Stability estimates for an inverse scattering problem at high frequencies

Journal of Mathematical Analysis and Applications

Volumn 400, Issue 2, 2013, Pages 525-540

  Ammari, Habib ^a   Bahouri, Hajer ^b   Ferreira, David Dos Santos ^c   Gallagher, Isabelle ^d  

^a ECOLE NORMALE SUPÉRIEURE   (France)

^b UNIVERSITÉ PARIS EST CRÉTEIL   (France)

^c UNIVERSITÉ PARIS 13   (France)

^d INSTITUT DE MATHÉMATIQUES DE JUSSIEU   (France)

Author keywords

Inverse scattering problem; Lipschitz stability; Resolution

Indexed keywords

EID: 84872038462     PISSN: 0022247X     EISSN: 10960813     Source Type: Journal    
DOI: 10.1016/j.jmaa.2012.10.066     Document Type: Article

References (48)

1. Alessandrini G. Stable determination of conductivity by boundary measurements. Appl. Anal. 1988, 27:153-172.
2. Alessandrini G. Examples of instability in inverse boundary-value problems. Inverse Problems 1997, 13:887-897.
3. Alessandrini G. Open issues of stability for the inverse conductivity problem. J. Inverse Ill-Posed Probl. 2007, 15:451-460.
4. John F. Continuous dependence on data for solutions of partial differential equations with a prescribed bound. Comm. Pure Appl. Math. 1960, 13:551-585.
5. Mandache N. Exponential instability in an inverse problem for the Schrödinger equation. Inverse Problems 2001, 17:1435-1444.
6. Ammari H., Garnier J., Jugnon V., Kang H. Direct reconstruction methods in ultrasound imaging of small anomalies. Lecture Notes in Mathematics 2011, vol. 2035:31-55. Springer-Verlag, Berlin.
7. Ammari H., Kang H. Reconstruction of Small Inhomogeneities from Boundary Measurements. Lecture Notes in Mathematics 2004, vol. 1846. Springer, Berlin.
8. Ammari H., Kang H., Kim E., Lee J.-Y. The generalized polarization tensors for resolved imaging. Part II: shape and electromagnetic parameters reconstruction of an electromagnetic inclusion from multistatic measurements. Math. Comp. 2012, 81:839-860.
9. de Hoop M.V., Qiu L., Scherzer O. Local analysis of inverse problems: Hölder stability and iterative reconstruction. Inverse Problems 2012, 28:045001.
10. Born M., Wolf E. Principles of Optics 1999, Cambridge University Press, Cambridge.
11. Bertero M., Boccacci P., Piana M. Resolution and super-resolution in inverse diffraction. Lecture Notes in Physics 1997, vol. 486:1-17. Springer, Berlin.
12. Chen Y. Inverse scattering via Heisenberg's uncertaintly principle. Inverse Problems 1997, 13:253-282.
13. Donoho D.L., Stark P.B. Uncertaintly principles and signal recovery. SIAM J. Appl. Math. 1989, 49:906-931.
14. Slepian D. Some comments on Fourier analysis, uncertainty and modeling. SIAM Rev. 1983, 25:379-393.
15. Ammari H. An Introduction to Mathematics of Emerging Biomedical Imaging. Math. & Appl. 2008, vol. 62. Springer-Verlag, Berlin.
16. Karp S.N. A convergent 'farfield' expansion for two-dimensional radiation functions. Comm. Pure Appl. Math. 1961, 14:427-434.
17. Novikov R.G. A multidimensional inverse spectral problem for the equation -δψ+(v(x)-Eu(x))ψ=0. Funct. Anal. Appl. 1988, 22:263-272.
18. Novikov R.G. Formulae and equations for finding scattering data from the Dirichlet-to-Neumann map with nonzero background potential. Inverse Problems 2005, 21:257-270.
19. Taylor M.E. Estimates for approximate solutions to acoustic inverse scattering problems. IMA Vol. Math. Appl. 1997, vol. 90:463-499. Springer, New York.
20. Ammari H., Garnier J., Kang H., Lim M., Sølna K. Multistatic imaging of extended targets. SIAM J. Imag. Sci. 2012, 5:564-600.
