Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.

IMPACT FACTOR 2018: 0.881
5-year IMPACT FACTOR: 1.170

CiteScore 2018: 0.91

SCImago Journal Rank (SJR) 2018: 0.430
Source Normalized Impact per Paper (SNIP) 2018: 0.969

Mathematical Citation Quotient (MCQ) 2018: 0.66

See all formats and pricing
More options …
Volume 26, Issue 2


A parameter choice strategy for a multilevel augmentation method in iterated Lavrentiev regularization

Chunmei Zeng / Xingjun Luo
  • Corresponding author
  • School of Mathematics and Computer Science, Gannan Normal University, Ganzhou 341000, P. R. China
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Suhua Yang / Fanchun Li
  • Department of Electronic Information Engineering, Jiangxi Vocational College of Applied Technology, Ganzhou 341000, P. R. China
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2018-01-10 | DOI: https://doi.org/10.1515/jiip-2017-0006


In this paper we apply the multilevel augmentation method to solve an ill-posed integral equation via the iterated Lavrentiev regularization. This method leads to fast solutions of discrete iterated Lavrentiev regularization. The convergence rates of the iterated Lavrentiev regularization are achieved by using a certain parameter choice strategy. Finally, numerical experiments are given to illustrate the efficiency of the method.

Keywords: Ill-posed integral equations; multilevel augmentation methods; a parameter choice strategy; iterated Lavrentiev regularization

MSC 2010: 65J20; 65R20


  • [1]

    Z. Chen, S. Cheng and H. Yang, Fast multilevel augmentation methods with compression technique for solving ill-posed integral equations, J. Integral Equations Appl. 23 (2011), no. 1, 39–70. Web of ScienceCrossrefGoogle Scholar

  • [2]

    Z. Chen, C. A. Micchelli and Y. Xu, The Petrov–Galerkin method for second kind integral equations. II. Multiwavelet schemes, Adv. Comput. Math. 7 (1997), no. 3, 199–233. CrossrefGoogle Scholar

  • [3]

    Z. Chen, C. A. Micchelli and Y. Xu, A multilevel method for solving operator equations, J. Math. Anal. Appl. 262 (2001), no. 2, 688–699. CrossrefGoogle Scholar

  • [4]

    Z. Chen, C. A. Micchelli and Y. Xu, Multiscale Methods for Fredholm Integral Equations, Cambridge Monogr. Appl. Comput. Math. 28, Cambridge University Press, Cambridge, 2015. Google Scholar

  • [5]

    Z. Chen, B. Wu and Y. Xu, Multilevel augmentation methods for solving operator equations, Numer. Math. J. Chinese Univ. (English Ser.) 14 (2005), no. 1, 31–55. Google Scholar

  • [6]

    Z. Chen, Y. Xu and H. Yang, A multilevel augmentation method for solving ill-posed operator equations, Inverse Problems 22 (2006), no. 1, 155–174. CrossrefGoogle Scholar

  • [7]

    V. Dicken and P. Maass, Wavelet–Galerkin methods for ill-posed problems, J. Inverse Ill-Posed Probl. 4 (1996), no. 3, 203–221. Google Scholar

  • [8]

    H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Math. Appl. 375, Kluwer Academic Publishers, Dordrecht, 1996. Google Scholar

  • [9]

    C. W. Groetsch, On the asymptotic order of accuracy of Tikhonov regularization, J. Optim. Theory Appl. 41 (1983), no. 2, 293–298. CrossrefGoogle Scholar

  • [10]

    C. W. Groetsch, The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind, Res. Notes Math. 105, Pitman, Boston, 1984. Google Scholar

  • [11]

    U. Hämarik, On the parameter choice in the regularized Ritz–Galerkin method, Eesti Tead. Akad. Toimetised Füüs. Mat. 42 (1993), no. 2, 133–143. Google Scholar

  • [12]

    U. Hämarik, R. Palm and T. Raus, Use of extrapolation in regularization methods, J. Inverse Ill-Posed Probl. 15 (2007), no. 3, 277–294. CrossrefGoogle Scholar

  • [13]

    U. Hämarik and T. Raus, On the choice of the regularization parameter in the case of the approximately given noise level of data, Numerical Mathematics and Advanced Applications, Springer, Berlin (2004), 400–409. Google Scholar

  • [14]

    U. Hämarik, T. Raus and R. Palm, On the analog of the monotone error rule for parameter choice in the (iterated) Lavrentiev regularization, Comput. Methods Appl. Math. 8 (2008), no. 3, 237–252. Google Scholar

  • [15]

    H. Harbrecht, S. Pereverzev and R. Schneider, Self-regularization by projection for noisy pseudodifferential equations of negative order, Numer. Math. 95 (2003), no. 1, 123–143. CrossrefGoogle Scholar

  • [16]

    B. Kaltenbacher, On the regularizing properties of a full multigrid method for ill-posed problems, Inverse Problems 17 (2001), no. 4, 767–788. CrossrefGoogle Scholar

