Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Error control and adaptivity for a variational model problem defined on functions of bounded variation


Author: Sören Bartels
Journal: Math. Comp. 84 (2015), 1217-1240
MSC (2010): Primary 65N30, 65N50
DOI: https://doi.org/10.1090/S0025-5718-2014-02893-7
Published electronically: October 23, 2014
MathSciNet review: 3315506
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We derive a fully computable, optimal a posteriori error estimate for the finite element approximation of a total variation regularized model problem and devise an adaptive refinement strategy. Numerical experiments reveal a significant improvement over related approximations on uniformly refined triangulations.


References [Enhancements On Off] (What's this?)

  • [Bar12] Sören Bartels, Total variation minimization with finite elements: convergence and iterative solution, SIAM J. Numer. Anal. 50 (2012), no. 3, 1162-1180. MR 2970738, https://doi.org/10.1137/11083277X
  • [BL03] Pavel Bělík and Mitchell Luskin, Approximation by piecewise constant functions in a BV metric, Math. Models Methods Appl. Sci. 13 (2003), no. 3, 373-393. Dedicated to Jim Douglas, Jr. on the occasion of his 75th birthday. MR 1977632 (2004f:65178), https://doi.org/10.1142/S0218202503002556
  • [BMR12] Sören Bartels, Alexander Mielke, and Tomáš Roubíček, Quasi-static small-strain plasticity in the limit of vanishing hardening and its numerical approximation, SIAM J. Numer. Anal. 50 (2012), no. 2, 951-976. MR 2914293, https://doi.org/10.1137/100819205
  • [Bra07] Dietrich Braess, Finite Elements, 3rd ed., Cambridge University Press, Cambridge, 2007. Theory, Fast Solvers, and Applications in Elasticity Theory; Translated from the German by Larry L. Schumaker. MR 2322235 (2008b:65142)
  • [Brè67] L. M. Brègman, A relaxation method of finding a common point of convex sets and its application to the solution of problems in convex programming, Z. Vyčisl. Mat. i Mat. Fiz. 7 (1967), 620-631 (Russian). MR 0215617 (35 #6457)
  • [BRH07] M. Burger, E. Resmerita, and L. He, Error estimation for Bregman iterations and inverse scale space methods in image restoration, Computing 81 (2007), no. 2-3, 109-135. MR 2354192 (2008k:94002), https://doi.org/10.1007/s00607-007-0245-z
  • [BS08] Susanne C. Brenner and L. Ridgway Scott, The Mathematical Theory of Finite Element Methods, 3rd ed., Texts in Applied Mathematics, vol. 15, Springer, New York, 2008. MR 2373954 (2008m:65001)
  • [BS12] Sören Bartels and Patrick Schreier, Local coarsening of simplicial finite element meshes generated by bisections, BIT 52 (2012), no. 3, 559-569. MR 2965291, https://doi.org/10.1007/s10543-012-0378-0
  • [BT09] Amir Beck and Marc Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imaging Sci. 2 (2009), no. 1, 183-202. MR 2486527 (2010d:35390), https://doi.org/10.1137/080716542
  • [CB02] Carsten Carstensen and Sören Bartels, Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. I. Low order conforming, nonconforming, and mixed FEM, Math. Comp. 71 (2002), no. 239, 945-969 (electronic). MR 1898741 (2003e:65212), https://doi.org/10.1090/S0025-5718-02-01402-3
  • [Cha04] Antonin Chambolle, An algorithm for total variation minimization and applications, J. Math. Imaging Vision 20 (2004), no. 1-2, 89-97. Special issue on mathematics and image analysis. MR 2049783 (2005m:49058), https://doi.org/10.1023/B:JMIV.0000011320.81911.38
