Fixed angle inverse scattering for sound speeds close to constant. (English) Zbl 1526.35320
Summary: We study the fixed angle inverse scattering problem of determining a sound speed from scattering measurements corresponding to a single incident wave. The main result shows that a sound speed close to constant can be stably determined by just one measurement. Our method is based on studying the linearized problem, which turns out to be related to the acoustic problem in photoacoustic imaging. We adapt the modified time-reversal method from P. Stefanov and G. Uhlmann [Inverse Probl. 25, No. 7, Article ID 075011, 16 p. (2009; Zbl 1177.35256)] to solve the linearized problem in a stable way, and we use this to give a local uniqueness result for the nonlinear inverse problem.
MSC:
| 35R30 | Inverse problems for PDEs |
| 35Q60 | PDEs in connection with optics and electromagnetic theory |
| 35J10 | Schrödinger operator, Schrödinger equation |
| 35L05 | Wave equation |
| 31B10 | Integral representations, integral operators, integral equations methods in higher dimensions |
| 78A40 | Waves and radiation in optics and electromagnetic theory |
Citations:
Zbl 1177.35256References:
| [1] | Acosta, S., A control approach to recover the wave speed (conformal factor) from one measurement, Inverse Probl. Imaging, 9 (2015), pp. 301-315, doi:10.3934/ipi.2015.9.301. · Zbl 1334.35414 |
| [2] | Acosta, S. and Montalto, C., Multiwave imaging in an enclosure with variable wave speed, Inverse Problems, 31 (2015), 12, doi:10.1088/0266-5611/31/6/065009. · Zbl 1320.65135 |
| [3] | Agranovsky, M., Kuchment, P., and Kunyansky, L., On reconstruction formulas and algorithms for the thermoacoustic tomography, in Photoacoustic Imaging and Spectroscopy, CRC Press, 2017, pp. 89-102. |
| [4] | Bao, G. and Symes, W. W., A trace theorem for solutions of linear partial differential equations, Math. Methods Appl. Sci., 14 (1991), pp. 553-562, doi:10.1002/mma.1670140803. · Zbl 0754.35023 |
| [5] | Barceló, J. A., Castro, C., Luque, T., Meroño, C. J., Ruiz, A., and de la Cruz Vilela, M., Uniqueness for the inverse fixed angle scattering problem, J. Inverse Ill-Posed Probl., 28 (2020), pp. 465-470, doi:10.1515/jiip-2019-0019. · Zbl 1446.35088 |
| [6] | Bayliss, A., Li, Y. Y., and Morawetz, C. S., Scattering by a potential using hyperbolic methods, Math. Comp., 52 (1989), pp. 321-338, doi:10.2307/2008470. · Zbl 0692.65068 |
| [7] | Bergh, J. and Löfström, J., Interpolation Spaces: An Introduction, , Springer Science & Business Media, 2012. · Zbl 0344.46071 |
| [8] | Burman, E., Feizmohammadi, A., and Oksanen, L., A finite element data assimilation method for the wave equation, Math. Comp., 89 (2020), pp. 1681-1709, doi:10.1090/mcom/3508. · Zbl 1436.65132 |
| [9] | Evans, L. C., Partial Differential Equations, 2nd ed., , American Mathematical Society, Providence, RI, 2010, doi:10.1090/gsm/019. · Zbl 1194.35001 |
| [10] | Finch, D. and Rakesh, Trace identities for solutions of the wave equation with initial data supported in a ball, Math. Methods Appl. Sci., 28 (2005), pp. 1897-1917, doi:10.1002/mma.647. · Zbl 1085.35092 |
| [11] | Hörmander, L., Linear Partial Differential Operators, 4th printing, Springer, 1976. · Zbl 0321.35001 |
| [12] | Hörmander, L., The Analysis of Linear Partial Differential Operators, Vols. I-IV, Springer-Verlag, Berlin, 1983-1985. · Zbl 0521.35001 |
| [13] | Hristova, Y., Time reversal in thermoacoustic tomography—an error estimate, Inverse Problems, 25 (2009), 14, doi:10.1088/0266-5611/25/5/055008. · Zbl 1167.35051 |
| [14] | Hristova, Y., Kuchment, P., and Nguyen, L., Reconstruction and time reversal in thermoacoustic tomography in acoustically homogeneous and inhomogeneous media, Inverse Problems, 24 (2008), 25, doi:10.1088/0266-5611/24/5/055006. · Zbl 1180.35563 |
| [15] | Ikehata, R., Local energy decay for linear wave equations with variable coefficients, J. Math. Anal. Appl., 306 (2005), pp. 330-348, doi:10.1016/j.jmaa.2004.12.056. · Zbl 1072.35112 |
| [16] | Kato, T., Quasi-linear equations of evolution, with applications to partial differential equations, in Spectral Theory and Differential Equations (Proc. Sympos., Dundee, 1974; dedicated to Konrad Jörgens), , Springer, 1974, pp. 25-70. · Zbl 0315.35077 |
| [17] | Kato, T. and Ponce, G., Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), pp. 891-907, doi:10.1002/cpa.3160410704. · Zbl 0671.35066 |
