Analysis of total variation flow and its finite element approximations
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 37 (2003) no. 3, p. 533-556

We study the gradient flow for the total variation functional, which arises in image processing and geometric applications. We propose a variational inequality weak formulation for the gradient flow, and establish well-posedness of the problem by the energy method. The main idea of our approach is to exploit the relationship between the regularized gradient flow (characterized by a small positive parameter ε, and the minimal surface flow [21] and the prescribed mean curvature flow [16]. Since our approach is constructive and variational, finite element methods can be naturally applied to approximate weak solutions of the limiting gradient flow problem. We propose a fully discrete finite element method and establish convergence to the regularized gradient flow problem as h,k0, and to the total variation gradient flow problem as h,k,ε0 in general cases. Provided that the regularized gradient flow problem possesses strong solutions, which is proved possible if the datum functions are regular enough, we establish practical a priori error estimates for the fully discrete finite element solution, in particular, by focusing on the dependence of the error bounds on the regularization parameter ε. Optimal order error bounds are derived for the numerical solution under the mesh relation k=O(h 2 ). In particular, it is shown that all error bounds depend on 1 ε only in some lower polynomial order for small ε.

Classification:  35B25,  35K57,  35Q99,  65M60,  65M12
Keywords: bounded variation, gradient flow, variational inequality, equations of prescribed mean curvature and minimal surface, fully discrete scheme, finite element method
     author = {Feng, Xiaobing and Prohl, Andreas},
     title = {Analysis of total variation flow and its finite element approximations},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {37},
     number = {3},
     year = {2003},
     pages = {533-556},
     doi = {10.1051/m2an:2003041},
     zbl = {1050.35004},
     mrnumber = {1994316},
     language = {en},
     url = {}
Feng, Xiaobing; Prohl, Andreas. Analysis of total variation flow and its finite element approximations. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 37 (2003) no. 3, pp. 533-556. doi : 10.1051/m2an:2003041.

[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). | MR 1857292 | Zbl 0957.49001

[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. | Zbl 0973.35109

[3] F. Andreu, C. Ballester, V. Caselles and J.M. Mazón, Minimizing total variation flow. Differential Integral Equations 14 (2001) 321-360. | Zbl 1020.35037

[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. | Zbl 1042.35018

[5] G. Bellettini and V. Caselles, The total variation flow in 𝐑 N . J. Differential Equations (accepted). | Zbl 1036.35099

[6] S.C. Brenner and L.R. Scott, The mathematical theory of finite element methods. Springer-Verlag, New York, 2nd ed. (2002). | MR 1894376 | Zbl 0804.65101

[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). | MR 348562 | Zbl 0252.47055

[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. | Zbl 0942.49014

[9] A. Chambolle and P.-L. Lions, Image recovery via total variation minimization and related problems. Numer. Math. 76 (1997) 167-188. | Zbl 0874.68299

[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). | MR 2011710 | Zbl 1035.94501

[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). | Zbl 0929.68118

[12] P.G. Ciarlet, The finite element method for elliptic problems. North-Holland Publishing Co., Amsterdam, Stud. Math. Appl. 4 (1978). | MR 520174 | Zbl 0383.65058

[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. | Zbl 0226.47038

[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. | Zbl 0898.65034

[15] C. Gerhardt, Boundary value problems for surfaces of prescribed mean curvature. J. Math. Pures Appl. 58 (1979) 75-109. | Zbl 0413.35024

[16] C. Gerhardt, Evolutionary surfaces of prescribed mean curvature. J. Differential Equations 36 (1980) 139-172. | Zbl 0485.35053

[17] D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order. Springer-Verlag, Berlin (2001). Reprint of the 1998 ed. | MR 1814364 | Zbl 1042.35002

[18] E. Giusti, Minimal surfaces and functions of bounded variation. Birkhäuser Verlag, Basel (1984). | MR 775682 | Zbl 0545.49018

[19] R. Hardt and X. Zhou, An evolution problem for linear growth functionals. Comm. Partial Differential Equations 19 (1994) 1879-1907. | Zbl 0811.35061

[20] C. Johnson and V. Thomée, Error estimates for a finite element approximation of a minimal surface. Math. Comp. 29 (1975) 343-349. | Zbl 0302.65086

[21] A. Lichnewsky and R. Temam, Pseudosolutions of the time-dependent minimal surface problem. J. Differential Equations 30 (1978) 340-364. | Zbl 0368.49016

[22] J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod (1969). | MR 259693 | Zbl 0189.40603

[23] R. Rannacher, Some asymptotic error estimates for finite element approximation of minimal surfaces. RAIRO Anal. Numér. 11 (1977) 181-196. | Numdam | Zbl 0356.35034

[24] L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms. Phys. D 60 (1992) 259-268. | Zbl 0780.49028

[25] J. Simon, Compact sets in the space L p (0,T;B). Ann. Mat. Pura Appl. 146 (1987) 65-96. | Zbl 0629.46031

[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). | MR 1736116 | Zbl 0939.49001