Publication Cover
Applicable Analysis
An International Journal
Volume 93, 2014 - Issue 5
173
Views
12
CrossRef citations to date
0
Altmetric
Articles

Inverse problem for a free transport equation using Carleman estimates

&
Pages 1073-1086 | Received 16 Nov 2012, Accepted 14 Jun 2013, Published online: 12 Jul 2013

References

  • Gerisch A, Kotschote M, Zacher R. Well-posedness of a quasilinear hyperbolic-parabolic system arising in mathematical biology. Nonlinear Differential Equations and Applications. 2007;14:593–624.
  • Dautray R, Lions J-L. Mathematical analysis and numerical methods for science and technology. Vol. 6. Berlin: Springer Verlag; 1993.
  • Fursikov AV, Imanuvilov OY. Controllability of evolution equations. Vol. 34, Lecture Notes Series. Seoul: Seoul National University; 1996.
  • Zhang X. Explicit observability inequalities for the wave equation with lowwer order terms by means of Carleman inequalities. SIAM Journal on Control and Optimization. 2000;39:812–834 (electronic).
  • Bukhgeim AL. Volterra equations and inverse problems. Inverse and Ill-Posed Problems Series. Utrecht: VSP; 1999.
  • Bukhgeim AL, Klibanov MV. Uniqueness in the large of a class of multidimensional inverse problems. Soviet Mathematics Doklady. 1981;17:244–247.
  • Klibanov MV. Inverse problems in the large and Carleman bounds. English Translation, Differential Equations. 1984;20:755–760.
  • Klibanov MV. Inverse problems and Carleman estimates. Inverse Problems. 1992;8:575–596.
  • Klibanov MV, Timonov AA. Carleman estimates for coefficient inverse problems and numerical applications. Inverse and Ill-Posed Problems Series. Utrecht: VSP; 2004.
  • Choulli M. Une introduction aux problèmes inverses elliptiques et paraboliques. Vol. 65, Mathématiques et Applications. Berlin: Springer-Verlag; 2009.
  • Klibanov MV. Carleman estimates for global uniqueness, stability and numerical methods for coefficient inverse problems, accepted by Journal of Inverse and Ill-Posed Problems. Available online of this journal as Ahead of Print. doi:10.1515/jiip-2012-0072
  • Yamamoto M. Carleman estimates for parabolic equations and applications. Topical Review Inverse Problems. 2009;25:123013.
  • Bal G, Ren K, Hielscher AH. Transport- and diffusion-based optical tomography in small domains: a comparative study. Applied Optics. 2007;27:6669–6679.
  • Davison B, Sykes JB. Neutron transport theory. Oxford University Press; 1957.
  • Case KM, Zweifel PF. Linear transport theory. Reading (MA): Addison-Wesley; 1967.
  • Landau LD, Lifshitz EM. Course of theoretical physics. Vol. 10, Physical Kinetics. New York: Pergamon Press; 1981.
  • Kharroubi M. Mathematical topics in neutron transport theory. Singapore: World Scientific; 1997.
  • Choulli M, Stefanov P. Inverse scattering and inverse boundary value problems for the linear Boltzmann equation. Communications in Partial Differential Equations. 1996;21:763–785.
  • Prilepko AI, Ivankov AL. Inverse problems for the determination of a coefficient and the right side of a non-stationary multivelocity transport equation with overdetermination at a point. Differential Equations. 1985;21:870–885.
  • Tamasan A. An inverse boundary value problem in two-dimensional transport. Inverse Problems. 2002;18(1):209–219.
  • Stefanov P. Inverse problems in transport theory. In: Inside out: inverse problems and applications. Vol. 47, Mathematical Sciences Research Institute Publications. Cambridge: Cambridge University Press; 2003. p. 111–131.
  • Bal G, Jolivet A. Time-dependent angularly averaged inverse transport. Inverse Problems. 2009;25:075010.
  • Bal G, Jolivet A. Stability for time-dependent inverse transport. SIAM Journal on Mathematical Analysis. 2010;42:679–700.
  • Bal G. Inverse transport theory and applications. Inverse Problems. 2009;25:053001, 48 pp.
  • Klibanov MV, Pamyatnykh SE. Global uniqueness for a coefficient inverse problem for the non-stationary transport equation via Carleman estimate. Journal of Mathematical Analysis and Applications. 2008;343:352–365.
  • Machida M, Yamamoto M. The Lipschitz stability for a coefficient inverse problem for the radiative transport equation; 2012. Available from: http://arxiv.org/abs/1212.6730
  • Klibanov MV, Pamyatnykh SE. Lipschitz stability of a non-standard problem for the non-stationary transport equation via Carleman estimate. Inverse Problems. 2006;22:881–890.
  • Klibanov MV, Yamamoto M. Exact controlability for the non stationary transport equation. SIAM Journal on Control and Optimization. 2007;46:2071–2195.
  • Baudouin L, de Buhan M, Ervedoza S. Global Carleman estimates for waves and application. Preprint. Available from: http://hal.archives-ouvertes.fr/hal-00633562/fr/
  • Baudouin L, Puel JP. Uniqueness and stability in an inverse problem for the Schrödinger equation. Inverse Problems. 2002;18:1537–1554.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.