  • [17]

    R. Kress, Linear Integral Equations, Appl. Math. Sci. 82, Springer, Berlin, 1989. Google Scholar

  • [18]

    X. Luo, F. Li and S. Yang, A posteriori parameter choice strategy for fast multiscale methods solving ill-posed integral equations, Adv. Comput. Math. 36 (2012), no. 2, 299–314. CrossrefWeb of ScienceGoogle Scholar

  • [19]

    X. J. Luo, F. C. Li and S. H. Yang, A fast iterative method for solving ill-posed integral equations based on an optimal projection method, Math. Numer. Sin. 33 (2011), no. 1, 1–14. Google Scholar

  • [20]

    P. Mathé and S. V. Pereverzev, Discretization strategy for linear ill-posed problems in variable Hilbert scales, Inverse Problems 19 (2003), no. 6, 1263–1277. CrossrefGoogle Scholar

  • [21]

    C. A. Micchelli and Y. Xu, Using the matrix refinement equation for the construction of wavelets on invariant sets, Appl. Comput. Harmon. Anal. 1 (1994), no. 4, 391–401. CrossrefGoogle Scholar

  • [22]

    C. A. Micchelli and Y. Xu, Reconstruction and decomposition algorithms for biorthogonal multiwavelets, Multidimens. Syst. Signal Process. 8 (1997), no. 1–2, 31–69. CrossrefGoogle Scholar

  • [23]

    S. V. Pereverzev and S. G. Solodkiĭ, On an approach to the discretization of the method of M. M. Lavrent’ev, Ukraïn. Mat. Zh. 48 (1996), no. 2, 212–219. Google Scholar

  • [24]

    S. V. Pereverzev and S. G. Solodkiĭ, Optimal discretization of ill-posed problems, Ukraïn. Mat. Zh. 52 (2000), no. 1, 106–121. Google Scholar

  • [25]

    R. Plato, The Galerkin scheme for Lavrentiev’s m-times iterated method to solve linear accretive Volterra integral equations of the first kind, BIT 37 (1997), no. 2, 404–423. CrossrefGoogle Scholar

  • [26]

    R. Plato, The Lavrentiev-regularized Galerkin method for linear accretive ill-posed problems, Matimyás Mat. 21 (1998), 57–66. Google Scholar

  • [27]

    R. Plato and G. Vainikko, On the regularization of the Ritz–Galerkin method for solving ill-posed problems, Tartu Riikl. Ül. Toimetised (1989), no. 863, 3–18. Google Scholar

  • [28]

    R. Plato and G. Vainikko, On the regularization of projection methods for solving ill-posed problems, Numer. Math. 57 (1990), no. 1, 63–79. CrossrefGoogle Scholar

  • [29]

    T. Raus, About discrepancy principle for solving ill-posed problems (in Russian), Acta Comment. Univ. Tartu. Math. 672 (1984), 16–26. Google Scholar

  • [30]

    T. Raus, The principle of the residual in the solution of ill-posed problems, Tartu Riikl. Ül. Toimetised (1984), no. 672, 16–26. Google Scholar

  • [31]

    S. G. Solodkiĭ, An economic approach to discretization of M. M. Lavrent’ev’s method, Sib. Math. J. 38 (1997), no. 2, 342–349. CrossrefGoogle Scholar

  • [32]

    S. G. Solodky and E. V. Lebedeva, On reduction of informational expenses in solving ill-posed problems with not exactly given input data, J. Inverse Ill-Posed Probl. 16 (2008), no. 2, 195–207. Web of ScienceGoogle Scholar

  • [33]

    A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems, V. H. Winston & Sons, Washington, 1977. Google Scholar

  • [34]

    G. M. Vainikko and A. Y. Veretennikov, Iterational Procedures in Ill-Posed Problems, Wiley, New York, 1985. Google Scholar

About the article

Received: 2017-01-09

Revised: 2017-11-21

Accepted: 2017-11-22

Published Online: 2018-01-10

Published in Print: 2018-04-01

Funding Source: National Natural Science Foundation of China

Award identifier / Grant number: 11761010

Award identifier / Grant number: 11361005

Award identifier / Grant number: 61502107

Award identifier / Grant number: 11661008

Funding Source: Natural Science Foundation of Jiangxi Province

Award identifier / Grant number: 20151BAB201011

Award identifier / Grant number: 20161BAB202069

Supported in part by the Natural Science Foundation of China under the grants 11761010, 11361005, 61502107 and 11661008, and the Natural Science Foundation of Jiangxi Province under the grants 20151BAB201011 and 20161BAB202069.

Citation Information: Journal of Inverse and Ill-posed Problems, Volume 26, Issue 2, Pages 153–170, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219, DOI: https://doi.org/10.1515/jiip-2017-0006.

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Comments (0)

Please log in or register to comment.
Log in