Numerics of acoustical 2D tomography based on the conservation laws

References

[1] L. Beilina and M. V. Klibanov, A posteriori error estimates for the adaptivity technique for the Tikhonov functional and global convergence for a coefficient inverse problem, Inverse Problems 26 (2010), no. 4, Article ID 045012. 10.1088/0266-5611/26/4/045012Search in Google Scholar

[2] L. Beilina and M. V. Klibanov, Synthesis of global convergence and adaptivity for a hyperbolic coefficient inverse problem in 3D, J. Inverse Ill-Posed Probl. 18 (2010), no. 1, 85–132. 10.1515/jiip.2010.003Search in Google Scholar

[3] L. Beilina and M. A. Shishlenin, Computational Comparison of Adaptive Hybrid FEM/FDM Method and Gelfand–Levitan–Krein Method for an Inverse Scattering Problem, preprints of Mathematics Department of Basel University (2006). Search in Google Scholar

[4] R. Bürger, A. Coronel and M. Sepúlveda, Numerical solution of an inverse problem for a scalar conservation law modelling sedimentation, Hyperbolic Problems: Theory, Numerics and Applications, Proc. Sympos. Appl. Math. 67, American Mathematical Society, Providence (2009), 445–454. 10.1090/psapm/067.2/2605240Search in Google Scholar

[5] V. A. Burov, V. B. Voloshinov, K. V. Dmitriev and N. V. Polikarpova, Acoustic waves in metamaterials, crystals, and anomalously refracting structures, Adv. Phys. Sci. 54 (2011), 1165–1170. Search in Google Scholar

[6] V. A. Burov, D. I. Zotov and O. D. Rumyantseva, Reconstruction of spatial distributions of sound velocity and absorption in soft biological tissues using model ultrasonic tomographic data, Acoust. Phys. 60 (2014), 479–91. 10.1134/S1063771014040022Search in Google Scholar

[7] V. A. Burov, D. I. Zotov and O. D. Rumyantseva, Reconstruction of the sound velocity and absorption spatial distributions in soft biological tissue phantoms from experimental ultrasound tomography data, Acoust. Phys. 61 (2015), 231–48. 10.1134/S1063771015020013Search in Google Scholar

[8] N. Duric, P. Littrup, C. Li, O. Roy, S. Schmidt, R. Janer, X. Cheng, J. Goll, O. Rama, L. Bey-Knight and W. Greenway, Breast ultrasound tomography: Bridging the gap to clinical practice, Proc. SPIE 8320 (2012), Article ID 83200. 10.1117/12.910988Search in Google Scholar

[9] N. Duric, P. Littrup, L. Poulo, A. Babkin, R. Pevzner, E. Holsapple, O. Rama and C. Glide, Detection of breast cancer with ultrasound tomography: first results with the computed ultrasound risk evaluation (CURE) prototype, Med. Phys. 34 (2007), no. 2, 773–785. 10.1118/1.2432161Search in Google Scholar PubMed

[10] V. M. Filatova, V. V. Nosikova and L. N. Pestov, Application of reverse time migration (RTM) procedure in ultrasound tomography, numerical modeling, Eurasian J. Math. Comp. Appl. 4 (2016), no. 4, 5–13. 10.32523/2306-6172-2016-4-4-5-13Search in Google Scholar

[11] V. M. Filatova, V. V. Nosikova, L. N. Pestov and A. G. Rudnitskii, Breast ultrasound tomography problem, simulation with noisy model, 2018 Days on Diffraction (DD), IEEE Press, Piscataway (2018), 106–111. 10.1109/DD.2018.8553531Search in Google Scholar

[12] S. K. Godunov, A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics (in Russian), Mat. Sb. (N.S.) 47(89) (1959), 271–306. Search in Google Scholar

[13] S. K. Godunov, S. P. Kiselev, I. M. Kulikov and V. I. Mali, Numerical and experimental simulation of wave formation during explosion welding (in Russian), Tr. Mat. Inst. Steklova 281 (2013), no. 1, 16–31; translation in Proc. Steklov Inst. Math. 281 (2013), 12–26. 10.1134/S0081543813040032Search in Google Scholar

[14] S. K. Godunov, A. V. Zabrodin, M. Y. Ivanov, A. N. Kraĭko and G. P. Prokopov, Numerical Solution for Multidimensional Problems of Gas Mechanics, Izdat. “Nauka”, Moscow, 1976. Search in Google Scholar

[15] A. V. Goncharsky and S. Y. Romanov, Iterative methods for solving coefficient inverse problems of wave tomography in models with attenuation, Inverse Problems 33 (2017), no. 2, Article ID 025003. 10.1088/1361-6420/33/2/025003Search in Google Scholar

[16] A. V. Goncharsky, S. Y. Romanov and S. Y. Seryozhnikov, Inverse problems of 3D ultrasonic tomography with complete and incomplete range data, Wave Motion 51 (2014), no. 3, 389–404. 10.1016/j.wavemoti.2013.10.001Search in Google Scholar

[17] A. V. Goncharsky, S. Y. Romanov and S. Y. Seryozhnikov, A computer simulation study of soft tissue characterization using low-frequency ultrasonic tomography, Ultrasonics 67 (2016), 136–150. 10.1016/j.ultras.2016.01.008Search in Google Scholar PubMed

[18] S. He and S. I. Kabanikhin, An optimization approach to a three-dimensional acoustic inverse problem in the time domain, J. Math. Phys. 36 (1995), no. 8, 4028–4043. 10.1063/1.530945Search in Google Scholar

