
-
Previous Article
Causal holography in application to the inverse scattering problems
- IPI Home
- This Issue
-
Next Article
Inverse obstacle scattering for elastic waves in three dimensions
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 |
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.
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. |
[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. |
[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. 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. |
[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. 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. |
[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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. |
[54] |
J. Sylvester and G. Uhlmann,
A global uniqueness theorem for an inverse boundary value problem, Ann. of Math. (2), 125 (1987), 153-169.
doi: 10.2307/1971291. |
[55] |
D. Tataru,
Unique continuation for solutions to PDE's; between Hörmander's theorem and Holmgren's theorem, Comm. Partial Differential Equations, 20 (1995), 855-884.
doi: 10.1080/03605309508821117. |
[56] |
B. E. Treeby and B. T. Cox, k-Wave: Matlab toolbox for the simulation and reconstruction of photoacoustic wave fields, Journal of Biomedical Optics., 15 (2010), 021314.
doi: 10.1117/1.3360308. |
[57] |
F. Zouari, E. Bl sten and M. S. Ghidaoui, Area reconstruction as a tool for blockage detection, In preparation. |
show all references
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. |
[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. |
[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. 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. |
[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. 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. |
[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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. |
[54] |
J. Sylvester and G. Uhlmann,
A global uniqueness theorem for an inverse boundary value problem, Ann. of Math. (2), 125 (1987), 153-169.
doi: 10.2307/1971291. |
[55] |
D. Tataru,
Unique continuation for solutions to PDE's; between Hörmander's theorem and Holmgren's theorem, Comm. Partial Differential Equations, 20 (1995), 855-884.
doi: 10.1080/03605309508821117. |
[56] |
B. E. Treeby and B. T. Cox, k-Wave: Matlab toolbox for the simulation and reconstruction of photoacoustic wave fields, Journal of Biomedical Optics., 15 (2010), 021314.
doi: 10.1117/1.3360308. |
[57] |
F. Zouari, E. Bl sten and M. S. Ghidaoui, Area reconstruction as a tool for blockage detection, In preparation. |






[1] |
Gen Nakamura, Michiyuki Watanabe. An inverse boundary value problem for a nonlinear wave equation. Inverse Problems and Imaging, 2008, 2 (1) : 121-131. doi: 10.3934/ipi.2008.2.121 |
[2] |
Xiaoli Feng, Meixia Zhao, Peijun Li, Xu Wang. An inverse source problem for the stochastic wave equation. Inverse Problems and Imaging, 2022, 16 (2) : 397-415. doi: 10.3934/ipi.2021055 |
[3] |
Luca Rondi. On the regularization of the inverse conductivity problem with discontinuous conductivities. Inverse Problems and Imaging, 2008, 2 (3) : 397-409. doi: 10.3934/ipi.2008.2.397 |
[4] |
Laurent Bourgeois, Dmitry Ponomarev, Jérémi Dardé. An inverse obstacle problem for the wave equation in a finite time domain. Inverse Problems and Imaging, 2019, 13 (2) : 377-400. doi: 10.3934/ipi.2019019 |
[5] |
Lucie Baudouin, Emmanuelle Crépeau, Julie Valein. Global Carleman estimate on a network for the wave equation and application to an inverse problem. Mathematical Control and Related Fields, 2011, 1 (3) : 307-330. doi: 10.3934/mcrf.2011.1.307 |
[6] |
Tony Liimatainen, Lauri Oksanen. Counterexamples to inverse problems for the wave equation. Inverse Problems and Imaging, 2022, 16 (2) : 467-479. doi: 10.3934/ipi.2021058 |
[7] |
Lauri Oksanen. Solving an inverse problem for the wave equation by using a minimization algorithm and time-reversed measurements. Inverse Problems and Imaging, 2011, 5 (3) : 731-744. doi: 10.3934/ipi.2011.5.731 |
[8] |
Bin Fan, Mejdi Azaïez, Chuanju Xu. An extension of the landweber regularization for a backward time fractional wave problem. Discrete and Continuous Dynamical Systems - S, 2021, 14 (8) : 2893-2916. doi: 10.3934/dcdss.2020409 |
[9] |
Max Gunzburger, Sung-Dae Yang, Wenxiang Zhu. Analysis and discretization of an optimal control problem for the forced Fisher equation. Discrete and Continuous Dynamical Systems - B, 2007, 8 (3) : 569-587. doi: 10.3934/dcdsb.2007.8.569 |
[10] |
Nguyen Huy Tuan, Mokhtar Kirane, Long Dinh Le, Van Thinh Nguyen. On an inverse problem for fractional evolution equation. Evolution Equations and Control Theory, 2017, 6 (1) : 111-134. doi: 10.3934/eect.2017007 |
[11] |
Anna Doubova, Enrique Fernández-Cara. Some geometric inverse problems for the linear wave equation. Inverse Problems and Imaging, 2015, 9 (2) : 371-393. doi: 10.3934/ipi.2015.9.371 |
[12] |
Jianli Xiang, Guozheng Yan. The uniqueness of the inverse elastic wave scattering problem based on the mixed reciprocity relation. Inverse Problems and Imaging, 2021, 15 (3) : 539-554. doi: 10.3934/ipi.2021004 |
[13] |
Vinicius Albani, Adriano De Cezaro, Jorge P. Zubelli. On the choice of the Tikhonov regularization parameter and the discretization level: A discrepancy-based strategy. Inverse Problems and Imaging, 2016, 10 (1) : 1-25. doi: 10.3934/ipi.2016.10.1 |
[14] |
Fernando Jiménez, Jürgen Scheurle. On some aspects of the discretization of the suslov problem. Journal of Geometric Mechanics, 2018, 10 (1) : 43-68. doi: 10.3934/jgm.2018002 |
[15] |
Kimitoshi Tsutaya. Scattering theory for the wave equation of a Hartree type in three space dimensions. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 2261-2281. doi: 10.3934/dcds.2014.34.2261 |
[16] |
Fabrice Planchon, John G. Stalker, A. Shadi Tahvildar-Zadeh. $L^p$ Estimates for the wave equation with the inverse-square potential. Discrete and Continuous Dynamical Systems, 2003, 9 (2) : 427-442. doi: 10.3934/dcds.2003.9.427 |
[17] |
Fabrice Planchon, John G. Stalker, A. Shadi Tahvildar-Zadeh. Dispersive estimate for the wave equation with the inverse-square potential. Discrete and Continuous Dynamical Systems, 2003, 9 (6) : 1387-1400. doi: 10.3934/dcds.2003.9.1387 |
[18] |
Gabriel Peyré, Sébastien Bougleux, Laurent Cohen. Non-local regularization of inverse problems. Inverse Problems and Imaging, 2011, 5 (2) : 511-530. doi: 10.3934/ipi.2011.5.511 |
[19] |
Philipp Hungerländer, Barbara Kaltenbacher, Franz Rendl. Regularization of inverse problems via box constrained minimization. Inverse Problems and Imaging, 2020, 14 (3) : 437-461. doi: 10.3934/ipi.2020021 |
[20] |
Michael V. Klibanov. A phaseless inverse scattering problem for the 3-D Helmholtz equation. Inverse Problems and Imaging, 2017, 11 (2) : 263-276. doi: 10.3934/ipi.2017013 |
2020 Impact Factor: 1.639
Tools
Metrics
Other articles
by authors
[Back to Top]