  • [CP11] Antonin Chambolle and Thomas Pock, A first-order primal-dual algorithm for convex problems with applications to imaging, J. Math. Imaging Vision 40 (2011), no. 1, 120-145. MR 2782122 (2012b:94004), https://doi.org/10.1007/s10851-010-0251-1
  • [ET99] Ivar Ekeland and Roger Témam, Convex Analysis and Variational Problems, Corrected reprint of the 1976 English edition, Classics in Applied Mathematics, vol. 28, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1999. Translated from the French. MR 1727362 (2000j:49001)
  • [HK04] M. Hintermüller and K. Kunisch, Total bounded variation regularization as a bilaterally constrained optimization problem, SIAM J. Appl. Math. 64 (2004), no. 4, 1311-1333. MR 2068672 (2005b:49062), https://doi.org/10.1137/S0036139903422784
  • [Nes07] Yu. Nesterov, Gradient methods for minimizing composite objective functions, ECore Discussion Paper 2007-96, 2007.
  • [NSV00] Ricardo H. Nochetto, Giuseppe Savaré, and Claudio Verdi, A posteriori error estimates for variable time-step discretizations of nonlinear evolution equations, Comm. Pure Appl. Math. 53 (2000), no. 5, 525-589. MR 1737503 (2000k:65142), https://doi.org/10.1002/(SICI)1097-0312(200005)53:5$ \langle $525::AID-CPA1$ \rangle $3.0.CO;2-M
  • [OBG$^+$05] Stanley Osher, Martin Burger, Donald Goldfarb, Jinjun Xu, and Wotao Yin, An iterative regularization method for total variation-based image restoration, Multiscale Model. Simul. 4 (2005), no. 2, 460-489 (electronic). MR 2162864 (2006c:49051), https://doi.org/10.1137/040605412
  • [Rep00] Sergey I. Repin, A posteriori error estimation for variational problems with uniformly convex functionals, Math. Comp. 69 (2000), no. 230, 481-500. MR 1681096 (2000i:49046), https://doi.org/10.1090/S0025-5718-99-01190-4
  • [Roc97] R. Tyrrell Rockafellar, Convex Analysis, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997. Reprint of the 1970 original; Princeton Paperbacks. MR 1451876 (97m:49001)
  • [ROF92] Leonid I. Rudin, Stanley Osher, and Emad Fatemi, Nonlinear total variation based noise removal algorithms, Physica D: Nonlinear Phenomena 60 (1992), no. 1-4, 259-268.
  • [Suq78] Pierre-Marie Suquet, Existence et Régularité des Solutions des Équations de la Plasticité, C. R. Acad. Sci. Paris Sér. A-B 286 (1978), no. 24, A1201-A1204 (French, with English summary). MR 501114 (80a:73033)
  • [Tho11] Marita Thomas, Quasistatic damage evolution with spatial BV-regularization, Discrete Contin. Dyn. Syst. Ser. S 6 (2013), no. 1, 235-255. MR 2983477, https://doi.org/10.3934/dcdss.2013.6.235
  • [WL11] Jingyue Wang and Bradley J. Lucier, Error bounds for finite-difference methods for Rudin-Osher-Fatemi image smoothing, SIAM J. Numer. Anal. 49 (2011), no. 2, 845-868. MR 2792398 (2012h:65244), https://doi.org/10.1137/090769594

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 65N30, 65N50

Retrieve articles in all journals with MSC (2010): 65N30, 65N50


Additional Information

Sören Bartels
Affiliation: Abteilung für Angewandte Mathematik, Albert-Ludwigs-Universität Freiburg, Hermann-Herder Str. 10, 79104 Freiburg i.Br., Germany
Email: bartels@mathematik.uni-freiburg.de

DOI: https://doi.org/10.1090/S0025-5718-2014-02893-7
Keywords: Total variation, functions of bounded variation, finite elements, adaptivity, error estimation
Received by editor(s): July 17, 2012
Received by editor(s) in revised form: March 11, 2013, July 24, 2013, and September 17, 2013
Published electronically: October 23, 2014
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society