Milan Journal of Mathematics

, Volume 82, Issue 1, pp 3–65 | Cite as

Mean Curvature, Threshold Dynamics, and Phase Field Theory on Finite Graphs

  • Yves van Gennip
  • Nestor Guillen
  • Braxton Osting
  • Andrea L. BertozziEmail author
Open Access


In the continuum, close connections exist between mean curvature flow, the Allen-Cahn (AC) partial differential equation, and the Merriman-Bence-Osher (MBO) threshold dynamics scheme. Graph analogues of these processes have recently seen a rise in popularity as relaxations of NP-complete combinatorial problems, which demands deeper theoretical underpinnings of the graph processes. The aim of this paper is to introduce these graph processes in the light of their continuum counterparts, provide some background, prove the first results connecting them, illustrate these processes with examples and identify open questions for future study.

We derive a graph curvature from the graph cut function, the natural graph counterpart of total variation (perimeter). This derivation and the resulting curvature definition differ from those in earlier literature, where the continuum mean curvature is simply discretized, and bears many similarities to the continuum nonlocal curvature or nonlocal means formulation. This new graph curvature is not only relevant for graph MBO dynamics, but also appears in the variational formulation of a discrete time graph mean curvature flow.

We prove estimates showing that the dynamics are trivial for both MBO and AC evolutions if the parameters (the time-step and diffuse interface scale, respectively) are sufficiently small (a phenomenon known as “freezing” or “pinning”) and also that the dynamics for MBO are nontrivial if the time step is large enough. These bounds are in terms of graph quantities such as the spectrum of the graph Laplacian and the graph curvature. Adapting a Lyapunov functional for the continuum MBO scheme to graphs, we prove that the graph MBO scheme converges to a stationary state in a finite number of iterations. Variations on this scheme have recently become popular in the literature as ways to minimize (continuum) nonlocal total variation.

Mathematics Subject Classification (2010)

34B45 35R02 53C44 53A10 49K15 49Q05 35K05 


Spectral graph theory Allen-Cahn equation Ginzburg-Landau functional Merriman-Bence-Osher threshold dynamics graph cut function total variation mean curvature flow nonlocal mean curvature gamma convergence graph coarea formula 


