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Sparsity and ℓ 1-Regularization

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Variational Source Conditions, Quadratic Inverse Problems, Sparsity Promoting Regularization

Part of the book series: Frontiers in Mathematics ((FM))

Abstract

In this section we describe the basic ideas of sparsity promoting regularization techniques, introduce ℓ 1-regularization and briefly discuss alternative methods. In addition we have a first look at several examples which will appear again later on in the text.

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References

  1. R.E. Megginson, An Introduction to Banach Space Theory, Graduate Texts in Mathematics, vol. 183 (Springer, New York, 1998)

    Book  Google Scholar 

  2. 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)

    Book  Google Scholar 

  3. J. Flemming, Generalized Tikhonov regularization and modern convergence rate theory in Banach spaces (Shaker Verlag, Aachen, 2012)

    Google Scholar 

  4. I. Daubechies, M. Defrise, C. De Mol, Communications on Pure and Applied Mathematics 57(11), 1413 (2004)

    Article  MathSciNet  Google Scholar 

  5. J. Flemming, M. Hegland, Applicable Analysis 94, 464 (2015)

    Article  MathSciNet  Google Scholar 

  6. J. Diestel, Sequences and Series in Banach Spaces, Graduate Texts in Mathematics, vol. 92 (Springer, New York, 1984)

    Book  Google Scholar 

  7. D. Gerth, B. Hofmann, arXiv.org arXiv:1711.08642 [math.FA] (2017). http://arxiv.org/abs/1711.08642

  8. R.I. Boţ, B. Hofmann, Eurasian Journal of Mathematical and Computer Applications 1(1), 29 (2013)

    Google Scholar 

  9. E.J. Candès, J.K. Romberg, T. Tao, Communications on Pure and Applied Mathematics 59(8), 1207 (2006)

    Article  MathSciNet  Google Scholar 

  10. M. Grasmair, M. Haltmeier, O. Scherzer, Applied Mathematics and Computation 218(6), 2693 (2011)

    Article  MathSciNet  Google Scholar 

  11. K. Bredies, D.A. Lorenz, Inverse Problems 25(8), 085011 (14pp) (2009)

    Google Scholar 

  12. M. Grasmair, Journal of Mathematical Analysis and Applications 365(1), 19 (2010)

    Article  MathSciNet  Google Scholar 

  13. D.H. Chen, B. Hofmann, J. Zou, Inverse Problems 33(1), 015004 (17pp) (2017)

    Google Scholar 

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Authors and Affiliations

  1. Fakultät für Mathematik, Technische Universität Chemnitz, Chemnitz, Germany

    Jens Flemming

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  1. Jens Flemming
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Flemming, J. (2018). Sparsity and ℓ 1-Regularization. In: Variational Source Conditions, Quadratic Inverse Problems, Sparsity Promoting Regularization. Frontiers in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-95264-2_9

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