| [18] | Klibanov, M. V. and Malinsky, J., Newton-Kantorovich method for three-dimensional potential inverse scattering problem and stability of the hyperbolic Cauchy problem with time-dependent data, Inverse Problems, 7 (1991), pp. 577-596, http://stacks.iop.org/0266-5611/7/577. · Zbl 0744.35065 |
| [19] | Krishnan, V. P., Rakesh, R., and Senapati, S., Stability for a formally determined inverse problem for a hyperbolic PDE with space and time dependent coefficients, SIAM J. Math. Anal., 53 (2021), pp. 6822-6846, doi:10.1137/21M1400596. · Zbl 1482.35275 |
| [20] | Kuchment, P. and Kunyansky, L., Mathematics of thermoacoustic tomography, European J. Appl. Math., 19 (2008), pp. 191-224, doi:10.1017/S0956792508007353. · Zbl 1185.35327 |
| [21] | Kuchment, P. and Kunyansky, L., Mathematics of photoacoustic and thermoacoustic tomography, in Handbook of Mathematical Methods in Imaging, Vols. 1, 2, 3, Springer, New York, 2015, pp. 1117-1167. · Zbl 1331.92076 |
| [22] | Lasiecka, I., Lions, J.-L., and Triggiani, R., Nonhomogeneous boundary value problems for second order hyperbolic operators, J. Math. Pures Appl., 65 (1986), pp. 149-192. · Zbl 0631.35051 |
| [23] | Ma, S. and Salo, M., Fixed angle inverse scattering in the presence of a Riemannian metric, J. Inverse Ill-Posed Probl., 30 (2022), pp. 495-520, doi:10.1515/jiip-2020-0119. · Zbl 1493.35114 |
| [24] | Meroño, C. J., Potenciano-Machado, L., and Salo, M., The fixed angle scattering problem with a first-order perturbation, Ann. Henri Poincaré, 22 (2021), pp. 3699-3746, doi:10.1007/s00023-021-01081-w. · Zbl 1485.35425 |
| [25] | Meroño, C. J., Recovery of Singularities in Inverse Scattering, Ph.D. thesis, Universidad Autónoma de Madrid, Madrid, 2018. · Zbl 1401.35347 |
| [26] | Rakesh and Salo, M., Fixed angle inverse scattering for almost symmetric or controlled perturbations, SIAM J. Math. Anal., 52 (2020), pp. 5467-5499, doi:10.1137/20M1319309. · Zbl 1453.35193 |
| [27] | Rakesh and Salo, M., The fixed angle scattering problem and wave equation inverse problems with two measurements, Inverse Problems, 36 (2020), 42, doi:10.1088/1361-6420/ab23a2. · Zbl 1439.35569 |
| [28] | Rakesh and Uhlmann, G., Uniqueness for the inverse backscattering problem for angularly controlled potentials, Inverse Problems, 30 (2014), 24, doi:10.1088/0266-5611/30/6/065005. · Zbl 1294.33016 |
| [29] | Ruiz, A., Recovery of the singularities of a potential from fixed angle scattering data, Comm. Partial Differential Equations, 26 (2001), pp. 1721-1738, doi:10.1081/PDE-100107457. · Zbl 1134.35386 |
| [30] | Shiota, T., An inverse problem for the wave equation with first order perturbation, Amer. J. Math., 107 (1985), pp. 241-251, doi:10.2307/2374463. · Zbl 0588.47012 |
| [31] | Smith, H. F., A parametrix construction for wave equations with \(C^{1,1}\) coefficients, Ann. Inst. Fourier (Grenoble), 48 (1998), pp. 797-835, http://www.numdam.org/item?id=AIF_199848_3_797_0. · Zbl 0974.35068 |
| [32] | Stefanov, P., Generic uniqueness for two inverse problems in potential scattering, Comm. Partial Differential Equations, 17 (1992), pp. 55-68, doi:10.1080/03605309208820834. · Zbl 0771.35082 |
| [33] | Stefanov, P. and Uhlmann, G., Linearizing non-linear inverse problems and an application to inverse backscattering, J. Funct. Anal., 256 (2009), pp. 2842-2866, doi:10.1016/j.jfa.2008.10.017. · Zbl 1169.65049 |
| [34] | Stefanov, P. and Uhlmann, G., Thermoacoustic tomography with variable sound speed, Inverse Problems, 25 (2009), 075011, doi:10.1088/0266-5611/25/7/075011. · Zbl 1177.35256 |
| [35] | Stefanov, P. and Uhlmann, G., Multiwave methods via ultrasound, in Inverse Problems and Applications: Inside Out. II, , Cambridge University Press, Cambridge, 2013, pp. 271-323. · Zbl 1316.35302 |
| [36] | Stein, E. M., Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, , Princeton University Press, Princeton, NJ, 1993. · Zbl 0821.42001 |
| [37] | Tamura, H., On the decay of local energy for wave equations with time-dependent potentials, J. Math. Soc. Japan, 33 (1981), pp. 605-618, doi:10.2969/jmsj/03340605. · Zbl 0463.35046 |
| [38] | Taylor, M. E., Partial Differential Equations I. Basic Theory, , 2nd ed., Springer, New York, 2011, doi:10.1007/978-1-4419-7055-8. · Zbl 1206.35002 |
| [39] | Taylor, M. E., Partial Differential Equations III. Nonlinear Equations, , 2nd ed., Springer, New York, 2011, doi:10.1007/978-1-4419-7049-7. · Zbl 1206.35004 |
| [40] | Vodev, G., Local energy decay of solutions to the wave equation for nontrapping metrics, Ark. Mat., 42 (2004), pp. 379-397, doi:10.1007/BF02385487. · Zbl 1061.58024 |
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