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The Quasi-reversibility Method to Numerically Solve an Inverse Source Problem for Hyperbolic Equations

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Abstract

We propose a numerical method to solve an inverse source problem of computing the initial condition of hyperbolic equations from the measurements of Cauchy data. This problem arises in thermo- and photo-acoustic tomography in a bounded cavity, in which the reflection of the wave makes the widely-used approaches, such as the time reversal method, not applicable. In order to solve this inverse source problem, we approximate the solution to the hyperbolic equation by its Fourier series with respect to a special orthonormal basis of \(L^2\). Then, we derive a coupled system of elliptic equations for the corresponding Fourier coefficients. We solve it by the quasi-reversibility method. The desired initial condition follows. We rigorously prove the convergence of the quasi-reversibility method as the noise level tends to 0. Some numerical examples are provided. In addition, we numerically prove that the use of the special basic above is significant.

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References

  1. Kruger, R.A., Liu, P., Fang, Y.R., Appledorn, C.R.: Photoacoustic ultrasound (PAUS)-reconstruction tomography. Med. Phys. 22, 1605 (1995)

    Google Scholar 

  2. Oraevsky, A., Jacques, S., Esenaliev, R., Tittel, F.: Laser-based optoacoustic imaging in biological tissues. Proc. SPIE 2134A, 122 (1994)

    Google Scholar 

  3. Kruger, R.A., Reinecke, D.R., Kruger, G.A.: Thermoacoustic computed tomography: technical considerations. Med. Phys. 26, 1832 (1999)

    Google Scholar 

  4. Do, N., Kunyansky, L.: Theoretically exact photoacoustic reconstruction from spatially and temporally reduced data. Inverse Probl. 34(9), 094004 (2018)

    MathSciNet  MATH  Google Scholar 

  5. Haltmeier, M.: Inversion of circular means and the wave equation on convex planar domains. Comput. Math. Appl. 65, 1025–1036 (2013)

    MathSciNet  MATH  Google Scholar 

  6. Natterer, F.: Photo-acoustic inversion in convex domains. Inverse Probl. Imaging 6, 315–320 (2012)

    MathSciNet  MATH  Google Scholar 

  7. Nguyen, L.V.: A family of inversion formulas in thermoacoustic tomography. Inverse Probl. Imaging 3, 649–675 (2009)

    MathSciNet  MATH  Google Scholar 

  8. Katsnelson, V., Nguyen, L.V.: On the convergence of time reversal method for thermoacoustic tomography in elastic media. Appl. Math. Lett. 77, 79–86 (2018)

    MathSciNet  MATH  Google Scholar 

  9. Hristova, Y.: Time reversal in thermoacoustic tomography-an error estimate. Inverse Probl. 25, 055008 (2009)

    MathSciNet  MATH  Google Scholar 

  10. Hristova, Y., Kuchment, P., Nguyen, L.V.: Reconstruction and time reversal in thermoacoustic tomography in acoustically homogeneous and inhomogeneous media. Inverse Probl. 24, 055006 (2008)

    MathSciNet  MATH  Google Scholar 

  11. Stefanov, P., Uhlmann, G.: Thermoacoustic tomography with variable sound speed. Inverse Probl. 25, 075011 (2009)

    MathSciNet  MATH  Google Scholar 

  12. Stefanov, P., Uhlmann, G.: Thermoacoustic tomography arising in brain imaging. Inverse Probl. 27, 045004 (2011)

    MathSciNet  MATH  Google Scholar 

  13. Clason, C., Klibanov, M.V.: The quasi-reversibility method for thermoacoustic tomography in a heterogeneous medium. SIAM J. Sci. Comput. 30, 1–23 (2007)

    MathSciNet  MATH  Google Scholar 

  14. Huang, C., Wang, K., Nie, L., Wang, L.V., Anastasio, M.A.: Full-wave iterative image reconstruction in photoacoustic tomography with acoustically inhomogeneous media. IEEE Trans. Med. Imaging 32, 1097–1110 (2013)

    Google Scholar 

  15. Paltauf, G., Nuster, R., Haltmeier, M., Burgholzer, P.: Experimental evaluation of reconstruction algorithms for limited view photoacoustic tomography with line detectors. Inverse Probl. 23, S81–S94 (2007)

    MathSciNet  MATH  Google Scholar 

  16. Paltauf, G., Viator, J.A., Prahl, S.A., Jacques, S.L.: Iterative reconstruction algorithm for optoacoustic imaging. J. Opt. Soc. Am. 112, 1536–1544 (2002)

    Google Scholar 

  17. Ammari, H., Bretin, E., Jugnon, E., Wahab, V.: Photoacoustic imaging for attenuating acoustic media. In: Ammari, H. (ed.) Mathematical Modeling in Biomedical Imaging II, pp. 57–84. Springer, Berlin (2012)

