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

Online
ISSN
1569-3945
See all formats and pricing
More options …
Volume 26, Issue 2

Issues

On ill-posedness concepts, stable solvability and saturation

Bernd HofmannORCID iD: http://orcid.org/0000-0001-7155-7605 / Robert Plato
Published Online: 2017-12-23 | DOI: https://doi.org/10.1515/jiip-2017-0090

Abstract

We consider different concepts of well-posedness and ill-posedness and their relations for solving nonlinear and linear operator equations in Hilbert spaces. First, the concepts of Hadamard and Nashed are recalled which are appropriate for linear operator equations. For nonlinear operator equations, stable respective unstable solvability is considered, and the properties of local well-posedness and ill-posedness are investigated. Those two concepts consider stability in image space and solution space, respectively, and both seem to be appropriate concepts for nonlinear operators which are not onto and/or not, locally or globally, injective. Several example situations for nonlinear problems are considered, including the prominent autoconvolution problems and other quadratic equations in Hilbert spaces. It turns out that for linear operator equations, well-posedness and ill-posedness are global properties valid for all possible solutions, respectively. The special role of the nullspace is pointed out in this case. Finally, non-injectivity also causes differences in the saturation behavior of Tikhonov and Lavrentiev regularization of linear ill-posed equations. This is examined at the end of this study.

Keywords: Ill-posedness; stable solvability; linear and nonlinear inverse problems; operator equations; regularization; saturation

MSC 2010: 47A52; 47J06; 65J20

References

  • [1]

    J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Systems Control Found. Appl. 2, Birkhäuser, Boston, 1990. Google Scholar

  • [2]

    R. I. Boţ and B. Hofmann, Conditional stability versus ill-posedness for operator equations with monotone operators in Hilbert space, Inverse Problems 32 (2016), no. 12, Article ID 125003. Web of ScienceGoogle Scholar

  • [3]

    R. I. Boţ, B. Hofmann and P. Mathé, Regularizability of ill-posed problems and the modulus of continuity, Z. Anal. Anwend. 32 (2013), no. 3, 299–312. CrossrefWeb of ScienceGoogle Scholar

  • [4]

    S. Bürger and B. Hofmann, About a deficit in low-order convergence rates on the example of autoconvolution, Appl. Anal. 94 (2015), no. 3, 477–493. Web of ScienceCrossrefGoogle Scholar

  • [5]

    A. L. Dontchev and R. T. Rockafellar, Implicit Functions and Solution Mappings. A View from Variational Analysis, Springer Monogr. Math., Springer, Dordrecht, 2009. Google Scholar

  • [6]

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

  • [7]

    G. Fleischer and B. Hofmann, On inversion rates for the autoconvolution equation, Inverse Problems 12 (1996), no. 4, 419–435. CrossrefGoogle Scholar

  • [8]

    J. Flemming, Regularization of autoconvolution and other ill-posed quadratic equations by decomposition, J. Inverse Ill-Posed Probl. 22 (2014), no. 4, 551–567. Web of ScienceGoogle Scholar

  • [9]

    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, Chemnitz, 2017. Google Scholar

  • [10]

    J. Flemming, B. Hofmann and I. Veselić, On 1-regularization in light of Nashed’s ill-posedness concept, Comput. Methods Appl. Math. 15 (2015), no. 3, 279–289. Google Scholar

  • [11]

    D. Gerth, B. Hofmann, S. Birkholz, S. Koke and G. Steinmeyer, Regularization of an autoconvolution problem in ultrashort laser pulse characterization, Inverse Probl. Sci. Eng. 22 (2014), no. 2, 245–266. CrossrefWeb of ScienceGoogle Scholar

  • [12]

    R. Gorenflo and B. Hofmann, On autoconvolution and regularization, Inverse Problems 10 (1994), no. 2, 353–373. CrossrefGoogle Scholar

  • [13]

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

  • [14]

    J. Hadamard, Lectures on Cauchy’s Problem in Linear Partial Differential Equations, Dover Publications, New York, 1953. Google Scholar

  • [15]

    B. Hofmann, Regularization for Applied inverse and Ill-Posed Problems, Teubner Texts in Math. 85, B. G. Teubner, Leipzig, 1986. Google Scholar

  • [16]

