-
Previous Article
Addendum to: "Variational source conditions for inverse Robin and flux problems by partial measurements"
- IPI Home
- This Issue
-
Next Article
Weighted area constraints-based breast lesion segmentation in ultrasound image analysis
Counterexamples to inverse problems for the wave equation
1. | Department of Mathematics and Statistics, University of Jyväskylä, Jyväskylä, Finland |
2. | Department of Mathematics and Statistics, University of Helsinki, Helsinki, Finland |
We construct counterexamples to inverse problems for the wave operator on domains in $ \mathbb{R}^{n+1} $, $ n \ge 2 $, and on Lorentzian manifolds. We show that non-isometric Lorentzian metrics can lead to same partial data measurements, which are formulated in terms certain restrictions of the Dirichlet-to-Neumann map. The Lorentzian metrics giving counterexamples are time-dependent, but they are smooth and non-degenerate. On $ \mathbb{R}^{n+1} $ the metrics are conformal to the Minkowski metric.
References:
[1] |
S. Alexakis, A. Feizmohammadi and L. Oksanen, Lorentzian Calderón problem under curvature bounds, Preprint arXiv: 2008.07508, 2020. |
[2] |
A. L. Besse, Einstein Manifolds, Classics in Mathematics. Springer-Verlag, Berlin, 2008. |
[3] |
S. N. Curry and A. R. Gover, An introduction to conformal geometry and tractor calculus,
with a view to applications in general relativity, volume 443 of London Math. Soc. Lecture
Note Ser., Cambridge Univ. Press, Cambridge, 2018, 86-170. |
[4] |
T. Daudé, N. Kamran and F. Nicoleau, A survey of non-uniqueness results for the anisotropic
Calder´on problem with disjoint data, In Nonlinear Analysis in Geometry and Applied Mathematics. Part 2, volume 2 of Harv. Univ. Cent. Math. Sci. Appl. Ser. Math., Int. Press,
Somerville, MA, 2018, 77-101. |
[5] |
T. Daudé, N. Kamran and F. Nicoleau,
Non-uniqueness results for the anisotropic Calderón problem with data measured on disjoint sets, Ann. Inst. Fourier (Grenoble), 69 (2019), 119-170.
doi: 10.5802/aif.3240. |
[6] |
T. Daudé, N. Kamran and F. Nicoleau,
On the hidden mechanism behind non-uniqueness for the anisotropic Calderón problem with data on disjoint sets, Ann. Henri Poincaré, 20 (2019), 859-887.
doi: 10.1007/s00023-018-00755-2. |
[7] |
T. Daudé, N. Kamran and F. Nicoleau,
The anisotropic Calderón problem for singular metrics of warped product type: The borderline between uniqueness and invisibility, J. Spectr. Theory, 10 (2020), 703-746.
doi: 10.4171/JST/310. |
[8] |
T. Daudé, N. Kamran and F. Nicoleau, On nonuniqueness for the anisotropic Calderón problem with partial data, Forum Math. Sigma, 8 (2020), Paper No. e7, 17 pp.
doi: 10.1017/fms.2020.1. |
[9] |
G. Eskin, Inverse hyperbolic problems with time-dependent coefficients, Comm. Partial Differential Equations, 32 (2007), 1737–1758.
doi: 10.1080/03605300701382340. |
[10] |
A. Greenleaf, M. Lassas and G. Uhlmann,
On nonuniqueness for Calderón's inverse problem, Math. Res. Lett., 10 (2003), 685-693.
doi: 10.4310/MRL.2003.v10.n5.a11. |
[11] |
L. Hörmander, The Analysis of Linear Partial Differential Operators. III, Classics in Mathematics. Springer, Berlin, 2007.
doi: 10.1007/978-3-540-49938-1. |
[12] |
A. Katchalov, Y. Kurylev and M. Lassas, Inverse Boundary Spectral Problems, volume 123 of Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2001.
doi: 10.1201/9781420036220. |
[13] |
C. Kenig and M. Salo,
The Calderón problem with partial data on manifolds and applications, Analysis and PDE, 6 (2013), 2003-2048.
doi: 10.2140/apde.2013.6.2003. |
[14] |
Y. Kian and L. Oksanen, Recovery of time-dependent coefficient on {R}iemannian manifold for hyperbolic equations, Int. Math. Res. Not. IMRN, (2019), 5087–5126.
doi: 10.1093/imrn/rnx263. |
[15] |
Y. Kian, Y. Kurylev, M. Lassas and L. Oksanen,
Unique recovery of lower order coefficients for hyperbolic equations from data on disjoint sets, J. Differential Equations, 267 (2019), 2210-2238.