  1. 1.
    R. A. Adams, Sobolev spaces, first ed., Pure and applied mathematics; a series of monographs and textbooks, vol. 65, Academic Press, Inc, New York, 1975.Google Scholar
  2. 2.
    S. M. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metallurgica 27 (1979), no. 6, 1085–1095.Google Scholar
  3. 3.
    F. Almgren, J. E. Taylor, and L. Wang, Curvature-driven flows: a variational approach, SIAM J. Control Optim. 31 (1993), no. 2, 387–438.Google Scholar
  4. 4.
    L. Ambrosio, N. Fusco, and D. Pallara, Functions of bounded variation and free discontinuity problems, first ed., Oxford Mathematical Monographs, Oxford University Press, Oxford, 2000.Google Scholar
  5. 5.
    G. Barles and C. Georgelin, A simple proof of convergence for an approximation scheme for computing motions by mean curvature, SIAM J. Numer. Anal. 32 (1995), no. 2, 484–500.Google Scholar
  6. 6.
    G. Barles, H. M. Soner, and P. E. Souganidis, Front propagation and phase field theory, SIAM J. Control Optim. 31 (1993), no. 2, 439–469.Google Scholar
  7. 7.
    A. Barrat, M. Barthelemy, R. Pastor-Satorras, and A. Vespignani, The architecture of complex weighted networks, Proceedings of the National Academy of Sciences of the United States of America 101 (2004), no. 11, 3747–3752.Google Scholar
  8. 8.
    P. W. Bates and A. Chmaj, A discrete convolution model for phase transitions, Arch. Ration. Mech. Anal. 150 (1999), no. 4, 281–305.Google Scholar
  9. 9.
    A. L. Bertozzi and A. Flenner, Diffuse interface models on graphs for analysis of high dimensional data, Multiscale Modeling and Simulation 10 (2012), no. 3, 1090–1118.Google Scholar
  10. 10.
    T. Biyikoglu, J. Leydold, and P. F. Stadler, Laplacian eigenvectors of graphs, Springer, 2007.Google Scholar
  11. 11.
    A. Björner, L. Lovász, and P. W. Shor, Chip-firing games on graphs, European J. Combin 12 (1991), no. 4, 283–291.Google Scholar
  12. 12.
    T. Bühler and M. Hein, Spectral clustering based on the graph p-laplacian, Proceedings of the 26th Annual International Conference on Machine Learning, ACM, 2009, 81–88.Google Scholar
  13. 13.
    M. Bradonjić and I. Saniee, Bootstrap percolation on random geometric graphs, Probability in the Engineering and Informational Sciences 28, (2014), no. 2, 169–181 (to appear).Google Scholar
  14. 14.
    A. Braides, Γ-convergence for beginners, first ed., Oxford Lecture Series inMathematics and its Applications, vol. 22, Oxford University Press, Oxford, 2002.Google Scholar
  15. 15.
    K. A. Brakke, The motion of a surface by its mean curvature, Mathematical Notes, vol. 20, Princeton University Press, Princeton, N.J., 1978.Google Scholar
  16. 16.
    L. Bronsard and R. V. Kohn, Motion by mean curvature as the singular limit of Ginzburg-Landau dynamics, J. Differential Equations 90 (1991), no. 2, 211–237.Google Scholar
  17. 17.
    L. Caffarelli, J.-M. Roquejoffre, and O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math. 63 (2010), no. 9, 1111–1144.Google Scholar
  18. 18.
    L. A. Caffarelli and P. E. Souganidis, Convergence of nonlocal threshold dynamics approximations to front propagation, Arch. Ration. Mech. Anal. 195 (2010), no. 1, 1–23.Google Scholar
  19. 19.
    G. Caginalp and E. Socolovsky, Phase field computations of single-needle crystals, crystal growth, and motion by mean curvature, SIAM Journal on Scientific Computing 15 (1994), no. 1, 106–126.Google Scholar
  20. 20.
    M. Calle, Mean curvature flow and minimal surfaces, Dissertation Department of Mathematics, New York University, UMI Number: 3283351 (2007).Google Scholar
  21. 21.
    O. Candogan, I. Menache, A. Ozdaglar, and P. A. Parrilo, Flows and decompositions of games: harmonic and potential games, Math. Oper. Res. 36 (2011), no. 3, 474–503.Google Scholar
  22. 22.
    M. C. Caputo and N. Guillen, Regularity for non-local almost minimal boundaries and applications, arXiv preprint 1003.2470 (2010).Google Scholar
  23. 23.
    Chalupa J., Leath P. L., Reich G. R.: Bootstrap percolation on a Bethe lattice. J. Phys. C 12, L31–L35 (1979)CrossRefGoogle Scholar
  24. 24.
    A. Chambolle, Total variation minimization and a class of binary MRF models, Energy minimization methods in computer vision and pattern recognition, Springer, 2005, 136– 152.Google Scholar
  25. 25.
    A. Chambolle and J. Darbon, On total variation minimization and surface evolution using parametric maximum flows, International Journal of Computer Vision 84 (2009), no. 3, 288–307.Google Scholar
  26. 26.
    A. Chambolle and M. Novaga, Convergence of an algorithm for the anisotropic and crystalline mean curvature flow, SIAM J. Math. Anal. 37 (2006), no. 6, 1978–1987.Google Scholar
  27. 27.
    T. F. Chan, S. Esedoḡlu, and M. Nikolova, Algorithms for finding global minimizers of image segmentation and denoising models, SIAM J. Appl. Math. 66 (2006), no. 5, 1632–1648.Google Scholar
  28. 28.
    R. Choksi, Y. van Gennip, and A. Oberman, Anisotropic total variation regularized L 1 approximation and denoising/deblurring of 2D bar codes, Inverse Probl. Imaging 5 (2011), no. 3, 591–617.Google Scholar
  29. 29.
    F. R. K. Chung, Spectral graph theory, AMS, 1997.Google Scholar
