Inverse Source Problem for a Wave Equation with Final Observation Data

  • Daijun Jiang
  • Yikan LiuEmail author
  • Masahiro Yamamoto
Conference paper
Part of the Mathematics for Industry book series (MFI, volume 26)


In this chapter, we study the inverse problem on recovering a spatial component of the source term in a wave equation by the final observation data. Employing the analytic Fredholm theory , we establish a generic well-posedness result concerning the uniqueness of our inverse source problem . Numerically, by treating a corresponding minimization problem, we investigate the variational equation for the minimizer and develop an iterative thresholding algorithm . One- and two-dimensional numerical experiments are implemented to demonstrate the robustness and accuracy of the proposed algorithm.



The work was supported by A3 Foresight Program “Modeling and Computation of Applied Inverse Problems,” Japan Society for the Promotion of Science (JSPS). The first author was financially supported by self-determined research funds of Central China Normal University from the colleges’ basic research and operation of MOE (No. CCNU14A05039), National Natural Science Foundation of China (Nos. 11326233, 11401241 and 11571265). The second and the third authors are partially supported by Grant-in-Aid for Scientific Research (S) 15H05740, JSPS.


  1. 1.
    Bukhgeim, A.L., Klibanov, M.V.: Global uniqueness of a class of multidimensional inverse problems. Sov. Math. Dokl. 24, 244–247 (1981)zbMATHGoogle Scholar
  2. 2.
    Daubechies, I., Defrise, M., De Mol, C.: An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Comm. Pure Appl. Math. 57, 1413–1457 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Fairweather, G., Mitchell, A.R.: A high accuracy alternating direction method for the wave equation. IMA J. Appl. Math. 1, 309–316 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Imanuvilov, O.Y., Yamamoto, M.: Global uniqueness and stability in determining coefficients of wave equations. Comm. Partial Differ. Equ. 26, 1409–1425 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Imanuvilov, O.Y., Yamamoto, M.: Global Lipschitz stability in an inverse hyperbolic problem by interior observation. Inverse Prob. 17, 717–728 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Isakov, V.: Inverse Problems for Partial Differential Equations. Springer, New York (2006)zbMATHGoogle Scholar
  7. 7.
    Jiang, D., Liu, Y., Yamamoto, M.: Inverse source problem for the hyperbolic equations with a time-dependent principal part. J. Differ. Equation. (in press)Google Scholar
  8. 8.
    Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1980)zbMATHGoogle Scholar
  9. 9.
    Klibanov, M.V., Timonov, A.: Carleman Estimates for Coefficient Inverse Problems and Numerical Applications. VSP, Utrecht (2004)CrossRefzbMATHGoogle Scholar
  10. 10.
    Lions, J.-L., Magenes, E.: Non-homogeneous Boundary Value Problems and Applications. Springer, Berlin (1972)CrossRefzbMATHGoogle Scholar
  11. 11.
    Liu, Y., Jiang, D., Yamamoto, M.: Inverse source problem for a double hyperbolic equation describing the three-dimensional time cone model. SIAM J. Appl. Math. 75, 2610–35 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Liu, Y., Xu, X., Yamamoto, M.: Growth rate modeling and identification in the crystallization of polymers. Inverse Prob. 28, 095008 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Liu, Y., Yamamoto, M.: On the multiple hyperbolic systems modelling phase transformation kinetics. Appl. Anal. 93, 1297–1318 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Puel, J.-P., Yamamoto, M.: Generic well-posedness in a multidimensional hyperbolic inverse problem. J. Inverse Ill-Posed Prob. 5, 55–84 (1997)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Yakhno, V.G.: A converse problem for a hyperbolic equation (in Russian). Mat. Zametki 26, 39–44 (1979)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Yamamoto, M.: Stability, reconstruction formula and regularization for an inverse source hyperbolic problem by a control method. Inverse Prob. 11, 481–496 (1995)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2017

Authors and Affiliations

  1. 1.School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical SciencesCentral China Normal UniversityWuhanPeople’s Republic of China
  2. 2.Graduate School of Mathematical SciencesThe University of TokyoTokyoJapan

Personalised recommendations