509.02645. doi: 10.4310/jdg/1542423627.

[40]

M. Lassas and L. Oksanen, Inverse problem for the Riemannian wave equation with Dirichlet data and Neumann data on disjoint sets, Duke Math. J., 163 (2014), 1071-1103. doi: 10.1215/00127094-2649534.

[41]

M. Lassas and L. Oksanen, Local reconstruction of a Riemannian manifold from a restriction of the hyperbolic Dirichlet-to-Neumann operator, in Inverse Problems and Applications, vol. 615 of Contemp. Math., Amer. Math. Soc., Providence, RI, 2014,223–231. doi: 10.1090/conm/615/12278.

[42]

S. Liu and L. Oksanen, A Lipschitz stable reconstruction formula for the inverse problem for the wave equation, Trans. Amer. Math. Soc., 368 (2016), 319-335. doi: 10.1090/tran/6332.

[43]

S. LuS. V. Pereverzev and R. Ramlau, An analysis of Tikhonov regularization for nonlinear ill-posed problems under a general smoothness assumption, Inverse Problems, 23 (2007), 217-230. doi: 10.1088/0266-5611/23/1/011.

[44]

P. Mathé and B. Hofmann, How general are general source conditions?, Inverse Problems, 24 (2008), 015009, 5pp. doi: 10.1088/0266-5611/24/1/015009.

[45]

A. Nachman, J. Sylvester and G. Uhlmann, An $n$-dimensional Borg-Levinson theorem, Comm. Math. Phys., 115 (1988), 595–605, URL http://projecteuclid.org/getRecord?id=euclid.cmp/1104161086. doi: 10.1007/BF01224129.

[46]

L. Oksanen, Inverse obstacle problem for the non-stationary wave equation with an unknown background, Comm. Partial Differential Equations, 38 (2013), 1492-1518. doi: 10.1080/03605302.2013.804550.

[47]

L. PestovV. Bolgova and O. Kazarina, Numerical recovering of a density by the BC-method, Inverse Probl. Imaging, 4 (2010), 703-712. doi: 10.3934/ipi.2010.4.703.

[48]

R. Ramlau, Regularization properties of Tikhonov regularization with sparsity constraints, Electron. Trans. Numer. Anal., 30 (2008), 54-74.

[49]

R. Ramlau and G. Teschke, A Tikhonov-based projection iteration for nonlinear ill-posed problems with sparsity constraints, Numer. Math., 104 (2006), 177-203. doi: 10.1007/s00211-006-0016-3.

[50]

E. Resmerita, Regularization of ill-posed problems in Banach spaces: convergence rates, Inverse Problems, 21 (2005), 1303-1314. doi: 10.1088/0266-5611/21/4/007.

[51]

O. Scherzer, The use of morozov's discrepancy principle for tikhonov regularization for solving nonlinear ill-posed problems, Computing, 51 (1993), 45-60. doi: 10.1007/BF02243828.

[52]

P. Stefanov and G. Uhlmann, Stability estimates for the hyperbolic Dirichlet to Neumann map in anisotropic media, J. Funct. Anal., 154 (1998), 330-358. doi: 10.1006/jfan.1997.3188.

[53]

P. Stefanov and G. Uhlmann, Recovery of a source term or a speed with one measurement and applications, Trans. Amer. Math. Soc., 365 (2013), 5737-5758. doi: 10.1090/S0002-9947-2013-05703-0.

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Figure 1.  The velocity function $ c_c $ used in the calibration of the regularization strategy
Figure 2.  The reconstruction error as a function of the multiplicative constant $ C_j = C_{reg1} $. Each curve corresponds to a noise level in (109). As expected, the reconstruction error is monotonous as a function of the noise level: The highest line corresponds to $ \epsilon_1^{(d)} = 0.1 $ and the lowest one to $ \epsilon_1^{(d)} = 0.01 $. We also observe that the reconstruction error becomes more sensitive to the choice of $ C_{reg1} $ as the noise level grows
Figure 3.  Two reconstructions (the solid blue lines) of a smooth velocity function $ c_s $ (the dashed lines). Top: Noise level $ \epsilon^{(d)}_1 = 0.1 $. Bottom: Noise level $ \epsilon^{(d)}_1 = 0.01 $
Figure 4.  The reconstruction error as a function of the noise level $ \epsilon^{(d)}_{1} $ (in log–log axes). We have used here 5 different realizations of the noise. The solid line is the average of these. Linear fitting (the dotted line) gives the estimated convergence the order 0.40
Figure 5.  Two reconstructions (the solid blue lines) of a piecewise constant velocity function $ c_p $ (the dashed lines). Top: Noise level $ \epsilon^{(d)}_1 = 0.1 $. Bottom: Noise level $ \epsilon^{(d)}_1 = 0.01 $
Figure 6.  Regularization parameter $ \alpha $, given by MDP as a function of the noise level (in log–log axes). The straight line represents the relation (111)
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