doi: 10.1016/j.jde.2019.03.008. |
[16] |
M. Lassas and T. Liimatainen, Conformal harmonic coordinates, To appear in Communications in Analysis and Geometry, Preprint arXiv: 1912.08030, 2019. |
[17] |
M. Lassas, T. Liimatainen and M. Salo, The Calderón problem for the conformal Laplacian, To appear in Communications in Analysis and Geometry, Preprint arXiv: 1612.07939, 2016. |
[18] |
M. Lassas and L. Oksanen, An inverse problem for a wave equation with sources and observations on disjoint sets, Inverse Problems, 26 (2010), 085012, 19 pp.
doi: 10.1088/0266-5611/26/8/085012. |
[19] |
M. Lassas and L. Oksanen,
Inverse problem for the Riemannian wave equation with Dirichlet data and {N}eumann data on disjoint sets, Duke Mathematical Journal, 163 (2014), 1071-1103.
doi: 10.1215/00127094-2649534. |
[20] |
M. Lassas, M. Taylor and G. Uhlmann,
The Dirichlet-to-Neumann map for complete Riemannian manifolds with boundary, Comm. Geom. Anal., 11 (2003), 207-222.
doi: 10.4310/CAG.2003.v11.n2.a2. |
[21] |
J. M. Lee and T. H. Parker,
The Yamabe problem, Bull. Amer. Math. Soc. (N.S.), 17 (1987), 37-91.
doi: 10.1090/S0273-0979-1987-15514-5. |
[22] |
W. R. B. Lionheart,
Conformal uniqueness results in anisotropic electrical impedance imaging, Inverse Problems, 13 (1997), 125-134.
doi: 10.1088/0266-5611/13/1/010. |
[23] |
J. B. Pendry, D. Schurig and D. R. Smith,
Controlling electromagnetic fields, Science, 312 (2006), 1780-1782.
doi: 10.1126/science.1125907. |
[24] |
Rakesh, Characterization of transmission data for Webster's horn equation, Inverse Problems, 16 (2000), L9–L24.
doi: 10.1088/0266-5611/16/2/102. |
[25] |
D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr and D. R. Smith,
Metamaterial electromagnetic cloak at microwave frequencies, Science, 314 (2006), 977-980.
doi: 10.1126/science.1133628. |
[26] |
G. Uhlmann,
Inverse problems: Seeing the unseen, Bull. Math. Sci., 4 (2014), 209-279.
doi: 10.1007/s13373-014-0051-9. |
show all references
References:
[1] |
S. Alexakis, A. Feizmohammadi and L. Oksanen, Lorentzian Calderón problem under curvature bounds, Preprint arXiv: 2008.07508, 2020. |
[2] |
A. L. Besse, Einstein Manifolds, Classics in Mathematics. Springer-Verlag, Berlin, 2008. |
[3] |
S. N. Curry and A. R. Gover, An introduction to conformal geometry and tractor calculus,
with a view to applications in general relativity, volume 443 of London Math. Soc. Lecture
Note Ser., Cambridge Univ. Press, Cambridge, 2018, 86-170. |
[4] |
T. Daudé, N. Kamran and F. Nicoleau, A survey of non-uniqueness results for the anisotropic
Calder´on problem with disjoint data, In Nonlinear Analysis in Geometry and Applied Mathematics. Part 2, volume 2 of Harv. Univ. Cent. Math. Sci. Appl. Ser. Math., Int. Press,
Somerville, MA, 2018, 77-101. |
[5] |
T. Daudé, N. Kamran and F. Nicoleau,
Non-uniqueness results for the anisotropic Calderón problem with data measured on disjoint sets, Ann. Inst. Fourier (Grenoble), 69 (2019), 119-170.
doi: 10.5802/aif.3240. |
[6] |
T. Daudé, N. Kamran and F. Nicoleau,
On the hidden mechanism behind non-uniqueness for the anisotropic Calderón problem with data on disjoint sets, Ann. Henri Poincaré, 20 (2019), 859-887.
doi: 10.1007/s00023-018-00755-2. |
[7] |
T. Daudé, N. Kamran and F. Nicoleau,
The anisotropic Calderón problem for singular metrics of warped product type: The borderline between uniqueness and invisibility, J. Spectr. Theory, 10 (2020), 703-746.
