Analysis of total variation flow and its finite element approximations

Mathematical Modelling and Numerical Analysis

Volumn 37, Issue 3, 2003, Pages 533-556

  Feng, Xiaobing ^a   Prohl, Andreas ^b  

^a UNIVERSITY OF TENNESSEE   (United States)

^b ETH ZURICH   (Switzerland)

Author keywords

Bounded variation; Equations of prescribed mean curvature and minimal surface; Finite element method; Fully discrete scheme; Gradient flow; Variational inequality

Indexed keywords

EID: 0038041043     PISSN: 0764583X     EISSN: None     Source Type: Journal    
DOI: 10.1051/m2an:2003041     Document Type: Article

References (26)

1. L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems. The Clarendon Press Oxford University Press, New York (2000).
2. F. Andreu, C. Ballester, V. Caselles and J.M. Mazón, The Dirichlet problem for the total variation flow. J. Funct. Anal. 180 (2001) 347-403.
3. F. Andreu, C. Ballester, V. Caselles and J.M. Mazón, Minimizing total variation flow. Differential Integral Equations 14 (2001) 321-360.
4. F. Andreu, V. Caselles, J.I. Díaz and J.M. Mazón, Some qualitative properties for the total variation flow. J. Funct. Anal. 188 (2002) 516-547.
5. N. J. Differential Equations (accepted).
6. S.C. Brenner and L.R. Scott, The mathematical theory of finite element methods. Springer-Verlag, New York, 2nd ed. (2002).
7. H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland Publishing Co., Amsterdam, North-Holland Math. Stud., No. 5. Notas de Matemática (50) (1973).
8. E. Casas, K. Kunisch and C. Pola, Regularization by functions of bounded variation and applications to image enhancement. Appl. Math. Optim. 40 (1999) 229-257.
9. A. Chambolle and P.-L. Lions, Image recovery via total variation minimization and related problems. Numer. Math. 76 (1997) 167-188.
10. T. Chan and J. Shen, On the role of the BV image model in image restoration. Tech. Report CAM 02-14, Department of Mathematics, UCLA (2002).
11. 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) 1964-1977 (electronic).
12. P.G. Ciarlet, The finite element method for elliptic problems. North-Holland Publishing Co., Amsterdam, Stud. Math. Appl. 4 (1978).
13. M.G. Crandall and T.M. Liggett, Generation of semi-groups of nonlinear transformations on general Banach spaces. Amer. J. Math. 93 (1971) 265-298.
14. D.C. Dobson and C.R. Vogel, Convergence of an iterative method for total variation denoising. SIAM J. Numer. Anal. 34 (1997) 1779-1791.
15. C. Gerhardt, Boundary value problems for surfaces of prescribed mean curvature. J. Math. Pures Appl. 58 (1979) 75-109.
16. C. Gerhardt, Evolutionary surfaces of prescribed mean curvature. J. Differential Equations 36 (1980) 139-172.
17. D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order. Springer-Verlag, Berlin (2001). Reprint of the 1998 ed.
18. E. Giusti, Minimal surfaces and functions of bounded variation. Birkhäuser Verlag, Basel (1984).
19. R. Hardt and X. Zhou, An evolution problem for linear growth functionals. Comm. Partial Differential Equations 19 (1994) 1879-1907.
20. C. Johnson and V. Thomée, Error estimates for a finite element approximation of a minimal surface. Math. Comp. 29 (1975) 343-349.
21. A. Lichnewsky and R. Temam, Pseudosolutions of the time-dependent minimal surface problem. J. Differential Equations 30 (1978) 340-364.
22. J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod (1969).
23. R. Rannacher, Some asymptotic error estimates for finite element approximation of minimal surfaces. RAIRO Anal. Numér. 11 (1977) 181-196.
24. L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms. Phys. D 60 (1992) 259-268.
25. p(0, T; B). Ann. Mat. Pura Appl. 146 (1987) 65-96.
26. M. Struwe, Applications to nonlinear partial differential equations and Hamiltonian systems, in Variational methods. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics (Results in Mathematics and Related Areas, 3rd Series. A Series of Modern Surveys in Mathematics), Vol. 34. Springer-Verlag, Berlin, 3rd ed. (2000).