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Free Discontinuities

  • Sören BartelsEmail author
Chapter
Part of the Springer Series in Computational Mathematics book series (SSCM, volume 47)

Abstract

The mathematical description of problems involving discontinuities requires using function spaces that extend the concept of weak derivatives. The gradients of functions of bounded variation are certain measures and the functions may jump across lower-dimensional subsets. The properties of this function space enable the mathematical modeling of fracture and crack formation of materials within the framework of the calculus of variations. Qualitatively, similar model problems arise in image processing to formulate denoising or segmentation of an image. Convergence, error estimation, iterative solution, and implementation of finite element discretizations of these model problems are investigated in this chapter.

References

  1. 1.
    Ambrosio, L.: Existence theory for a new class of variational problems. Arch. Ration. Mech. Anal. 111(4), 291–322 (1990). http://dx.doi.org/10.1007/BF00376024
  2. 2.
    Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs, The Clarendon Press. Oxford University Press, New York (2000)Google Scholar
  3. 3.
    Ambrosio, L., Tortorelli, V.M.: Approximation of functionals depending on jumps by elliptic functionals via \(\varGamma \)-convergence. Commun. Pure Appl. Math. 43(8), 999–1036 (1990). http://dx.doi.org/10.1002/cpa.3160430805
  4. 4.
    Attouch, H., Buttazzo, G., Michaille, G.: Variational Analysis in Sobolev and BV Spaces. MPS/SIAM Series on Optimization, vol. 6. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2006)Google Scholar
  5. 5.
    Aubert, G.: Mathematical Problems in Image Processing. Applied Mathematical Sciences, vol. 147, 2nd edn. Springer, New York (2006)Google Scholar
  6. 6.
    Bartels, S.: Total variation minimization with finite elements: convergence and iterative solution. SIAM J. Numer. Anal. 50(3), 1162–1180 (2012). http://dx.doi.org/10.1137/11083277X
  7. 7.
    Bartels, S.: Broken Sobolev space iteration for total variation regularized minimization problems (2013). PreprintGoogle Scholar
  8. 8.
    Bartels, S., Nochetto, R.H., Salgado, A.J.: Discrete total variation flows without regularization. SIAM J. Numer. Anal. 52(1), 363–385 (2014). http://dx.doi.org/10.1137/120901544
  9. 9.
    Braides, A.: Approximation of Free-Discontinuity Problems. Lecture Notes in Mathematics, vol. 1694. Springer, Berlin (1998)Google Scholar
  10. 10.
    Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis. 40(1), 120–145 (2011). http://dx.doi.org/10.1007/s10851-010-0251-1
  11. 11.
    Ekeland, I., Témam, R.: Convex Analysis and Variational Problems. Classics in Applied Mathematics, vol. 28, English edn. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1999). http://dx.doi.org/10.1137/1.9781611971088
  12. 12.
    Hintermüller, M., Kunisch, K.: Total bounded variation regularization as a bilaterally constrained optimization problem. SIAM J. Appl. Math. 64(4), 1311–1333 (2004). http://dx.doi.org/10.1137/S0036139903422784
  13. 13.
    Kawohl, B.: From Mumford-Shah to Perona-Malik in image processing. Math. Methods Appl. Sci. 27(15), 1803–1814 (2004). http://dx.doi.org/10.1002/mma.564
  14. 14.
    Rockafellar, R.T.: Convex Analysis. Princeton Mathematical Series, vol. 28. Princeton University Press, Princeton (1970)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Abteilung für Angewandte MathematikAlbert-Ludwigs-Universität FreiburgFreiburgGermany

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