doi: 10.4171/JST/310. |
[8] |
T. Daudé, N. Kamran and F. Nicoleau, On nonuniqueness for the anisotropic Calderón problem with partial data, Forum Math. Sigma, 8 (2020), Paper No. e7, 17 pp.
doi: 10.1017/fms.2020.1. |
[9] |
G. Eskin, Inverse hyperbolic problems with time-dependent coefficients, Comm. Partial Differential Equations, 32 (2007), 1737–1758.
doi: 10.1080/03605300701382340. |
[10] |
A. Greenleaf, M. Lassas and G. Uhlmann,
On nonuniqueness for Calderón's inverse problem, Math. Res. Lett., 10 (2003), 685-693.
doi: 10.4310/MRL.2003.v10.n5.a11. |
[11] |
L. Hörmander, The Analysis of Linear Partial Differential Operators. III, Classics in Mathematics. Springer, Berlin, 2007.
doi: 10.1007/978-3-540-49938-1. |
[12] |
A. Katchalov, Y. Kurylev and M. Lassas, Inverse Boundary Spectral Problems, volume 123 of Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2001.
doi: 10.1201/9781420036220. |
[13] |
C. Kenig and M. Salo,
The Calderón problem with partial data on manifolds and applications, Analysis and PDE, 6 (2013), 2003-2048.
doi: 10.2140/apde.2013.6.2003. |
[14] |
Y. Kian and L. Oksanen, Recovery of time-dependent coefficient on {R}iemannian manifold for hyperbolic equations, Int. Math. Res. Not. IMRN, (2019), 5087–5126.
doi: 10.1093/imrn/rnx263. |
[15] |
Y. Kian, Y. Kurylev, M. Lassas and L. Oksanen,
Unique recovery of lower order coefficients for hyperbolic equations from data on disjoint sets, J. Differential Equations, 267 (2019), 2210-2238.
doi: 10.1016/j.jde.2019.03.008. |
[16] |
M. Lassas and T. Liimatainen, Conformal harmonic coordinates, To appear in Communications in Analysis and Geometry, Preprint arXiv: 1912.08030, 2019. |
[17] |
M. Lassas, T. Liimatainen and M. Salo, The Calderón problem for the conformal Laplacian, To appear in Communications in Analysis and Geometry, Preprint arXiv: 1612.07939, 2016. |
[18] |
M. Lassas and L. Oksanen, An inverse problem for a wave equation with sources and observations on disjoint sets, Inverse Problems, 26 (2010), 085012, 19 pp.
doi: 10.1088/0266-5611/26/8/085012. |
[19] |
M. Lassas and L. Oksanen,
Inverse problem for the Riemannian wave equation with Dirichlet data and {N}eumann data on disjoint sets, Duke Mathematical Journal, 163 (2014), 1071-1103.
doi: 10.1215/00127094-2649534. |
[20] |
M. Lassas, M. Taylor and G. Uhlmann,
The Dirichlet-to-Neumann map for complete Riemannian manifolds with boundary, Comm. Geom. Anal., 11 (2003), 207-222.
doi: 10.4310/CAG.2003.v11.n2.a2. |
[21] |
J. M. Lee and T. H. Parker,
The Yamabe problem, Bull. Amer. Math. Soc. (N.S.), 17 (1987), 37-91.
doi: 10.1090/S0273-0979-1987-15514-5. |
[22] |
W. R. B. Lionheart,
Conformal uniqueness results in anisotropic electrical impedance imaging, Inverse Problems, 13 (1997), 125-134.
doi: 10.1088/0266-5611/13/1/010. |
[23] |
J. B. Pendry, D. Schurig and D. R. Smith,
Controlling electromagnetic fields, Science, 312 (2006), 1780-1782.
doi: 10.1126/science.1125907. |
[24] |
Rakesh, Characterization of transmission data for Webster's horn equation, Inverse Problems, 16 (2000), L9–L24.
doi: 10.1088/0266-5611/16/2/102. |
[25] |
D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr and D. R. Smith,
Metamaterial electromagnetic cloak at microwave frequencies, Science, 314 (2006), 977-980.
doi: 10.1126/science.1133628. |
[26] |
G. Uhlmann,
Inverse problems: Seeing the unseen, Bull. Math. Sci., 4 (2014), 209-279.
