Multistage Iterative Method to Tackle Inverse Problems of Wave Tomography

Alexander V. Goncharsky; Sergey Y. Romanov; Sergey Y. Seryozhnikov

doi:10.14529/jsfi220106

Authors

Alexander V. Goncharsky Lomonosov Moscow State University; Moscow Center of Fundamental and Applied Mathematics, Moscow, Russia https://orcid.org/0000-0001-9953-611X
Sergey Y. Romanov Lomonosov Moscow State University; Moscow Center of Fundamental and Applied Mathematics, Moscow, Russia https://orcid.org/0000-0001-7882-2061
Sergey Y. Seryozhnikov Lomonosov Moscow State University; Moscow Center of Fundamental and Applied Mathematics, Moscow, Russia https://orcid.org/0000-0002-4106-2692

DOI:

https://doi.org/10.14529/jsfi220106

Keywords:

ultrasound tomography, coefficient inverse problem, gradient method, numerical simulation

Abstract

This paper is concerned with developing the methods for solving inverse problems of lowfrequency ultrasound tomography under scalar wave models using supercomputer technologies. Unlike X-ray tomography, the inverse problem considered is posed as a problem of minimizing a non-convex residual functional. The multistage iterative method (MSM) is proposed as a method for obtaining an approximate solution to the inverse problem. Convergence of the method to the exact solution is achieved via the use of low-frequency sounding signals at the initial stages of the iterative method. The method is illustrated on model problems focused on ultrasound tomographic diagnostics of soft tissues in medicine. Finite-difference time-domain method is used to solve the wave equation, which accounts for most of the computational complexity of the method. The multistage method reduces the computation time, since the initial stages use low-resolution finite difference grids. The effectiveness of the MSM method is investigated on GPU and SIMD-capable CPU computing platforms. Numerical simulations showed that modern processors equipped with AVX-512 FPUs are capable of solving small-scale problems of wave tomography. For large-scale tasks, GPUs equipped with fast on-board memory are preferred. The numerical algorithm is data-parallel and well-suited for GPU architecture. The proposed method can be used in medical imaging and nondestructive testing applications.

References

Radon, J.: Uber die Bestimmung von Funktionen durch ihre Integralwerte langs gewisser Mannigfaltigkeiten. Berichte uber die Verhandlungen der Koniglich-Sachsischen Akademie der Wissenschaften zu Leipzig, Mathematisch-Physische Klasse [Reports on the Proceedings of the Royal Saxonian Academy of Sciences at Leipzig, Mathematical and Physical Section], Leipzig: Teubner 69, 262–277 (1917)

Tikhonov, A.N.: Solution of incorrectly formulated problems and the regularization method. Soviet. Math. Dokl. 4, 1035–1038 (1963)

Tikhonov, A.N.: Regularization of incorrectly posed problems. Soviet Math. Dokl. 4, 1624–1627 (1963)

Tikhonov, A.N., Goncharsky, A.V., Stepanov, V.V., Yagola, A.G.: Numerical Methods for the Solution of Ill-Posed Problems. Springer, Dordrecht (1995). https://doi.org/10.1007/978-94-015-8480-7

Bakushinsky, A., Goncharsky, A.: Ill-Posed Problems: Theory and Applications. Kluwer Academic Publishers. Springer, Dordrecht (1994). https://doi.org/10.1007/978-94-011-1026-6

Vinokurov, V. A.: Regularizability of Functions. In: Ill-posed problems in the Natural Sciences, pp. 52–70, Mir, Moscow (1987)

Lavrentiev, M.M., Romanov, V.G., Shishatskii, S.P.: Ill-Posed Problems of Mathematical Physics and Analysis. American Mathematical Society, Providence, (1986)

Ramm, A.G.: Non-uniqueness of the solution to an inverse problem in geophysics. Inverse problems 2, 123–125 (1986)

Engl, H.W., Kunish, K., Neubaer, A.: Convergence rate for Tikhonov regularization of nonlinear ill-posed problem. Inverse problems 4, 532–540 (1989)

Groetch, C.W.: The theory of Tikhonov regularization for Fredholm equations of the first kind. SIAM Review 28, 116–118 (1986). https://doi.org/10.1137/1028033

Nashed, M.Z.: Ill-posed Problems: Theory and Practice. Reidel, Dordrecht (1985)

Klibanov, M.V., Timonov, A.A.: Carleman Estimates for Coefficient Inverse Problems and Numerical Applications. Walter de Gruyter GmbH (2004). https://doi.org/10.1515/9783110915549

Romanov, V.G., Kabanikhin, S.I. Inverse Problems for Maxwell’s Equations. VSP, Utrecht, (1994)

Goncharsky, A.V., Romanov, S.Y., Seryozhnikov, S.Y.: Inverse problems of 3D ultrasonic tomography with complete and incomplete range data. Wave Motion 51(3), 389–404 (2014). https://doi.org/10.1016/j.wavemoti.2013.10.001

Goncharsky, A.V., Romanov, S.Y., Seryozhnikov, S.Y.: Comparison of the capabilities of GPU clusters and general-purpose supercomputers for solving 3D inverse problems of ultrasound tomography. Journal of Parallel and Distributed Computing 133, 77–92 (2019). https://doi.org/10.1016/j.jpdc.2019.06.008

Natterer, F., Sielschott, H., Dorn, O., et al.: Frechet derivatives for some bilinear inverse problems. SIAM J. Appl. Math. 62, 2092–2113 (2002). https://doi.org/10.1137/s0036139901386375