    MATH  Google Scholar 

  18. Ammari, H., Bretin, E., Garnier, J., Wahab, V.: Time reversal in attenuating acoustic media. Contemp. Math. 548, 151–163 (2011)

    MathSciNet  MATH  Google Scholar 

  19. Haltmeier, M., Nguyen, L.V.: Reconstruction algorithms for photoacoustic tomography in heterogeneous damping media. J. Math. Imaging Vis. 61, 1007–1021 (2019)

    MathSciNet  MATH  Google Scholar 

  20. Acosta, S., Palacios, B.: Thermoacoustic tomography for an integro-differential wave equation modeling attenuation. J. Differ. Equ. 5, 1984–2010 (2018)

    MathSciNet  MATH  Google Scholar 

  21. Burgholzer, P., Grün, H., Haltmeier, M., Nuster, R., Paltauf, G.: Compensation of acoustic attenuation for high-resolution photoa-coustic imaging with line detectors. Proc. SPIE 6437, 643724 (2007)

    Google Scholar 

  22. Homan, A.: Multi-wave imaging in attenuating media. Inverse Probl. Imaging 7, 1235–1250 (2013)

    MathSciNet  MATH  Google Scholar 

  23. Kowar, R.: On time reversal in photoacoustic tomography for tissue similar to water. SIAM J. Imaging Sci. 7, 509–527 (2014)

    MathSciNet  MATH  Google Scholar 

  24. Kowar, R., Scherzer, O.: Photoacoustic imaging taking into account attenuation. In: Ammari, H. (ed.) Mathematics and Algorithms in Tomography II. Lecture Notes in Mathematics, pp. 85–130. Springer, Berlin (2012)

    MATH  Google Scholar 

  25. Nachman, A.I., Smith, J.F., III., Waag, R.C.: An equation for acoustic propagation in inhomogeneous media with relaxation losses. J. Acoust. Soc. Am. 88, 1584–1595 (1990)

    Google Scholar 

  26. Cox, B., Arridge, S., Beard, P.: Photoacoustic tomography with a limited-aperture planar sensor and a reverberant cavity. Inverse Probl. 23, S95 (2007)

    MathSciNet  MATH  Google Scholar 

  27. Cox, B., Beard, P.: Photoacoustic tomography with a single detector in a reverberant cavity. J. Acoust. Soc. Am. 123, 3371–3371 (2008)

    Google Scholar 

  28. Kunyansky, L., Holman, B., Cox, B.: Photoacoustic tomography in a rectangular reflecting cavity. Inverse Probl. 29, 125010 (2013)

    MathSciNet  MATH  Google Scholar 

  29. Nguyen, L.V., Kunyansky, L.: A dissipative time reversal technique for photo-acoustic tomography in a cavity. SIAM J. Imaging Sci. 9, 748–769 (2016)

    MathSciNet  MATH  Google Scholar 

  30. Lattès, R., Lions, J.L.: The Method of Quasireversibility: Applications to Partial Differential Equations. Elsevier, New York (1969)

    MATH  Google Scholar 

  31. Bécache, E., Bourgeois, L., Franceschini, L., Dardé, J.: Application of mixed formulations of quasi-reversibility to solve ill-posed problems for heat and wave equations: the 1d case. Inverse Probl. Imaging 9(4), 971–1002 (2015)

    MathSciNet  MATH  Google Scholar 

  32. Bourgeois, L.: A mixed formulation of quasi-reversibility to solve the Cauchy problem for Laplace’s equation. Inverse Probl. 21, 1087–1104 (2005)

    MathSciNet  MATH  Google Scholar 

  33. Bourgeois, L.: Convergence rates for the quasi-reversibility method to solve the Cauchy problem for Laplace’s equation. Inverse Probl, 22, 413–430 (2006)

    MathSciNet  MATH  Google Scholar 

  34. Bourgeois, L., Dardé, J.: A duality-based method of quasi-reversibility to solve the Cauchy problem in the presence of noisy data. Inverse Probl. 26, 095016 (2010)

    MathSciNet  MATH  Google Scholar 

  35. Dardé, J.: Iterated quasi-reversibility method applied to elliptic and parabolic data completion problems. Inverse Probl. Imaging 10, 379–407 (2016)

    MathSciNet  MATH  Google Scholar 

  36. Klibanov, M.V., Kuzhuget, A.V., Kabanikhin, S.I., Nechaev, D.: A new version of the quasi-reversibility method for the thermoacoustic tomography and a coefficient inverse problem. Appl. Anal. 87, 1227–1254 (2008)

    MathSciNet  MATH  Google Scholar 

  37. Klibanov, M.V.: Carleman estimates for global uniqueness, stability and numerical methods for coefficient inverse problems. J. Inverse Ill-Posed Probl. 21, 477–560 (2013)

