Can the nonlocal characterization of sobolev spaces by bourgain et al. be useful for solving variational problems?

SIAM Journal on Numerical Analysis

Volumn 47, Issue 2, 2009, Pages 844-860

  Aubert, Gilles ^a   Kornprobst, Pierre ^b  

^a UNIVERSITÉ DE NICE SOPHIA ANTIPOLIS   (France)

^b INRIA   (France)

Author keywords

BV; Calculus of variation; Functional analysis; Integral approximations; Nonlocal formulations; Sobolev spaces; Variational approach

Indexed keywords

FUNCTIONAL ANALYSIS;

BV; CALCULUS OF VARIATION; INTEGRAL APPROXIMATIONS; NONLOCAL FORMULATIONS; SOBOLEV SPACES; VARIATIONAL APPROACH;

AEROSPACE VEHICLES;

EID: 60049086524     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/070696751     Document Type: Article

References (19)

1. L. AMBROSIO, N. FUSCO, AND D. PALLARA, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Math. Monogr., Oxford University Press, New York, 2000.
2. G. AUBERT AND P. KORNPROBST, Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations, 2nd ed., Appl. Math. Sci. 147, Springer-Verlag, New York, 2006.
3. M. BERTERO AND P. BOCCACCI, Introduction to Inverse Problems in Imaging, Institute of Physics Publishing, Bristol, 1998.
4. D. P. BERTSEKAS, Nonlinear Programming, 2nd ed., Athena Scientific, Nashua, NH, 1999.
5. J. BOURGAIN, H. BREZIS, AND P. MIRONESCU, Another look at Sobolev spaces, in Optimal Control and Partial Differential Equations, J. L. Menaldi, E. Rofman, and A. Sulem, eds., IOS Press, 2001, pp. 439-455.
6. A. BUADES, B. COLL, AND J. M. MOREL, Neighborhood filters and PDE's, Numer. Math, 105 (2006), pp. 1-34.
7. A. CHAMBOLLE, An algorithm for total variation minimization and applications, J. Math. Imaging Vision, 20 (2004), pp. 89-97.
8. T. F. CHAN, G. H. GOLUB, AND P. MULET, A nonlinear primal-dual method for total variation-based image restoration, SIAM J. Sci. Comput., 20 (1999), pp. 1964-1977.
9. E. DARVE, Méthodes multipôles rapides: Résolution des équations de Maxwell par formulations intégrales, Ph.D. thesis, Université de Paris 6, 1999.
10. L. C. EVANS AND R. F. GARIEPY, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, FL, 1992.
11. G. GILBOA, J. DARBON, S. OSHER, AND T. F. CHAN, Nonlocal Convex Functional for Image Regularization, Technical Report 06-57, UCLA CAM Report, UCLA, Los Angeles, CA, 2006.
12. G. GLIBOA AND S. OSHER, Nonlocal linear image regularization and supervised segmentation, Multiscale Model. Simul., 6 (2007), pp. 595-630.
13. M. HINTERMÜLLER AND K. KUNISCH, Total bounded variation regularization as a, bilaterally constrained optimization problem, SIAM J. Appl. Math., 64 (2004), pp. 1311-1333.
14. M. HINTERMÜLLER AND G. STADLER, An infeasible primal-dual algorithm for total bounded variation-based inf-convolution-type image restoration, SIAM J. Sci. Comput., 28 (2006), pp. 1-23.
15. A. KIRSCH, An Introduction to the Mathematical Theory of Inverse Problems, Appl. Math. Sci. 120, Springer-Verlag, New York, 1996.
16. A. PONCE, A new approach to Sobolev spaces and connections to γ-convergence, Calc. Var. Partial Differential Equations, 19 (2004), pp. 229-255.
17. N. Z. SHOR., Minimization Methods for Nondifferentiable Functions, Springer Ser. Comput. Math. 3, Springer-Verlag, Berlin, 1985.
18. C. R. VOGEL AND M. E. OMAN, Fast, robust total variation-based, reconstruction of noisy, blurred images, IEEE Trans. Image Process., 7 (1998), pp. 813-824.
19. P. WEISS, L. BLANC-FÉRAUD, AND G. AUBERT, Efficient schemes for total variation minimization under constraints in image processing, SIAM J. Sci. Comput., to appear.