# Approximation of maximal Cheeger sets by projection

Guillaume Carlier; Myriam Comte; Gabriel Peyré

ESAIM: Mathematical Modelling and Numerical Analysis (2008)

- Volume: 43, Issue: 1, page 139-150
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topCarlier, Guillaume, Comte, Myriam, and Peyré, Gabriel. "Approximation of maximal Cheeger sets by projection." ESAIM: Mathematical Modelling and Numerical Analysis 43.1 (2008): 139-150. <http://eudml.org/doc/194441>.

@article{Carlier2008,

abstract = {
This article deals with the numerical computation of the Cheeger constant and the approximation of the maximal Cheeger set of a given subset of
$\{\mathbb R\}^d$. 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.
},

author = {Carlier, Guillaume, Comte, Myriam, Peyré, Gabriel},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Cheeger sets; Cheeger constant; total variation minimization; projections.; Cheeger set; Cheeger constants; projections; landslide modeling; continuous maximal flow problem},

language = {eng},

month = {10},

number = {1},

pages = {139-150},

publisher = {EDP Sciences},

title = {Approximation of maximal Cheeger sets by projection},

url = {http://eudml.org/doc/194441},

volume = {43},

year = {2008},

}

TY - JOUR

AU - Carlier, Guillaume

AU - Comte, Myriam

AU - Peyré, Gabriel

TI - Approximation of maximal Cheeger sets by projection

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2008/10//

PB - EDP Sciences

VL - 43

IS - 1

SP - 139

EP - 150

AB -
This article deals with the numerical computation of the Cheeger constant and the approximation of the maximal Cheeger set of a given subset of
${\mathbb R}^d$. 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.

LA - eng

KW - Cheeger sets; Cheeger constant; total variation minimization; projections.; Cheeger set; Cheeger constants; projections; landslide modeling; continuous maximal flow problem

UR - http://eudml.org/doc/194441

ER -

## References

top- F. Alter and V. Caselles, Uniqueness of the Cheeger set of a convex body. Preprint (2007) available at . Zbl1167.52005URIhttp://cvgmt.sns.it
- F. Alter, V. Caselles and A. Chambolle, Evolution of characteristic functions of convex sets in the plane by the minimizing total variation flow. Interfaces Free Bound.7 (2005) 29–53. Zbl1084.49003
- L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs. Oxford University Press, New York (2000). Zbl0957.49001
- B. Appleton and H. Talbot, Globally minimal surfaces by continuous maximal flows. IEEE Trans. Pattern Anal. Mach. Intell.28 (2006) 106–118.
- G. Bellettini, V. Caselles, A. Chambolle and M. Novaga, Crystalline mean curvature flow of convex sets. Arch. Ration. Mech. Anal.179 (2006) 109–152. Zbl1148.53049
- G. Buttazzo, G. Carlier and M. Comte, On the selection of maximal Cheeger sets. Differential Integral Equations20 (2007) 991–1004. Zbl1212.49019
- G. Carlier and M. Comte, On a weighted total variation minimization problem. J. Funct. Anal.250 (2007) 214–226. Zbl1120.49011
- V. Caselles, A. Chambolle and M. Novaga, Uniqueness of the Cheeger set of a convex body. Pacific J. Math.232 (2007) 77–90. Zbl1221.35171
- A. Chambolle, An algorithm for total variation minimization and applications, Special issue on mathematics and image analysis. J. Math. Imaging Vision20 (2004) 89–97.
- A. Chambolle and P.-L. Lions, Image recovery via total variation minimization. Numer. Math.76 (1997) 167–188. Zbl0874.68299
- P.-L. Combettes, A block-iterative surrogate constraint splitting method for quadratic signal recovery. IEEE Trans. Signal Process.51 (2003) 1771–1782.
- P.-L. Combettes and J.-C. Pesquet, image restoration subject to a total variation constraint. IEEE Trans. Image Process.13 (2004) 1213–1222.
- N. Cristescu, 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 publication101 (2000) 86–98.
- F. Demengel, Théorèmes d'existence pour des équations avec l'opérateur “1-Laplacien”, première valeur propre de $-{\Delta}_{1}$. C. R. Math. Acad. Sci. Paris334 (2002) 1071–1076.
- F. Demengel, Some existence's results for noncoercive “1-Laplacian” operator. Asymptotic Anal.43 (2005) 287–322. Zbl1192.35036
- G. Duvaut and J.-L. Lions, Les inéquations en mécanique et en physique. Dunod, Paris (1972). Zbl0298.73001
- I. Ekeland and R. Temam, Convex Analysis and Variational Problems, Classics in Mathematics. Society for Industrial and Applied Mathematics, Philadelphia (1999). Zbl0939.49002
- L.C. Evans and R. Gariepy, Measure Theory and Fine Properties of Functions. CRC Press (1992). Zbl0804.28001
- R. Hassani, I.R. Ionescu and T. Lachand-Robert, Shape optimization and supremal minimization approaches in landslides modeling. Appl. Math. Opt.52 (2005) 349–364. Zbl1081.49030
- P. Hild, I.R. Ionescu, T. Lachand-Robert and I. Rosca, The blocking of an inhomogeneous Bingham fluid. Applications to landslides. ESAIM: M2AN36 (2002) 1013–1026. Zbl1057.76004
- I.R. Ionescu and T. Lachand-Robert, Generalized Cheeger sets related to landslides. Calc. Var. Partial Differential Equations23 (2005) 227–249. Zbl1062.49036
- R. Nozawa, Max-flow min-cut theorem in an anisotropic network. Osaka J. Math.27 (1990) 805–842. Zbl0723.90020
- L.I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms. Physica D60 (1992) 259–268. Zbl0780.49028
- G. Strang, Maximal flow through a domain. Math. Programming26 (1983) 123–143. Zbl0513.90026
- G. Strang, Maximum flows and minimum cuts in the plane. J. Global Optimization (to appear). Zbl1207.49042

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.