    MathSciNet  MATH  Google Scholar 

  38. Klibanov, M.V., Santosa, F.: A computational quasi-reversibility method for Cauchy problems for Laplace’s equation. SIAM J. Appl. Math. 51, 1653–1675 (1991)

    MathSciNet  MATH  Google Scholar 

  39. Klibanov, M.V., Malinsky, J.: Newton-Kantorovich method for 3-dimensional potential inverse scattering problem and stability for the hyperbolic Cauchy problem with time dependent data. Inverse Probl. 7, 577–596 (1991)

    MathSciNet  MATH  Google Scholar 

  40. Klibanov, M.V.: Carleman estimates for the regularization of ill-posed Cauchy problems. Appl. Numer. Math. 94, 46–74 (2015)

    MathSciNet  MATH  Google Scholar 

  41. Klibanov, M.V.: Convexification of restricted Dirichlet to Neumann map. J. Inverse Ill-Posed Probl. 25(5), 669–685 (2017)

    MathSciNet  MATH  Google Scholar 

  42. Nguyen, L.H., Li, Q., Klibanov, M.V.: A convergent numerical method for a multi-frequency inverse source problem in inhomogenous media. Inverse Probl. Imaging 13, 1067–1094 (2019)

    MathSciNet  MATH  Google Scholar 

  43. Li, Q., Nguyen, L.H.: Recovering the initial condition of parabolic equations from lateral Cauchy data via the quasi-reversibility method. Inverse Probl. Sci. Eng. 28, 580–598 (2020)

    MathSciNet  Google Scholar 

  44. Klibanov, M.V., Nguyen, L.H.: PDE-based numerical method for a limited angle X-ray tomography. Inverse Probl. 35, 045009 (2019)

    MathSciNet  MATH  Google Scholar 

  45. Khoa, V.A., Klibanov, M.V., Nguyen, L.H.: Convexification for a 3D inverse scattering problem with the moving point source. SIAM J. Imaging Sci. 13(2), 871–904 (2020)

    MathSciNet  MATH  Google Scholar 

  46. Klibanov, M.V., Le, T.T., Nguyen, L.H.: Convergent numerical method for a linearized travel time tomography problem with incomplete data. SIAM J. Sci. Comput. 42, B1173–B1192 (2020)

    MATH  Google Scholar 

  47. Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics, American Mathematical Society, Providence (2010)

    MATH  Google Scholar 

  48. Klibanov, M.V., Nguyen, D.-L.: Convergence of a series associated with the convexification method for coefficient inverse problems. arXiv:2004.05660 (2020)

  49. Lavrent’ev, M.M., Romanov, V.G., Shishatski, S.P.: Ill-Posed Problems of Mathematical Physics and Analysis. Translations of Mathematical Monographs, AMS, Providence (1986)

    Google Scholar 

  50. Khoa, V.A., Bidney, G.W., Klibanov, M.V., Nguyen, L.H., Nguyen, L., Sullivan, A., Astratov, V.N.: Convexification and experimental data for a 3D inverse scattering problem with the moving point source. Inverse Probl. 36, 085007 (2020)

    MathSciNet  MATH  Google Scholar 

  51. Nguyen, L.H.: A new algorithm to determine the creation or depletion term of parabolic equations from boundary measurements. Comput. Math. Appl. 80, 2135–2149 (2020)

    MathSciNet  MATH  Google Scholar 

  52. Khoa, V.A., Bidney, G.W., Klibanov, M.V., Nguyen, L.H., Nguyen, L., Sullivan, A., Astratov, V.N.: An inverse problem of a simultaneous reconstruction of the dielectric constant and conductivity from experimental backscattering data. Inverse Probl. Sci. Eng. (2020). https://doi.org/10.1080/17415977.2020.1802447

    Article  Google Scholar 

  53. Smirnov, A.V., Klibanov, M.V., Nguyen, L.H.: On an inverse source problem for the full radiative transfer equation with incomplete data. SIAM J. Sci. Comput. 41, B929–B952 (2019)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank Dr. Michael V. Klibanov for many fruitful discussions.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC, 28223, USA

    Thuy T. Le, Loc H. Nguyen & William Powell

  2. Department of Mathematics, Purdue University, West Lafayette, IN, 47907, USA

    Thi-Phong Nguyen

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  1. Thuy T. Le
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  3. Thi-Phong Nguyen
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Correspondence to Thi-Phong Nguyen.

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Thuy Le and Loc Nguyen are supported by US Army Research Laboratory and US Army Research Office Grant W911NF-19-1-0044.

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Le, T.T., Nguyen, L.H., Nguyen, TP. et al. The Quasi-reversibility Method to Numerically Solve an Inverse Source Problem for Hyperbolic Equations. J Sci Comput 87, 90 (2021). https://doi.org/10.1007/s10915-021-01501-3

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  • DOI: https://doi.org/10.1007/s10915-021-01501-3

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