Abstract
Our focus is on the stable approximate solution of linear operator equations based on noisy data by using ℓ 1-regularization as a sparsity-enforcing version of Tikhonov regularization. We summarize recent results on situations where the sparsity of the solution slightly fails. In particular, we show how the recently established theory for weak*-to-weak continuous linear forward operators can be extended to the case of weak*-to-weak* continuity. This might be of interest when the image space is non-reflexive. We discuss existence, stability and convergence of regularized solutions. For injective operators, we will formulate convergence rates by exploiting variational source conditions. The typical rate function obtained under an ill-posed operator is strictly concave and the degree of failure of the solution sparsity has an impact on its behavior. Linear convergence rates just occur in the two borderline cases of proper sparsity, where the solutions belong to ℓ 0, and of well-posedness. For an exemplary operator, we demonstrate that the technical properties used in our theory can be verified in practice. In the last section, we briefly mention the difficult case of oversmoothing regularization where x † does not belong to ℓ 1.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
S.W. Anzengruber, B. Hofmann, R. Ramlau, On the interplay of basis smoothness and specific range conditions occurring in sparsity regularization. Inverse Prob. 29, 125002, 21 (2013)
S.W. Anzengruber, B. Hofmann, P. Mathé, Regularization properties of the sequential discrepancy principle for Tikhonov regularization in Banach spaces. Appl. Anal. 93, 1382–1400 (2014)
R.I. Boţ, B. Hofmann, The impact of a curious type of smoothness conditions on convergence rates in ℓ 1-regularization. Eurasian J. Math. Comput. Appl. 1, 29–40 (2013)
K. Bredies, D.A. Lorenz, Regularization with non-convex separable constraints. Inverse Prob. 25, 085011, 14 (2009)
M. Burger, J. Flemming, B. Hofmann, Convergence rates in ℓ 1-regularization if the sparsity assumption fails. Inverse Prob. 29, 025013, 16 (2013)
D. Chen, B. Hofmann, J. Zou, Elastic-net regularization versus ℓ 1-regularization for linear inverse problems with quasi-sparse solutions. Inverse Prob. 33(1), 015004, 17 (2017)
J. Cheng, M. Yamamoto, One new strategy for a priori choice of regularizing parameters in Tikhonov’s regularization. Inverse Prob. 16(4), L31–L38 (2000)
I. Daubechies, M. Defrise, C. De Mol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Comm. Pure Appl. Math. 57(11), 1413–1457 (2004)
J. Flemming, Convergence rates for ℓ 1-regularization without injectivity-type assumptions. Inverse Prob. 32(9), 095001, 19 (2016)
J. Flemming, Quadratic inverse problems and sparsity promoting regularization – two subjects, some links between them and an application in laser optics. Habilitation thesis, Technische Universität Chemnitz, Germany, 2017
J. Flemming, D. Gerth, Injectivity and weak∗-to-weak continuity suffice for convergence rates in ℓ 1-regularization. J. Inv. Ill-Posed Prob. (2017, Published ahead of print). https://doi.org/10.1515/jiip-2017-0008
J. Flemming, M. Hegland, Convergence rates in ℓ 1-regularization when the basis is not smooth enough. Appl. Anal. 94, 464–476 (2015)
J. Flemming, B. Hofmann, I. Veselić, On ℓ 1-regularization in light of Nashed’s ill-posedness concept. Comput. Methods Appl. Math. 15(3), 279–289 (2015)
J. Flemming, B. Hofmann, I. Veselić, A unified approach to convergence rates for ℓ 1-regularization and lacking sparsity. J. Inverse Ill-Posed Prob. 24(2), 139–148 (2016)
D. Gerth, Tikhonov regularization with oversmoothing penalties. Preprint (2016). https://www.tu-chemnitz.de/mathematik/preprint/2016/PREPRINTtextunderscore08.pdf
D. Gerth, Convergence rates for ℓ 1-regularization when the exact solution is not in ℓ 1. Article in preparation (2017)
D. Gerth, Convergence rates for ℓ 1-regularization without the help of a variational inequality. Electron. Trans. Numer. Anal. 46, 233–244 (2017)
S. Goldberg, E. Thorp, On some open questions concerning strictly singular operators. Proc. Am. Math. Soc. 14, 334–336 (1963)
M. Grasmair, Well-posedness and convergence rates for sparse regularization with sublinear l q penalty term. Inverse Prob. Imaging 3(3), 383–387 (2009)
M. Grasmair, Generalized Bregman distances and convergence rates for non-convex regularization methods. Inverse Prob. 26, 115014, 16 (2010)
M. Grasmair, M. Haltmeier, O. Scherzer, Sparse regularization with l q penalty term. Inverse Prob. 24(5), 055020, 13 (2008)
M. Grasmair, M. Haltmeier, O. Scherzer, Necessary and sufficient conditions for linear convergence of ℓ 1-regularization. Comm. Pure Appl. Math. 64, 161–182 (2011)
B. Hofmann, P. Mathé, Parameter choice in Banach space regularization under variational inequalities. Inverse Prob. 28, 104006, 17 (2012)
B. Hofmann, B. Kaltenbacher, C. Pöschl, O. Scherzer, A convergence rates result for Tikhonov regularization in Banach spaces with non-smooth operators. Inverse Prob. 23, 987–1010 (2007)
D.A. Lorenz, Convergence rates and source conditions for Tikhonov regularization with sparsity constraints. J. Inverse Ill-Posed Prob. 16, 463–478 (2008)
R.E. Megginson, An Introduction to Banach Space Theory. Graduate Texts in Mathematics, vol. 183 (Springer, New York, 1998)
M.Z. Nashed, A new approach to classification and regularization of ill-posed operator equations, in Inverse and Ill-Posed Problems (Sankt Wolfgang, 1986). Notes and Reports in Mathematics in Science and Engineering, vol. 4 (Academic, Boston, MA, 1987), pp. 53–75
R. Ramlau, Regularization properties of Tikhonov regularization with sparsity constraints. Electron. Trans. Numer. Anal. 30, 54–74 (2008)
R. Ramlau, E. Resmerita, Convergence rates for regularization with sparsity constraints. Electron. Trans. Numer. Anal. 37, 87–104 (2010)
W. Rudin, Functional Analysis. International Series in Pure and Applied Mathematics, 2nd edn. (McGraw-Hill, New York, 1991)
O. Scherzer (Ed.), Handbook of Mathematical Methods in Imaging, 2nd edn. (Springer, New York, 2011)
O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier, F. Lenzen, Variational Methods in Imaging. Applied Mathematical Sciences, vol. 167 (Springer, New York, 2009)
T. Schuster, B. Kaltenbacher, B. Hofmann, K.S. Kazimierski, Regularization Methods in Banach Spaces. Radon Series on Computational and Applied Mathematics, vol. 10 (Walter de Gruyter, Berlin/Boston, 2012)
Acknowledgements
The research of the authors was financially supported by Deutsche Forschungsgemeinschaft (DFG-grant HO 1454/10-1).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG
About this chapter
Cite this chapter
Gerth, D., Hofmann, B. (2018). On ℓ 1-Regularization Under Continuity of the Forward Operator in Weaker Topologies. In: Hofmann, B., Leitão, A., Zubelli, J. (eds) New Trends in Parameter Identification for Mathematical Models. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-70824-9_4
Download citation
DOI: https://doi.org/10.1007/978-3-319-70824-9_4
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-70823-2
Online ISBN: 978-3-319-70824-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)