    B. Hofmann, Ill-posedness and local ill-posedness concepts in Hilbert spaces, Optimization 48 (2000), no. 2, 219–238. CrossrefGoogle Scholar

  • [17]

    B. Hofmann and O. Scherzer, Local ill-posedness and source conditions of operator equations in Hilbert spaces, Inverse Problems 14 (1998), no. 5, 1189–1206. CrossrefGoogle Scholar

  • [18]

    V. K. Ivanov, V. V. Vasin and V. P. Tanana, Theory of Linear Ill-Posed Problems and Its Applications, 2nd ed., Inverse Ill-posed Probl. Ser., VSP, Utrecht, 2002. Google Scholar

  • [19]

    A. Kirsch and A. Rieder, Seismic tomography is locally ill-posed, Inverse Problems 30 (2014), no. 12, Article ID 125001. Web of ScienceGoogle Scholar

  • [20]

    A. Kirsch and A. Rieder, Inverse problems for abstract evolution equations with applications in electrodynamics and elasticity, Inverse Problems 32 (2016), no. 8, Article ID 085001. Web of ScienceGoogle Scholar

  • [21]

    R. Krämer and P. Mathé, Modulus of continuity of Nemytskiĭ operators with application to the problem of option pricing, J. Inverse Ill-Posed Probl. 16 (2008), no. 5, 435–461. Google Scholar

  • [22]

    D. Lorenz and N. Worliczek, Necessary conditions for variational regularization schemes, Inverse Problems 29 (2013), no. 7, Article ID 075016. Web of ScienceGoogle Scholar

  • [23]

    S. Lu and J. Flemming, Convergence rate analysis of Tikhonov regularization for nonlinear ill-posed problems with noisy operators, Inverse Problems 28 (2012), no. 10, Article ID 104003. Web of ScienceGoogle Scholar

  • [24]

    M. Z. Nashed, A new approach to classification and regularization of ill-posed operator equations, Inverse and Ill-Posed Problems (Sankt Wolfgang 1986), Notes Rep. Math. Sci. Engrg. 4, Academic Press, Boston (1987), 53–75. Google Scholar

  • [25]

    A. Neubauer, On converse and saturation results for Tikhonov regularization of linear ill-posed problems, SIAM J. Numer. Anal. 34 (1997), no. 2, 517–527. CrossrefGoogle Scholar

  • [26]

    R. Plato, Iterative and parametric methods for linear ill-posed equations, Habilitation thesis, Fachbereich Mathematik, TU Berlin, Berlin, 1995. Google Scholar

  • [27]

    R. Plato, Converse results, saturation and quasi-optimality for Lavrentiev regularization of accretive problems, SIAM J. Numer. Anal. 55 (2017), no. 3, 1315–1329. CrossrefWeb of ScienceGoogle Scholar

  • [28]

    O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier and F. Lenzen, Variational Methods in Imaging, Appl. Math. Sci. 167, Springer, New York, 2009. Google Scholar

  • [29]

    T. Schuster, B. Kaltenbacher, B. Hofmann and K. S. Kazimierski, Regularization Methods in Banach Spaces, Radon Ser. Comput. Appl. Math. 10, De Gruyter, Berlin, 2012. Google Scholar

  • [30]

    U. Tautenhahn, On the method of Lavrentiev regularization for nonlinear ill-posed problems, Inverse Problems 18 (2002), no. 1, 191–207. CrossrefGoogle Scholar

  • [31]

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

About the article

Received: 2017-09-19

Accepted: 2017-12-13

Published Online: 2017-12-23

Published in Print: 2018-04-01


Funding Source: Deutsche Forschungsgemeinschaft

Award identifier / Grant number: HO 1454/10-1

The first author gratefully acknowledges support by the German Research Foundation (DFG) under grant HO 1454/10-1.


Citation Information: Journal of Inverse and Ill-posed Problems, Volume 26, Issue 2, Pages 287–297, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219, DOI: https://doi.org/10.1515/jiip-2017-0090.

Export Citation

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

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]
Robert Plato and Bernd Hofmann
Journal of Optimization Theory and Applications, 2019, Volume 182, Number 2, Page 525
[2]
Hongwu Zhang and Xiaoju Zhang
Inverse Problems in Science and Engineering, 2018, Page 1

Comments (0)

Please log in or register to comment.
Log in