doi: 10.1007/s13373-014-0051-9. |
[1] |
Xiaosheng Li, Gunther Uhlmann. Inverse problems with partial data in a slab. Inverse Problems and Imaging, 2010, 4 (3) : 449-462. doi: 10.3934/ipi.2010.4.449 |
[2] |
Fatemeh Ahangari. Conformal deformations of a specific class of lorentzian manifolds with non-irreducible holonomy representation. Numerical Algebra, Control and Optimization, 2019, 9 (4) : 401-412. doi: 10.3934/naco.2019039 |
[3] |
Fioralba Cakoni, Rainer Kress. Integral equations for inverse problems in corrosion detection from partial Cauchy data. Inverse Problems and Imaging, 2007, 1 (2) : 229-245. doi: 10.3934/ipi.2007.1.229 |
[4] |
Sebastian Acosta. A control approach to recover the wave speed (conformal factor) from one measurement. Inverse Problems and Imaging, 2015, 9 (2) : 301-315. doi: 10.3934/ipi.2015.9.301 |
[5] |
Suman Kumar Sahoo, Manmohan Vashisth. A partial data inverse problem for the convection-diffusion equation. Inverse Problems and Imaging, 2020, 14 (1) : 53-75. doi: 10.3934/ipi.2019063 |
[6] |
Li Liang. Increasing stability for the inverse problem of the Schrödinger equation with the partial Cauchy data. Inverse Problems and Imaging, 2015, 9 (2) : 469-478. doi: 10.3934/ipi.2015.9.469 |
[7] |
Soumen Senapati, Manmohan Vashisth. Stability estimate for a partial data inverse problem for the convection-diffusion equation. Evolution Equations and Control Theory, 2021 doi: 10.3934/eect.2021060 |
[8] |
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 |
[9] |
Lok Ming Lui, Chengfeng Wen, Xianfeng Gu. A conformal approach for surface inpainting. Inverse Problems and Imaging, 2013, 7 (3) : 863-884. doi: 10.3934/ipi.2013.7.863 |
[10] |
Frank Natterer. Incomplete data problems in wave equation imaging. Inverse Problems and Imaging, 2010, 4 (4) : 685-691. doi: 10.3934/ipi.2010.4.685 |
[11] |
Toshiyuki Suzuki. Semilinear Schrödinger evolution equations with inverse-square and harmonic potentials via pseudo-conformal symmetry. Communications on Pure and Applied Analysis, 2021, 20 (12) : 4347-4377. doi: 10.3934/cpaa.2021163 |
[12] |
Zuxing Xuan. On conformal measures of parabolic meromorphic functions. Discrete and Continuous Dynamical Systems - B, 2015, 20 (1) : 249-257. doi: 10.3934/dcdsb.2015.20.249 |
[13] |
Peter Haïssinsky, Kevin M. Pilgrim. Examples of coarse expanding conformal maps. Discrete and Continuous Dynamical Systems, 2012, 32 (7) : 2403-2416. doi: 10.3934/dcds.2012.32.2403 |
[14] |
Nicholas Hoell, Guillaume Bal. Ray transforms on a conformal class of curves. Inverse Problems and Imaging, 2014, 8 (1) : 103-125. doi: 10.3934/ipi.2014.8.103 |
[15] |
Tomasz Szarek, Mariusz Urbański, Anna Zdunik. Continuity of Hausdorff measure for conformal dynamical systems. Discrete and Continuous Dynamical Systems, 2013, 33 (10) : 4647-4692. doi: 10.3934/dcds.2013.33.4647 |
[16] |
Hans Henrik Rugh. On dimensions of conformal repellers. Randomness and parameter dependency. Discrete and Continuous Dynamical Systems, 2012, 32 (7) : 2553-2564. doi: 10.3934/dcds.2012.32.2553 |
[17] |
Mario Roy, Mariusz Urbański. Multifractal analysis for conformal graph directed Markov systems. Discrete and Continuous Dynamical Systems, 2009, 25 (2) : 627-650. doi: 10.3934/dcds.2009.25.627 |
[18] |
Marcelo M. Disconzi. On the existence of solutions and causality for relativistic viscous conformal fluids. Communications on Pure and Applied Analysis, 2019, 18 (4) : 1567-1599. doi: 10.3934/cpaa.2019075 |
[19] |
Domenico Mucci. Maps into projective spaces: Liquid crystal and conformal energies. Discrete and Continuous Dynamical Systems - B, 2012, 17 (2) : 597-635. doi: 10.3934/dcdsb.2012.17.597 |
[20] |
Rossen I. Ivanov. Conformal and Geometric Properties of the Camassa-Holm Hierarchy. Discrete and Continuous Dynamical Systems, 2007, 19 (3) : 545-554. doi: 10.3934/dcds.2007.19.545 |
2020 Impact Factor: 1.639
Tools
Metrics
Other articles
by authors
[Back to Top]