IDEAS home Printed from https://ideas.repec.org/p/zbw/sfb373/200250.html
   My bibliography  Save this paper

Adaptive wavelet Galerkin methods for linear inverse problems

Author

Listed:
  • Cohen, Albert
  • Hoffmann, Marc
  • Reiß, Markus

Abstract

We introduce and analyse numerical methods for the treatment of inverse problems, based on an adaptive wavelet Galerkin discretization. These methods combine the theoretical advantages of the wavelet-vaguelette decomposition (WVD) in terms of optimally adapting to the unknown smoothness of the solution, together with the numerical simplicity of Galerkin methods. Two strategies are proposed: the first one simply combines a thresholding algorithm on the data with a Galerkin inversion on a fixed liner space, while the second one performs the inversion through an adaptive procedure in which a smaller space adapted to the solution is iteratively constructed. For both methods, we recover the same minimax rates achieved by WVD for various function classes modeling the solution.

Suggested Citation

  • Cohen, Albert & Hoffmann, Marc & Reiß, Markus, 2002. "Adaptive wavelet Galerkin methods for linear inverse problems," SFB 373 Discussion Papers 2002,50, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
  • Handle: RePEc:zbw:sfb373:200250
    as

    Download full text from publisher

    File URL: https://www.econstor.eu/bitstream/10419/65296/1/726723541.pdf
    Download Restriction: no

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Chen, Xiaohong & Reiss, Markus, 2011. "On Rate Optimality For Ill-Posed Inverse Problems In Econometrics," Econometric Theory, Cambridge University Press, vol. 27(03), pages 497-521, June.
    2. Xiaohong Chen & Timothy Christensen, 2013. "Optimal Uniform Convergence Rates for Sieve Nonparametric Instrumental Variables Regression," Papers 1311.0412, arXiv.org.
    3. Xiaohong Chen & Timothy Christensen, 2013. "Optimal Uniform Convergence Rates for Sieve Nonparametric Instrumental Variables Regression," Cowles Foundation Discussion Papers 1923, Cowles Foundation for Research in Economics, Yale University.
    4. Marteau Clement & Loubes Jean-Michel, 2012. "Adaptive estimation for an inverse regression model with unknown operator," Statistics & Risk Modeling, De Gruyter, vol. 29(3), pages 215-242, August.
    5. Anne Vanhems & Jean-Michel Loubes, 2004. "Saturation spaces for regularization methods in inverse problems," Econometric Society 2004 North American Summer Meetings 380, Econometric Society.
    6. Xiaohong Chen & Demian Pouzo, 2008. "Estimation of nonparametric conditional moment models with possibly nonsmooth moments," CeMMAP working papers CWP12/08, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    7. Xiaohong Chen & Timothy M. Christensen, 2013. "Optimal uniform convergence rates for sieve nonparametric instrumental variables regression," CeMMAP working papers CWP56/13, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    8. Xiaohong Chen & Demian Pouzo, 2008. "Estimation of Nonparametric Conditional Moment Models with Possibly Nonsmooth Moments," Cowles Foundation Discussion Papers 1650, Cowles Foundation for Research in Economics, Yale University, revised Oct 2008.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:zbw:sfb373:200250. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (ZBW - German National Library of Economics). General contact details of provider: http://edirc.repec.org/data/sfhubde.html .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.