PHaseless inverse scattering problems in three dimensions

SIAM Journal on Applied Mathematics

Volumn 74, Issue 2, 2014, Pages 392-410

  Klibanov, Michael V ^a  

^a UNIVERSITY OF NORTH CAROLINA AT CHARLOTTE   (United States)

Author keywords

Finding complex zeros; Inverse scattering problems without the phase information; Uniqueness theorems

Indexed keywords

MATHEMATICAL MODELS; MATHEMATICAL TECHNIQUES;

COMPLEX ZEROS; FREQUENCY DOMAINS; INVERSE SCATTERING PROBLEMS; PHASE INFORMATION; SCATTERED FIELD; THREE DIMENSIONS; UNIQUENESS THEOREM;

SCATTERING;

EID: 84903296110     PISSN: 00361399     EISSN: None     Source Type: Journal    
DOI: 10.1137/130926250     Document Type: Article

References (39)

1. T. Aktosun and P.E. Sacks, Inverse problem on the line without phase information, Inverse Problems, 14 (1998), pp. 211-224.
2. N.F. Berk and C.F. Majkrzak, Statistical analysis of phase-inversion neutron specular reflectivity, Langmuir, 25 (2009), pp. 4132-4144.
3. L. Beilina and M.V. Klibanov, Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems, Springer, New York, 2012.
4. A.L. Bukhgeim and M.V. Klibanov, Uniqueness in the large of a class of multidimensional inverse problems, Soviet Math. Dokl., 17 (1981), pp. 244-247.
5. A.L. Bukhgeim, Introduction in the Theory of Inverse Problems, VSP, Utrecht, 2000.
6. P. Colwell, Blaschke Products: Bounded Analytic Functions, University of Michigan Press, Ann Arbor, MI, 1985.
7. Z. Dai and P.K. Lamm, Local regularization for the nonlinear inverse autoconvolution problem, SIAM J. Numer. Anal., 46 (2008), pp. 832-868.
8. D. Dobson, Phase reconstruction via nonlinear least squares, Inverse Problems, 8 (1992), pp. 541-557.
9. J.R. Fienup, Phase retrieval algorithms: A personal tour [invited], Appl. Opt., 52 (2013), pp. 45-56.
10. J.R. Fienup, J.C. Maron, T.J. Schulz, and J.H. Seldin, Hubble space telescope characterization by using phase retrieval algorithms, Appl. Opt., 32 (1993), pp. 1747-1768.
11. D. Gerth, B. Hofman, S. Birkholz, S. Koke, and G. Steinmeyer, Regularization of an autoconvolution problem in ultrashort laser pulse characterization, Inverse Problems Sci. Engrg., 22 (2014), pp. 245-266.
12. D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, New York, 1984.
13. N.E. Hurt, Phase Retrieval and Zero Crossings: Mathematical Methods in Image Reconstruction, Kluwer Academic, Dodrecht, 2002.
14. V. Isakov, Inverse Problems for Partial Differential Equations, 2nd ed., Springer, New York, 2005.
15. V. Isakov, Inverse obstacle problems, Inverse Problems, 25 (2009), 123002.
16. O. Ivanyshyn, Shape reconstruction of acoustic obstacles from the modulus of the far field pattern, Inverse Problems Imaging, 1 (2007), pp. 609-622.
17. O. Ivanyshyn and R. Kress, Identification of sound-soft 3D obstacles from phaseless data, Inverse Problems Imaging, 4 (2010), pp. 131-149.
18. O. Ivanyshyn and R. Kress, Inverse scattering for surface impedance from phase-less far field data, J. Comput. Phys., 230 (2011), pp. 3443-3452.
19. M.V. Klibanov, Determination of a function with compact support from the absolute value of its Fourier transform, and an inverse scattering problem, Differential Equations, 22 (1987), pp. 1232-1240.
20. M.V. Klibanov, Uniqueness of the determination of distortions of a crystal lattice by the Xray diffraction method in a continuous dynamical model, Differential Equations, 25 (1989), pp. 520-527.
21. M.V. Klibanov and P.E. Sacks, Phaseless inverse scattering and the phase problem in optics, J. Math. Physics, 33 (1992), pp. 3813-3821.
22. M.V. Klibanov, P.E. Sacks, and A.V. Tikhonravov, The phase retrieval problem. Topical review, Inverse Problems, 11 (1995), pp. 1-28.
23. M.V. Klibanov, On the recovery of a 2-D function from the modulus of its Fourier transform, J. Math. Anal. Appl., 323 (2006), pp. 818-843.
24. M.V. Klibanov and A. Timonov, Carleman Estimates for Coefficient Inverse Problems and Numerical Applications, VSP, Utrecht, 2004.
25. M. V. Klibanov, Carleman estimates for global uniqueness, stability and numerical methods for coefficient inverse problems, J. Inverse Ill-Posed Probl., 21 (2013), pp. 477-560.
26. R. Kress and W. Rundell, Inverse obstacle scattering with modulus of the far field pattern as data, in Inverse Problems in Medical Imaging and Nondestructive Testing (Oberwolfach, 1996), Springer, Vienna, 1997, pp. 75-92.
27. M.F.C. Ladd and R.A. Palmer, Structure Determination by X-Ray Crystallography, Plenum Press, New York, 1993.
28. O.A. Ladyzhenskaya, Boundary Value Problems of Mathematical Physics, Springer, New York, 1985.
29. M.M. Lavrentiev, V.G. Romanov, and S.P. Shishatskii, Ill-Posed Problems of Mathematical Physics and Analysis, AMS, Providence, RI, 1986.
30. Z.T. Nazarchuk, R.O. Hryniv, and A.T. Synyavsky, Reconstruction of the impedance Schrodinger equation from the modulus of the reflection coefficients, Wave Motion, 49 (2012), pp. 719-736.
31. R.G. Newton, Inverse Schrodinger Scattering in Three Dimensions, Springer, New York, 2011.
32. R.G. Novikov, Multidimensional inverse spectral problem for the equation-ψ + (v(x) Eu(x))ψ = 0, Funct. Anal. Appl., 22 (1988), pp. 263-272.
33. R.G. Novikov, The inverse scattering problem on a fixed energy level for the two-dimensional Schrodinger operator, J. Funct. Anal., 103 (1992), pp. 409-463.
34. V.G. Romanov, Integral Geometry and Inverse Problems for Hyperbolic Equations, Springer-Verlag, Berlin, 1974.
35. V.G. Romanov, Inverse Problems of Mathematical Physics, VNU Science Press, Utrecht, 1986.
36. B. Sixou, V. Davidoiu, M. Langer, and F. Peyrin, Absorption and phase retrieval with Tikhonov and joint sparsity regularizations, Inverse Problems Imaging, 7 (2013), pp. 267-282.
37. B.R. Vainberg, Principles of radiation, limiting absorption and limiting amplitude in the general theory of partial differential equations, Russian Math. Surv., 21 (1966), pp. 115-193.
38. B.R. Vainberg, Asymptotic Methods in Equations of Mathematical Physics, Gordon and Breach, New York, 1989.
39. X.-L. Zhou and S.-H. Chen, Theoretical foundation of X-ray and neutron reflectometry, Phys. Rep., 257 (1995), pp. 223-348.