21. H. Ammari, J. Garnier, K. Sølna, Resolution and stability analysis in full aperture, linearized conductivity and wave imaging, Proc. Amer. Math. Soc. (in press).
22. Zhao H. Analysis of the response matrix for an extended target. SIAM J. Appl. Math. 2004, 64:725-745.
23. Bao G., Chen Y., Ma F. Regularity and stability for the scattering map of a linearized inverse medium problem. J. Math. Anal. Appl. 2000, 247:255-271.
24. Chen Y., Rokhlin V. On the inverse scattering problem for the Helmholtz equation in one dimension. Inverse Problems 1992, 8:365-391.
25. Coifman R., Goldberg M., Hrycak T., Israeli M., Rokhlin V. An improved operator expansion algorithm for direct and inverse scattering computations. Waves Random Media 1999, 9:441-457.
26. Isakov V. Increased stability in the Cauchy problem for some elliptic equations. International Math. Series 2008, vol. 6:339-362. Springer. C. Bardos, A. Fursikov (Eds.).
27. Isakov V., Kindermann S. Subspaces of stability in the Cauchy problem for the Helmholtz equation. Methods Appl. Anal. 2011, 18:1-29.
28. Bao G., Junshan J., Triki F. A multi-frequency inverse source problem. J. Differential Equations 2010, 249:3443-3465.
29. Bao G., Junshan J., Triki F. Numerical solution of the inverse source problem for the Helmholtz equation with multiple frequency data. Contemp. Math. 2011, 548:45-60.
30. Nagayasu S., Uhlmann G., Wang J.-N. Depth dependent stability estimates in electrical impedance tomography. Inverse Problems 2009, 25:075001.
31. S. Nagayasu, G. Uhlmann, J.-N. Wang, Increasing stability in an inverse problem for the acoustic equation, Preprint.
32. Derveaux G., Papanicolaou G., Tsogka C. Resolution and denoising in near-field imaging. Inverse Problems 2006, 22:1437-1456.
33. Bao G., Yun K. On the stability of an inverse problem for the wave equation. Inverse Problems 2009, 25:045003. (p. 7).
34. Melrose R.B. Geometric Scattering Theory 1995, Cambridge University Press.
35. Burq N. Semi-classical estimates for the resolvent in nontrapping geometries. Int. Math. Res. Not. 2002, 5:221-241.
36. Agmon S. Spectral properties of Schrödinger operators and scattering theory. Ann. Sc. Norm. Super Pisa Cl. Sci. 1975, 2:151-218.
37. J.M. Melenk, On generalized finite element methods, Ph.D. Thesis, University of Maryland, 1995.
38. Engquist B., Majda A. Absorbing boundary conditions for the numerical simulation of waves. Math. Comp. 1977, 31:629-651.
39. Keller J.B., Givoli D. Exact nonreflecting boundary conditions. J. Comput. Phys. 1989, 82:172-192.
40. Sylvester J., Uhlmann G. A global uniqueness theorem for an inverse boundary value problem. Ann. of Math. 1987, 125:153-169.
41. Nachman A.I. Reconstruction from boundary measurements. Ann. of Math. 1988, 128:531-576.
42. Boumenir A., Tuan V.K. Sampling eigenvalues in Hardy spaces. SIAM J. Numer. Anal. 2007, 45:473-483.
43. Rybkin A., Tuan V.K. A new interpolation formula for the Titchmarsh-Weyl m-function. Proc. Amer. Math. Soc. 2009, 137:4177-4185.
44. H. Ammari, J. Garnier, K. Sølna, Limited view resolving power of conductivity imaging from boundary measurements, SIAM J. Math. Anal. (in press).
45. Carney P.S., Schotland J.C. Near-field tomography. Math. Sci. Res. Inst. Publ. 2003, vol. 47:133-168. Cambridge Press, Cambridge.
46. Lebedev N.N. Special Functions and their Applications 1965, Prentice Hall.
47. Olver F.W.J. Asymptotics and Special Functions 1974, Academic Press.
48. Abramowitz M., Stegun I. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series 1964, vol. 55:xiv+1046.