Advertisement

Inverse Problems

  • Victor Isakov
Chapter
  • 2k Downloads
Part of the Applied Mathematical Sciences book series (AMS, volume 127)

Abstract

In this chapter, we formulate basic inverse problems and indicate their applications. The choice of these problems is not random. We think that it represents their interconnections and some hierarchy.

Keywords

Inverse Problem Hyperbolic Equation Inverse Scattering Integral Geometry Cauchy Data 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. [Be3]
    Belishev, M.I. Recent progress in the boundary control method. Inverse Problems, 23 (2007), R1-R67.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [Ber]
    Berard, P.H. Spectral geometry: direct and inverse problems. NY, Springer-Verlag, 1986.CrossRefzbMATHGoogle Scholar
  3. [Bu1]
    Bukhgeim, A. L. Recovering a potential from Cauchy data in the two-dimensional case. J. Inverse Ill-Posed Probl., 16 (2008), 19–33.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [C]
    Calderon, A.P. On an inverse boundary value problem. “Seminar on numerical Analysis and Its Applications to Continuum Physics”, Rio de Janeiro, 1980, 65–73.Google Scholar
  5. [ChS]
    Chadan, K., Sabatier, P.C. Inverse Problems in Quantum Scattering Theory. Springer-Verlag, Berlin Heidelberg New York, 1989.CrossRefzbMATHGoogle Scholar
  6. [CoK]
    Colton, D., Kirsch, A. A simple method for solving inverse scattering problems in the resonance region. Inverse Problems, 12 (1996), 383–395.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [CoKr]
    Colton, D., Kress, R. Inverse Acoustic and Electromagnetic Scattering Theory. Appl. Math. Sc., 93, Springer Verlag, 2013.Google Scholar
  8. [CouH]
    Courant, R., Hilbert, D. Methods of Mathematical Physics. Vol. II. Wiley, 1962.zbMATHGoogle Scholar
  9. [DS]
    DeHoop, M., Stolk, C.C. Microlocal analysis of seismic inverse scattering in anisotropic elastic media. Comm. Pure Appl. Math., 55 (2002), 261–301.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [DGI1]
    DeTurck, D., Gordon, C. Isospectral deformations, I. Riemann structures on two-step nil spaces. Comm. Pure Appl. Math., 40 (1987), 367–387.Google Scholar
  11. [DGI2]
    DeTurck, D., Gordon, C. Isospectral deformations, II. Trace formulas, metrics, and potentials. Comm. Pure Appl. Math., 42 (1989), 1067–1095.Google Scholar
  12. [EnHN]
    Engl, H.W., Hanke, M., Neubauer, A. Regularization of Inverse Problems. Kluwer, Dordrecht, 1996.CrossRefzbMATHGoogle Scholar
  13. [ER1]
    Eskin, G., Ralston, J. On Isospectral Periodic Potentials in \(\mathbb{R}^{n}\), I, II. Comm. Pure Appl. Math., 37 (1984), 647–676, 715–753.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [F]
    Faddeev, L.D. The inverse problem in the quantum theory of scattering, II. J. Soviet Math., 5 (1976), #3, Consultants Bureau, New York, 1976, 334–396.Google Scholar
  15. [GoWW]
    Gordon, C., Webb, D., Wolpert, S. One cannot hear the shape of a drum. Bull. of AMS, 27 (1992), 134–138..MathSciNetCrossRefzbMATHGoogle Scholar
  16. [GorV]
    Gorenflo, R., Vessela, S. Abel Integral Equation. Lect. Notes Math., 1461, Springer-Verlag, 1991.Google Scholar
  17. [Gui]
    Guillemin, V. Inverse spectral results on two-dimensional tori. J. of AMS, 3 (1990), 375–387.MathSciNetzbMATHGoogle Scholar
  18. [GuM]
    Guillemin, V., Melrose, R. An inverse spectral result for elliptical regions in \(\mathbb{R}^{2}\). Adv. Math., 32 (1979), 128–148.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [Her]
    Herglotz, G. Ueber die analytische Fortsetzung des Potentials ins Innere der anziehenden Massen. Teubner-Verlag, Leipzig, 1914.zbMATHGoogle Scholar
  20. [I1]
    Inverse problems and related topics. Nakamura, G., Saitoh, S., Seo, J.-K., Yamamoto, M., editors. Chapman and Hall/CRC, 2000.Google Scholar
  21. [I2]
