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Adaptive wavelet Galerkin methods for linear inverse problems

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  • 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
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    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(3), pages 497-521, June.
    2. 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.
    3. 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.
    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. Pawan Gupta & Marianna Pensky, 2018. "Solution of Linear Ill-Posed Problems Using Random Dictionaries," Sankhya B: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 80(1), pages 178-193, May.
    6. Anne Vanhems & Jean-Michel Loubes, 2004. "Saturation spaces for regularization methods in inverse problems," Econometric Society 2004 North American Summer Meetings 380, Econometric Society.

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