This article deals with the numerical computation of the Cheeger constant and the approximation of the maximal Cheeger set of a given subset of . This problem is motivated by landslide modelling as well as by the continuous maximal flow problem. Using the fact that the maximal Cheeger set can be approximated by solving a rather simple projection problem, we propose a numerical strategy to compute maximal Cheeger sets and Cheeger constants.
Keywords: Cheeger sets, Cheeger constant, total variation minimization, projections
@article{M2AN_2009__43_1_139_0,
author = {Carlier, Guillaume and Comte, Myriam and Peyr\'e, Gabriel},
title = {Approximation of maximal {Cheeger} sets by projection},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {139--150},
year = {2009},
publisher = {EDP Sciences},
volume = {43},
number = {1},
doi = {10.1051/m2an/2008040},
mrnumber = {2494797},
zbl = {1161.65046},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2008040/}
}
TY - JOUR AU - Carlier, Guillaume AU - Comte, Myriam AU - Peyré, Gabriel TI - Approximation of maximal Cheeger sets by projection JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2009 SP - 139 EP - 150 VL - 43 IS - 1 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2008040/ DO - 10.1051/m2an/2008040 LA - en ID - M2AN_2009__43_1_139_0 ER -
%0 Journal Article %A Carlier, Guillaume %A Comte, Myriam %A Peyré, Gabriel %T Approximation of maximal Cheeger sets by projection %J ESAIM: Modélisation mathématique et analyse numérique %D 2009 %P 139-150 %V 43 %N 1 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/m2an/2008040/ %R 10.1051/m2an/2008040 %G en %F M2AN_2009__43_1_139_0
Carlier, Guillaume; Comte, Myriam; Peyré, Gabriel. Approximation of maximal Cheeger sets by projection. ESAIM: Modélisation mathématique et analyse numérique, Tome 43 (2009) no. 1, pp. 139-150. doi: 10.1051/m2an/2008040
[1] and , Uniqueness of the Cheeger set of a convex body. Preprint (2007) available at http://cvgmt.sns.it. | Zbl | MR
[2] , and , Evolution of characteristic functions of convex sets in the plane by the minimizing total variation flow. Interfaces Free Bound. 7 (2005) 29-53. | Zbl | MR
[3] , and , Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs. Oxford University Press, New York (2000). | Zbl | MR
[4] and , Globally minimal surfaces by continuous maximal flows. IEEE Trans. Pattern Anal. Mach. Intell. 28 (2006) 106-118.
[5] , , and , Crystalline mean curvature flow of convex sets. Arch. Ration. Mech. Anal. 179 (2006) 109-152. | Zbl | MR
[6] , and , On the selection of maximal Cheeger sets. Differential Integral Equations 20 (2007) 991-1004. | MR
[7] and , On a weighted total variation minimization problem. J. Funct. Anal. 250 (2007) 214-226. | Zbl | MR
[8] , and , Uniqueness of the Cheeger set of a convex body. Pacific J. Math. 232 (2007) 77-90. | MR
[9] , An algorithm for total variation minimization and applications, Special issue on mathematics and image analysis. J. Math. Imaging Vision 20 (2004) 89-97. | MR
[10] and , Image recovery via total variation minimization. Numer. Math. 76 (1997) 167-188. | Zbl | MR
[11] , A block-iterative surrogate constraint splitting method for quadratic signal recovery. IEEE Trans. Signal Process. 51 (2003) 1771-1782. | MR
[12] and , image restoration subject to a total variation constraint. IEEE Trans. Image Process. 13 (2004) 1213-1222.
[13] , A model of stability of slopes in Slope Stability 2000, in Proceedings of Sessions of Geo-Denver 2000, D.V. Griffiths, G.A. Fenton, T.R. Martin Eds., Geotechnical special publication 101 (2000) 86-98.
[14] , Théorèmes d’existence pour des équations avec l’opérateur “1-Laplacien”, première valeur propre de . C. R. Math. Acad. Sci. Paris 334 (2002) 1071-1076. | Zbl | MR
[15] , Some existence's results for noncoercive “1-Laplacian” operator. Asymptotic Anal. 43 (2005) 287-322. | MR | Zbl
[16] and , Les inéquations en mécanique et en physique. Dunod, Paris (1972). | Zbl | MR
[17] and , Convex Analysis and Variational Problems, Classics in Mathematics. Society for Industrial and Applied Mathematics, Philadelphia (1999). | Zbl | MR
[18] and , Measure Theory and Fine Properties of Functions. CRC Press (1992). | Zbl | MR
[19] , and , Shape optimization and supremal minimization approaches in landslides modeling. Appl. Math. Opt. 52 (2005) 349-364. | Zbl | MR
[20] , , and , The blocking of an inhomogeneous Bingham fluid. Applications to landslides. ESAIM: M2AN 36 (2002) 1013-1026. | Zbl | MR | Numdam | EuDML
[21] and , Generalized Cheeger sets related to landslides. Calc. Var. Partial Differential Equations 23 (2005) 227-249. | Zbl | MR
[22] , Max-flow min-cut theorem in an anisotropic network. Osaka J. Math. 27 (1990) 805-842. | Zbl | MR
[23] , and , Nonlinear total variation based noise removal algorithms. Physica D 60 (1992) 259-268. | Zbl
[24] , Maximal flow through a domain. Math. Programming 26 (1983) 123-143. | Zbl | MR
[25] , Maximum flows and minimum cuts in the plane. J. Global Optimization (to appear). | MR | Zbl
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