Skip to main content
SpringerLink
Log in

Minimization of the Tikhonov functional in Banach spaces smooth and convex of power type by steepest descent in the dual

  • Published:
Computational Optimization and Applications Aims and scope Submit manuscript

Abstract

For Tikhonov functionals of the form Ψ(x)=‖Ax−y‖ r Y +α‖x‖ q X we investigate a steepest descent method in the dual of the Banach space X. We show convergence rates for the proposed method and present numerical tests.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alber, Y., Iusem, A., Solodov, M.: Minimization of nonsmooth convex functionals in Banach spaces. J. Convex Anal. 4(2), 235–255 (1997)

    MATH  MathSciNet  Google Scholar 

  2. Bonesky, T., Kazimierski, K., Maass, P., Schöpfer, F., Schuster, T.: Minimization of Tikhonov functionals in Banach spaces. Abstr. Appl. Anal. 2008, Article ID 192,679, 19 pages (2008). doi:10.1155/2008/192679

  3. Bredies, K., Lorenz, D.: Iterated hard shrinkage for minimization problems with sparsity constraints. Preprint Series of the DFG priority program 1114 (2006)

  4. Bredies, K., Lorenz, D., Maass, P.: A generalized conditional gradient method and its connection to an iterative shrinkage method. Comput. Optim. Appl. 42(2), 173–193 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bregman, L.M.: The relaxation method for finding common points of convex sets and its application to the solution of problems in convex programming. USSR Comput. Math. Math. Phys. 7, 200–217 (1967)

    Article  Google Scholar 

  6. Burger, M., Osher, S.: Convergence rates of convex variational regularization. Inverse Probl. 20(5), 1411–1420 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  7. Cioranescu, I.: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Kluwer Academic, Dordrecht (1990)

    MATH  Google Scholar 

  8. Daubechies, I., Defrise, M., Mol, C.D.: An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Commun. Pure Appl. Math. 57(11), 1413–1457 (2004)

    Article  MATH  Google Scholar 

  9. Deville, R., Godefroy, G., Zizler, V.: Smoothness and Renormings in Banach spaces. Pitman Monographs and Surveys in Pure and Applied Mathematics. Longman Scientific & Technical, Harlow (1993)

    MATH  Google Scholar 

  10. Dunn, J.C.: Rates of convergence for conditional gradient algorithms near singular and nonsingular extremals. SIAM J. Control Optim. 17, 187–211 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  11. Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. North-Holland, Amsterdam (1976)

    MATH  Google Scholar 

  12. Hanner, O.: On the uniform convexity of L p and ℓ p . Ark. Mat. 3, 239–244 (1956)

    Article  MATH  MathSciNet  Google Scholar 

  13. Hofmann, B., Kaltenbacher, B., Pöschl, C., Scherzer, O.: A convergence rates result for Tikhonov regularization in Banach spaces with non-smooth operators. Inverse Probl. 23(3), 987–1010 (2007). http://stacks.iop.org/0266-5611/23/987

    Article  MATH  Google Scholar 

  14. Resmerita, E.: Regularization of ill-posed problems in Banach spaces: convergence rates. Inverse Probl. 21(4), 1303–1314 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  15. Resmerita, E., Scherzer, O.: Error estimates for non-quadratic regularization and the relation to enhancement. Inverse Probl. 22(3), 801–814 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  16. Schöpfer, F., Louis, A., Schuster, T.: Nonlinear iterative methods for linear ill-posed problems in Banach spaces. Inverse Probl. 22, 311–329 (2006)

    Article  MATH  Google Scholar 

  17. Xu, Z.B., Roach, G.: Characteristic inequalities of uniformly convex and uniformly smooth Banach spaces. J. Math. Anal. Appl. 157, 189–210 (1991)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Center for Industrial Mathematics (ZeTeM), University of Bremen, 28334, Bremen, Germany

    Kamil S. Kazimierski

Authors
  1. Kamil S. Kazimierski
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kamil S. Kazimierski.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kazimierski, K.S. Minimization of the Tikhonov functional in Banach spaces smooth and convex of power type by steepest descent in the dual. Comput Optim Appl 48, 309–324 (2011). https://doi.org/10.1007/s10589-009-9257-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-009-9257-2

Keywords

Search

Navigation