Multipoint Formulas for Phase Recovering from Phaseless Scattering Data

Abstract

We give formulas for phase recovering from appropriate monochromatic phaseless scattering data at 2n points in dimension \(d=3\) and in dimension \(d=2\). These formulas are recurrent and explicit and their precision is proportional to n. By this result we continue studies of Novikov (Bulletin des Sciences Mathématiques 139(8):923–936, 2015), where formulas of such a type were given for \(n=1\), \(d\ge 2\).

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Agaltsov, A.D., Hohage, T., Novikov, R.G.: An iterative approach to monochromatic phaseless inverse scattering. Inverse Prob. 35(2), 24001 (2019). ( 24 pp.)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Born, M.: Quantenmechanik der Stossvorgange. Z. Angew. Phys. 38(11–12), 803–827 (1926)

    MATH  Google Scholar 

  3. 3.

    Chadan, K., Sabatier, P.C.: Inverse Problems in Quantum Scattering Theory, 2nd edn. Springer, Berlin (1989)

    Google Scholar 

  4. 4.

    Faddeev, L.D., Merkuriev, S.P.: Quantum Scattering Theory for Multi-particle Systems, Mathematical Physics and Applied Mathematics, 11. Kluwer Academic Publishers Group, Dordrecht (1993)

    Google Scholar 

  5. 5.

    Hohage, T., Novikov, R.G.: Inverse wave propagation problems without phase information. Inverse Prob. 35(7), 070301 (2019). (4pp.)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Ivanyshyn, O., Kress, R.: Identification of sound-soft 3D obstacles from phaseless data. Inverse Probl. Imaging 4, 131–149 (2010)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Jonas, P., Louis, A.K.: Phase contrast tomography using holographic measurements. Inverse Prob. 20(1), 75–102 (2004)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Klibanov, M.V.: Phaseless inverse scattering problems in three dimensions. SIAM J. Appl. Math. 74(2), 392–410 (2014)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Klibanov, M.V., Romanov, V.G.: Reconstruction procedures for two inverse scattering problems without the phase information. SIAM J. Appl. Math. 76(1), 178–196 (2016)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Klibanov, M.V., Koshev, N.A., Nguyen, D.-L., Nguyen, L.H., Brettin, A., Astratov, V.N.: A numerical method to solve a phaseless coefficient inverse problem from a single measurement of experimental data. SIAM J. Imaging Sci. 11(4), 2339–2367 (2018)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Melrose, R.B.: Geometric Scattering Theory. Cambridge University Press, Stanford (1995)

    Google Scholar 

  12. 12.

    Novikov, R.G.: Inverse scattering without phase information. Séminaire Laurent Schwartz—EDP et applications, Exp. No. 16, 13 p (2014–2015)

  13. 13.

    Novikov, R.G.: An iterative approach to non-overdetermined inverse scattering at fixed energy. Sbornik 206(1), 120–134 (2015)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Novikov, R.G.: Formulas for phase recovering from phaseless scattering data at fixed frequency. Bulletin des Sciences Mathématiques 139(8), 923–936 (2015)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Novikov, R.G.: Phaseless inverse scattering in the one-dimensional case. Eurasian J. Math. Comput. Appl. 3(1), 63–69 (2015)

    Google Scholar 

  16. 16.

    Novikov, R.G., Galchenkova, M.A.: Phase recovering from phaseless scattering data at a few points, Report of stage M2. (2018)

  17. 17.

    Palamodov, V.: A fast method of reconstruction for X-ray phase contrast imaging with arbitrary Fresnel number. arXiv:1803.08938v1 (2018)

  18. 18.

    Romanov, V.G.: Inverse problems without phase information that use wave interference. Sib. Math. J. 59(3), 494–504 (2018)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

The author thanks B. Berndtsson and M. V. Klibanov for remarks on results of [12, 14], which stimulated studies of the present work.

Author information

Affiliations

Authors

Corresponding author

Correspondence to R. G. Novikov.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Novikov, R.G. Multipoint Formulas for Phase Recovering from Phaseless Scattering Data. J Geom Anal 31, 1965–1991 (2021). https://doi.org/10.1007/s12220-019-00329-6

Download citation

Keywords

  • Schrödinger equation
  • Helmholtz equation
  • Monochromatic scattering data
  • Phase recovering
  • Phaseless inverse scattering

Mathematics Subject Classification

  • 35J10
  • 35P25
  • 35R30
  • 81U40