Beilina, L., Klibanov, M.V., Kokurin, M.Y.: Adaptivity with relaxation for ill-posed problems and global convergence for a coefficient inverse problem. J. Math. Sci. 167, 279–325 (2010). https://doi.org/10.1007/s10958-010-9921-1

Goncharsky, A.V., Romanov, S.Y.: Iterative Methods for Solving Coefficient Inverse Problems of Wave Tomography in Models with Attenuation. Inverse Probl. 33(2), 025003 (2017). https://doi.org/10.1088/1361-6420/33/2/025003

Global Optimization: From Theory to Implementation. Edition by Liberti, L., Maculan, N. Springer (2006)

Gel’fand, I.M., Tsetlin, M.L.: Some methods of control for complex systems. Russian Mathematical Surveys 17, 95–117 (1962). https://doi.org/10.1070/rm1962v017n01abeh001124

Sulimov, A.V., Zheltkov, D.A., Oferkin, I.V., et al.: Tensor Train Global Optimization: Application to Docking in the Configuration Space with a Large Number of Dimensions. Communications in Computer and Information Science (CCIS), vol. 793, pp. 151–167. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-71255-0_12

Duric, N., Littrup, P., Li, C., et al.: Breast ultrasound tomography: bridging the gap to clinical practice. Proc. SPIE, 8320, 83200O (2012). https://doi.org/10.1117/12.910988

Jirik, R., Peterlik, I., Ruiter, N., et al.: Sound-speed image reconstruction in sparse-aperture 3D ultrasound transmission tomography. IEEE Trans. Ultrason. Ferroelectr. Freq. Control. 59, 254–264 (2012). https://doi.org/10.1109/tuffc.2012.2185

Goncharsky, A.V., Romanov, S.Y., Seryozhnikov, S.Y.: A computer simulation study of soft tissue characterization using low-frequency ultrasonic tomography. Ultrasonics 67, 136–150 (2016). https://doi.org/10.1016/j.ultras.2016.01.008

Shen, Y., Shamout, F.E., Oliver, J.R., et al.: Artificial intelligence system reduces falsepositive findings in the interpretation of breast ultrasound exams. Nat. Commun. 12, 5645 (2021). https://doi.org/10.1038/s41467-021-26023-2

Natterer, F.: Sonic imaging. In: Scherzer O. (eds) Handbook of Mathematical Methods in Imaging, pp. 1253–1278. Springer, New York (2015). https://doi.org/10.1007/978-1-4939-0790-8_37

Bakushinsky, A., Goncharsky, A., Romanov, S., Seatzu, S.: On the identification of velocity in seismics and in acoustic sounding. Pubblicazioni IAGA, Series “Problemi non ben posti e inversi” 71 (1994)

Bakushinskii, A.B., Kozlov, A.I., Kokurin, M.Yu.: One Inverse Problem for a Three-Dimensional Wave Equation. Computational Mathematics and Mathematical Physics 43(8), 1149–1158 (2003)

Bakushinskii, A.B., Leonov, A.S.: Low-cost numerical method for solving a coefficient inverse problem for the wave equation in three-dimensional space. Computational Mathematics and Mathematical Physics 58, 548–561 (2018). https://doi.org/10.1134/s0965542518040073

Klibanov, M.V., Li, J., Zhang, W.: Linear Lavrent’ev Integral Equation for the Numerical Solution of a Nonlinear Coefficient Inverse Problem. SIAM J. Appl. Math. 81(5), 1954–1978 (2021). https://doi.org/10.1137/20M1376558

Hamilton, B., Bilbao, S.: Fourth-order and optimised finite difference schemes for the 2-D wave equation. In: Proc. of the 16th Int. Conference on Digital Audio Effects (DAFx-13). pp. 363–395. Springer (2013)

Voevodin, V., Antonov, A., Nikitenko, D., et al.: Supercomputer Lomonosov-2: Large Scale, Deep Monitoring and Fine Analytics for the User Community. Supercomputing Frontiers and Innovations 6(2), 4–11 (2019). https://doi.org/10.14529/jsfi190201

Goncharsky, A., Seryozhnikov, S.: Supercomputer technology for ultrasound tomographic image reconstruction: mathematical methods and experiment. In: Voevodin, V., Sobolev, S. (eds) Supercomputing. RuSCDays 2018. Communications in Computer and Information Science, vol. 965, pp. 401–413. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-05807-4_34

Romanov, S. Supercomputer simulation study of the convergence of iterative methods for solving inverse problems of 3D acoustic tomography with the data on a cylindrical surface // In: Voevodin, V., Sobolev, S. (eds) Supercomputing. RuSCDays 2018. Communications in Computer and Information Science, vol. 965, pp. 388–400. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-05807-4_33

Rocki, K., Essendelft, D.V., Sharapov, I.: Fast stencil-code computation on a wafer-scale processor. In: Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis (2020)

Lechleiter, A., Schlasche, J.W.: Identifying Lame parameters from time-dependent elastic wave. Inverse Problems in Science and Engineering 25(1), 2–26 (2017). https://doi.org/10.1080/17415977.2015.1132713

He, J., Rao, J., Fleming, J.D., et al.: Numerical ultrasonic full waveform inversion (FWI) for complex structures in coupled 2D solid/fluid media. Smart Materials and Structures 30, 085044 (2021). https://doi.org/10.1088/1361-665X/ac0f44