    Inside Out, Inverse Problems and Applications Uhlmann, G., editor. MSRI Publications, 47. Cambridge Univ. Press, 2003.Google Scholar
  22. [I3]
    Inverse Problems: Theory and Applications Alessandrini, G., Uhlmann, G., editors. Contemp. Math., 333, AMS, Providence, RI, 2003.Google Scholar
  23. [Is4]
    Isakov, V. Inverse Source Problems. Math. Surveys and Monographs Series, Vol. 34, AMS, Providence, R.I., 1990.Google Scholar
  24. [KKL]
    Katchalov, A., Kurylev, Y., Lassas, M. Inverse boundary spectral problems. Chapman and Hall-CRC, 2000.zbMATHGoogle Scholar
  25. [Ke]
    Keller, J. Inverse Problems. Amer. Math. Monthly, 83 (1976), 107–118.MathSciNetCrossRefGoogle Scholar
  26. [La]
    Langer, R.E. An inverse problem in differential equations. Bull. Amer. Math. Soc., 39 (1933), 814–820.MathSciNetCrossRefzbMATHGoogle Scholar
  27. [LaRS]
    Lavrentiev, M.M., Romanov, V.G., Shishatskij, S.P. Ill-posed problems of mathematical physics and analysis. Transl. of Math. Monographs, 64, AMS, Providence, R.I., 1986.Google Scholar
  28. [LaxP2]
    Lax, P., Phillips, R. Scattering Theory. Academic Press, 1989.zbMATHGoogle Scholar
  29. [Mar]
    Marchenko, V.A. Sturm-Liouville Operators and Applications. Birkhäuser, 1986.CrossRefzbMATHGoogle Scholar
  30. [Ne]
    New Analytic and Geometric Methods in Inverse Problems. Lectures given at the EMS Summer School and Conference held in Edinburgh, Scotland, 2000. Springer-Verlag, 2004.Google Scholar
  31. [No]
    Novikov, P.S. Sur le problème inverse du potentiel. Dokl. Akad. Nauk SSSR, 18 (1938), 165–168.zbMATHGoogle Scholar
  32. [OsPS]
    Osgood, B., Phillips, R., Sarnak, P. Compact isospectral sets of surfaces. J. Funct. Anal., 80 (1988), 212–234.MathSciNetCrossRefzbMATHGoogle Scholar
  33. [P]
    Payne, L.E. Improperly Posed Problems in Partial Differential Equations. regional Conf. Series in Applied Mathematics, SIAM, Philadelphia, 1975.Google Scholar
  34. [PoT]
    Pöschel, J., Trubowitz, E. Inverse Spectral Theory. Academic Press, Orlando, 1986.zbMATHGoogle Scholar
  35. [Pr]
    Prilepko, A.I. Über die Existenz and Eindeutigkeit von Lösungen inverser Probleme. Math Nach., 63 (1974), 135–153.CrossRefzbMATHGoogle Scholar
  36. [PrOV]
    Prilepko, A.I., Orlovskii, D.G., Vasin, I.A. Methods for solving inverse problems in mathematical physics. Marcel Dekker, New York-Basel, 2000.Google Scholar
  37. [Pro]
    Protter, M. Can one hear the shape of a drum? SIAM Review, 29 (1987), 185–196.MathSciNetCrossRefzbMATHGoogle Scholar
  38. [Sc]
    Scattering. Ed. Pike, R., Sabatier, P. Academic Press, 2002.Google Scholar
  39. [Sh]
    Sharafutdinov, V.A. Integral Geometry of Tensor Fields. VSP, Utrecht, 1994.CrossRefzbMATHGoogle Scholar
  40. [SyU2]
    Sylvester, J., Uhlmann, G. Inverse Boundary Value Problems at the Boundary - Continuous Dependence. Comm. Pure Appl. Math., 41 (1988) 197–221.MathSciNetCrossRefzbMATHGoogle Scholar
  41. [Tat2]
    Tataru, D. Unique continuation for solutions to PDE’s; between Hormander’s Theorem and Holmgren’s Theorem. Comm. Part. Diff. Equat., 20 (1995), 855–884.CrossRefzbMATHGoogle Scholar
  42. [TiA]
    Tikhonov, A.N., Arsenin, V.Ya. Solutions of ill-posed problems. Transl. from Russian, John Wiley & Sons, New York - Toronto, 1977.Google Scholar
  43. [U]
    Uhlmann, G. Developments in inverse problems since Calderon’s fundamental paper. Harmonic analysis and partial differential equations. Chicago lectures in Mathematics, (1999), 295–345.Google Scholar
  44. [U1]
    Uhlmann, G. Electrical impedance tomography and Calderon’s problem. Inverse Problems, 25 (2009), 123011.MathSciNetCrossRefzbMATHGoogle Scholar
  45. [Vi]
    Vigneras, M.-F. Varieétés riemanniennes isospectrales et non isométriques. Ann. Math., 91 (1980), 21–32.CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Victor Isakov
    • 1
  1. 1.Department of Mathematics and StatisticsWichita State UniversityWichitaUSA

Personalised recommendations