\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

On Tikhonov-type regularization with approximated penalty terms

Abstract Full Text(HTML) Figure(2) Related Papers Cited by
  • In this paper, we deal with (nonlinear) ill-posed problems that are regularized by minimizing Tikhonov-type functionals. If the minimization is tedious for some penalty term $ P_0 $, we approximate it by a family of penalty terms $ ({P_\beta}) $ having nicer properties and analyze what happens as $ \beta\to 0 $.

    We investigate the discrepancy principle for the choice of the regularization parameter and apply all results to linear problems with sparsity constraints. Numerical results show that the proposed method yields good results.

    Mathematics Subject Classification: Primary: 47J06, 47A52; Secondary: 65J20.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  Exact solution and reconstructions for noise level $ \delta = 1 $%

    Figure 2.  Exact solution and reconstructions for noise level $ \delta = 10 $%

  • [1] A. B. Bakushinskii and A. Goncharsky, Iterative Methods for the Solution of Incorrect Problems, Nauka, Moscow, 1989, In Russian.
    [2] A. B. Bakushinskii and M. Y. Kokurin, Iterative Methods for Approximate Solution of Inverse Problems, vol. 577 of Mathematics and Its Applications, Springer, Dordrecht, 2004.
    [3] H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Kluwer, Dordrecht, 1996.
    [4] S. F. Gilyazov and N. L. Gol'dman, Regularization of Ill-Posed Problems by Iteration Methods, Kluwer, Dordrecht, 2000.
    [5] M. Hanke, A regularization Levenberg-Marquardt scheme, with applications to inverse groundwater filtration problems, Inverse Problems, 13 (1997), 79-95.  doi: 10.1088/0266-5611/13/1/007.
    [6] P. C. Hansen and J. S. Jørgensen, AIR tools II: Algebraic iterative reconstruction methods, improved implementation, Numer. Algorithms, 79 (2018), 107-137.  doi: 10.1007/s11075-017-0430-x.
    [7] F. Hinterer, S. Hubmer and R. Ramlau, A note on the minimization of a Tikhonov functional with $\ell^1$-penalty, Inverse Problems, 36 (2020), 074001 19pp. doi: 10.1088/1361-6420/ab89c2.
    [8] K. Ito and B. Jin, Inverse Problems: Tikhonov Theory and Algorithms, vol. 22 of Series on Applied Mathematics, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015.
    [9] B. Kaltenbacher, A. Neubauer and O. Scherzer, Iterative Regularization Methods for Nonlinear Ill-Posed Problems, no. 6 in Radon Series on Computational and Applied Mathematics, Walter de Gruyter GmbH & Co. KG, Berlin, 2008. doi: 10.1515/9783110208276.
    [10] C. C. Paige and M. A. Saunders, LSQR: An algorithm for sparse linear equations and sparse least squares, ACM Trans. Math. Software, 8 (1982), 43-71.  doi: 10.1145/355984.355989.
    [11] C. Pöschl, E. Resmerita and O. Scherzer, Discretization of variational regularization in Banach spaces, Inverse Problems, 26 (2010), 105017. doi: 10.1088/0266-5611/26/10/105017.
    [12] R. Ramlau and C. A. Zarzer, On the minimization of a Tikhonov functional with a non-convex sparsity constraint, Electron. Trans. Numer. Anal., 39 (2012), 467-505. 
    [13] O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier and F. Lenzen, Variational Methods in Imaging, vol. 167 of Applied Mathematical Sciences, Springer, New York, 2009.
    [14] T. Schuster, B. Kaltenbacher, B. Hofmann and K. S. Kazimierski, Regularization Methods In Banach Spaces, vol. 10 of Radon Series on Computational and Applied Mathematics, de Gruyter, Berlin, 2012. doi: 10.1515/9783110255720.
    [15] A. N. Tikhonov, A. S. Leonov and A. G. Yagola, Nonlinear Ill-Posed Problems, no. 14 in Vol. 1 and 2, Applied Mathematics and Mathematical Computation, Chapman & Hall, London, 1998. doi: 10.1007/978-94-017-5167-4.
  • 加载中

Figures(2)

SHARE

Article Metrics

HTML views(560) PDF downloads(355) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return