Advanced search
Publication Cover
Inverse Problems in Science and Engineering Volume 23, 2015 - Issue 2
Submit an article Journal homepage
343
Views
3
CrossRef citations to date
0
Altmetric
Articles

Spectral method for ill-posed problems based on the balancing principle

Xiangtuan XiongDepartment of Mathematics, Northwest Normal University, Lanzhou, China.Correspondence[email protected]
,
Xiaoyan FanDepartment of Mathematics, Northwest Normal University, Lanzhou, China.
&
Ming LiCollege of Mathematics, Taiyuan University of Technology, Taiyuan, China.
Pages 292-306 | Received 08 May 2013, Accepted 10 Feb 2014, Published online: 20 Mar 2014

References

  • Magnanini R, Papi G. An inverse problem for the Helmholtz equation. Inverse Probl. 1985;1:357–370.
  • Sondhi M. Reconstruction of objects from their sound-diffraction patterns. J. Acoust. Soc. Am. 1969;46:1158–1164.
  • Bertero M, De CM. Stability problems in inverse diffraction. IEEE Trans. Antenn. Propag. 1981;AP-29:368–372.
  • Santosa F. A level-set approach for inverse problems involving obstacles. Control Optim. Cal. Var. 1996;1:17–33.
  • Carasso AS. Overcoming Hölder continuity in ill-posed continuation problems. SIAM J. Numer. Anal. 1994;31:1535–1557.
  • Xiong XT, Wang JX, Li M. An optimal method for fractional heat conduction problem backward in time. Appl. Anal. 2012;91:823–840.
  • Fu CL, Zhang YX, Cheng H, Ma YJ. The a posteriori Fourier method for solving ill-posed problems. Inverse Probl. 2012;28:095002. 26 p.
  • Engl HW, Hanke M, Neubauer A. Regularization of inverse problems. Dordrecht: Kluwer Academic Publisher; 1996.
  • Xiong XT, Fu CL. Two approximate methods of a Cauchy problem for the Helmholtz equation. Comput. Appl. Math. 2007;26:285–307.
  • Reginska T, Tautenhahn U. Conditional stability estimates and regularization with applications to Cauchy problems for the Helmholtz equation. Numer. Funct. Anal. Optim. 2009;30:1065–1097.
  • Pereverzev S, Shock E. On the adaptive selection of the parameter in regularization of ill-posed problems. SIAM J. Numer. Anal. 2006;75:2060–2076.
  • Xiong XT, Zhao XC, Wang JX. Spectral Galerkin method and its application to a Cauchy problem of Helmholtz equation. Numer. Algor. 2013;63:691–711.
  • Xiong XT, Fu CL, Qian Z. On three spectral regularization methods for a backward heat conduction problem. J. Korean Math. Soc. 2007;44:1281–1290.
  • Nair MT. Linear operator equations: approximation and regularization. Singapore: World Scientific; 2009.
  • Tautenhahn U. Optimality for linear ill-posed problems under general source conditions. Numer. Funct. Anal. Optim. 1998;19:377–398.
  • Hào DN. Methods for inverse heat conduction problems. Frankfurt am Main, New York (NY): Peter Lang; 1998.
  • Kaipio JP, Somersalo E. Statistical and computational inverse problems. New York (NY): Springer, Applied Mathematical Sciences; 2005.

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:

Order Reprints Request Corporate Permissions

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:

Request Academic Permissions

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.

Related research

People also read lists articles that other readers of this article have read.

Recommended articles lists articles that we recommend and is powered by our AI driven recommendation engine.

Cited by lists all citing articles based on Crossref citations.
Articles with the Crossref icon will open in a new tab.