[19] H. Holden, F. S. Priuli and N. H. Risebro, On an inverse problem for scalar conservation laws, Inverse Problems 30 (2014), no. 3, Article ID 035015. 10.1088/0266-5611/30/3/035015Search in Google Scholar

[20] R. Jirik, I. Peterlik, N. Ruiter, J. Fousek, R. Dapp, M. Zapf and J. Jan, Sound-speed image reconstruction in sparse-aperture 3D ultrasound transmission tomography, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 59 (2012), no. 2, 254–264. 10.1109/TUFFC.2012.2185Search in Google Scholar PubMed

[21] S. I. Kabanikhin, Definitions and examples of inverse and ill-posed problems, J. Inverse Ill-Posed Probl. 16 (2008), no. 4, 317–357. 10.1515/JIIP.2008.019Search in Google Scholar

[22] S. I. Kabanikhin, K. T. Iskakov, B. B. Sholpanbaev, M. A. Shishlenin and D. K. Tokseit, Development of a mathematical model for signal processing using laboratory data, Bull. Karaganda Univ. Math. 92 (2018), no. 4, 148–157. 10.31489/2018M4/148-157Search in Google Scholar

[23] S. I. Kabanikhin, N. S. Novikov, I. V. Oseledets and M. A. Shishlenin, Fast Toeplitz linear system inversion for solving two-dimensional acoustic inverse problem, J. Inverse Ill-Posed Probl. 23 (2015), no. 6, 687–700. 10.1515/jiip-2015-0083Search in Google Scholar

[24] S. I. Kabanikhin, D. B. Nurseitov, M. A. Shishlenin and B. B. Sholpanbaev, Inverse problems for the ground penetrating radar, J. Inverse Ill-Posed Probl. 21 (2013), no. 6, 885–892. 10.1515/jip-2013-0057Search in Google Scholar

[25] S. I. Kabanikhin, K. K. Sabelfeld, N. S. Novikov and M. A. Shishlenin, Numerical solution of an inverse problem of coefficient recovering for a wave equation by a stochastic projection methods, Monte Carlo Methods Appl. 21 (2015), no. 3, 189–203. 10.1515/mcma-2015-0103Search in Google Scholar

[26] S. I. Kabanikhin, K. K. Sabelfeld, N. S. Novikov and M. A. Shishlenin, Numerical solution of the multidimensional Gelfand–Levitan equation, J. Inverse Ill-Posed Probl. 23 (2015), no. 5, 439–450. 10.1515/jiip-2014-0018Search in Google Scholar

[27] S. I. Kabanikhin, O. Scherzer and M. A. Shishlenin, Iteration methods for solving a two dimensional inverse problem for a hyperbolic equation, J. Inverse Ill-Posed Probl. 11 (2003), no. 1, 87–109. 10.1515/156939403322004955Search in Google Scholar

[28] S. I. Kabanikhin and M. A. Shishlenin, Quasi-solution in inverse coefficient problems, J. Inverse Ill-Posed Probl. 16 (2008), no. 7, 705–713. 10.1515/JIIP.2008.043Search in Google Scholar

[29] S. I. Kabanikhin and M. A. Shishlenin, Numerical algorithm for two-dimensional inverse acoustic problem based on Gel’fand–Levitan–Krein equation, J. Inverse Ill-Posed Probl. 18 (2010), no. 9, 979–995. 10.1515/jiip.2011.016Search in Google Scholar

[30] H. Kang and K. Tanuma, Inverse problems for scalar conservation laws, Inverse Problems 21 (2005), no. 3, 1047–1059. 10.1088/0266-5611/21/3/015Search in Google Scholar

[31] I. M. Kulikov, N. S. Novikov and M. A. Shishlenin, Mathematical modelling of ultrasound wave propagation in two dimensional medium: Direct and inverse problem, Siberian Elect. Math. Rep. 12 (2015), C219–C228. Search in Google Scholar

[32] V. G. Romanov and S. I. Kabanikhin, Inverse Problems for Maxwell’s Equations, VSP, Utrecht, 1994. 10.1515/9783110900101Search in Google Scholar

[33] S. S. Titarenko, I. M. Kulikov, I. G. Chernykh, M. A. Shishlenin, O. I. Krivorotko, D. A. Voronov and M. Hildyard, Multilevel parallelization: Grid methods for solving direct and inverse problems, Commun. Comput. Inform. Sci. 687 (2016), 118–131. 10.1007/978-3-319-55669-7_10Search in Google Scholar

[34] V. Vasin, Irregular nonlinear operator equations: Tikhonov’s regularization and iterative approximation, J. Inverse Ill-Posed Probl. 21 (2013), no. 1, 109–123. 10.1515/jip-2012-0084Search in Google Scholar

[35] V. V. Vasin, The method of quasi-solutions by Ivanov is the effective method of solving ill-posed problems, J. Inverse Ill-Posed Probl. 16 (2008), no. 6, 537–552. 10.1515/JIIP.2008.028Search in Google Scholar

[36] J. Wiskin, D. Borup, M. Andre, S. Johnson, J. Greenleaf, Y. Parisky and J. Klock, Threedimensional nonlinear inverse scattering: Quantitative transmission algorithms, refraction corrected reflection, scanner design, and clinical results, Proc. Mtgs. Acoust. 19 (2013), Article ID 075001. 10.1121/1.4800267Search in Google Scholar

[37] J. Wiskin, D. Borup, E. Iuanow, J. Klock and M. Lenox, 3-D nonlinear acoustic inverse scattering: Algorithm and quantitative results, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 64 (2017), no. 8, 1161–1174. 10.1109/TUFFC.2017.2706189Search in Google Scholar PubMed PubMed Central