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

Jussi Korpela; Matti Lassas; Lauri Oksanen

doi:10.3934/ipi.2019027

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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

Jussi Korpela ^1, , Matti Lassas ^1,2, and Lauri Oksanen ^1,2,

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.

Keywords: Inverse problem, regularization theory, wave equation, discretization.

Mathematics Subject Classification: Primary: 35R30; Secondary: 35L05.

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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References:

[1]	M. Anderson, A. Katsuda, Y. Kurylev, M. 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. Google Scholar
[2]	L. Baudouin, M. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[5]	M. I. Belishev, An approach to multidimensional inverse problems for the wave equation, Dokl. Akad. Nauk SSSR, 297 (1987), 524-527. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[10]	M. I. Belishev, I. B. Ivanov, I. 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. Google Scholar
[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. Google Scholar
[12]	K. Bingham, Y. Kurylev, M. 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. Google Scholar
[13]	N. Bissantz, T. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[16]	M. F. Dahl, A. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[20]	B. Hofmann, B. Kaltenbacher, C. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[27]	A. Katchalov, Y. Kurylev, M. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[30]	A. Katsuda, Y. 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. Google Scholar
[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. Google Scholar
[32]	M. V. Klibanov, A. E. Kolesov, L. 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. Google Scholar
[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. Google Scholar
[34]	K. Knudsen, M. Lassas, J. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[38]	Y. Kurylev, M. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[43]	S. Lu, S. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[47]	L. Pestov, V. 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. Google Scholar
[48]	R. Ramlau, Regularization properties of Tikhonov regularization with sparsity constraints, Electron. Trans. Numer. Anal., 30 (2008), 54-74. Google Scholar
[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. Google Scholar
[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. Google Scholar
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Figure 1. The velocity function $ c_c $ used in the calibration of the regularization strategy

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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

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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 $

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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

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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 $

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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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2020 Impact Factor: 1.639

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American Institute of Mathematical Sciences

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

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American Institute of Mathematical Sciences

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

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