  30. 30.
    T. H. Colding and W. P. Minicozzi II, Minimal surfaces and mean curvature flow, arXiv preprint 1102.1411 (2011).Google Scholar
  31. 31.
    G. Dal Maso, An introduction to Γ-convergence, first ed., Progress in Nonlinear Differential Equations and Their Applications, vol. 8, Birkhäuser, Boston, 1993.Google Scholar
  32. 32.
    X. Desquesnes, A. Elmoataz, and O. Lézoray, PDEs level sets on weighted graphs, Image Processing (ICIP), 2011 18th IEEE International Conference on, IEEE, 2011, 3377–3380.Google Scholar
  33. 33.
    X. Desquesnes, A. Elmoataz, and O. Lézoray, Eikonal equation adaptation on weighted graphs: Fast geometric diffusion process for local and non-local image and data processing, Journal of Mathematical Imaging and Vision (2012), 1–20.Google Scholar
  34. 34.
    X. Desquesnes, A. Elmoataz, and O. Lézoray, Generalized fronts propagation on weighted graphs, Proceedings of ALGORITMY, 2012, 371–381.Google Scholar
  35. 35.
    X. Desquesnes, A. Elmoataz, and O. Lézoray, PdEs-based morphology on graphs for cytological slides segmentation and clustering, Biomedical Imaging (ISBI), 2012 9th IEEE International Symposium on, IEEE, 2012, 1619–1622.Google Scholar
  36. 36.
    X. Desquesnes, A. Elmoataz, O. Lézoray, and Vinh-Thong Ta, Efficient algorithms for image and high dimensional data processing using eikonal equation on graphs, Advances in Visual Computing, Springer, 2010, 647–658.Google Scholar
  37. 37.
    R. Durrett, Random graph dynamics, vol. 20, Cambridge university press, 2007.Google Scholar
  38. 38.
    K. Ecker, Regularity theory for mean curvature flow, vol. 57, Springer, 2004.Google Scholar
  39. 39.
    I. Ekeland and R. Temam, Convex analysis and variational problems, North-Holland Publishing Co., Amsterdam, 1976, Translated from the French, Studies in Mathematics and its Applications, Vol. 1.Google Scholar
  40. 40.
    A. Elmoataz, X. Desquesnes, and O. Lézoray, Non-local morphological PDEs and p- Laplacian equation on graphs with applications in image processing and machine learning, IEEE Journal of Selected Topics in Signal Processing 6 (2012), no. 7, 764–779.Google Scholar
  41. 41.
    A. Elmoataz, O. Lezoray, and S. Bougleux, Nonlocal discrete regularization on weighted graphs: a framework for image and manifold processing, Image Processing, IEEE Transactions on 17 (2008), no. 7, 1047–1060.Google Scholar
  42. 42.
    S. Esedoḡlu and F. Otto, Threshold dynamics for networks with arbitrary surface tensions (to appear).Google Scholar
  43. 43.
    S. Esedoḡlu, S. J. Ruuth, and R. Tsai, Threshold dynamics for high order geometric motions, Interfaces and Free Boundaries 10 (2008), no. 3, 263–282.Google Scholar
  44. 44.
    S. Esedoḡlu, S. Ruuth, and R. Tsai, Diffusion generated motion using signed distance functions, Journal of Computational Physics 229 (2010), no. 4, 1017–1042.Google Scholar
  45. 45.
    S. Esedoḡlu and Y.-H. R. Tsai, Threshold dynamics for the piecewise constant Mumford-Shah functional, Journal of Computational Physics 211 (2006), no. 1, 367– 384.Google Scholar
  46. 46.
    L. C. Evans, Convergence of an algorithm for mean curvature motion, Indiana Univ. Math. J. 42 (1993), no. 2, 533–557.Google Scholar
  47. 47.
    L. C. Evans and R. F. Gariepy, Measure theory and fine properties of functions, first ed., Studies in Advanced Mathematics, CRC Press LLC, Boca Raton, Florida, 1992.Google Scholar
  48. 48.
    Fiedler M.: Algebraic connectivity of graphs. Czech. Math. J. 23, 298–305 (1973)MathSciNetGoogle Scholar
  49. 49.
    J. Friedman and J. P. Tillich, Calculus on graphs, arXiv preprint cs/0408028 (2004).Google Scholar
  50. 50.
    T. Funaki and H. Spohn, Motion by mean curvature from the Ginzburg-Landau ∇ϕ interface model, Comm. Math. Phys. 185 (1997), no. 1, 1–36.Google Scholar
  51. 51.
    C. Garcia-Cardona, E. Merkurjev, A. L. Bertozzi, A. Flenner, and A. Percus, Fast multiclass segmentation using diffuse interface methods on graphs, submitted (2013).Google Scholar
  52. 52.
    R. Ghosh, K. Lerman, T. Surachawala, K. Voevodski, and S. Teng, Non-conservative diffusion and its application to social network analysis, arXiv preprint 1102.4639 (2011).Google Scholar
  53. 53.
    R. Ghosh and K. Lerman, Rethinking centrality: the role of dynamical processes in social network analysis, arXiv preprint 1209.4616 (2012).Google Scholar
  54. 54.
    G. Gilboa and S. Osher, Nonlocal operators with applications to image processing, Multiscale Modeling & Simulation 7 (2009), no. 3, 1005–1028.Google Scholar
  55. 55.
    E. Giusti, Minimal surfaces and functions of bounded variation, Monographs in Mathematics 80, Springer Science+Business Media, New York, 1984.Google Scholar
  56. 56.
    L. J. Grady and J. R. Polimeni, Discrete calculus, Springer-Verlag London Ltd., 2010.Google Scholar
  57. 57.
    S. Guattery and G. L. Miller, On the quality of spectral separators, SIAM J. Matrix Anal. Appl. 19 (1998), no. 3, 701–719.Google Scholar
  58. 58.
    Hele-Shaw H. S.: Flow of water. Nature 58, 520 (1898)CrossRefGoogle Scholar
  59. 59.
    R. A. Horn and C. R. Johnson, Matrix analysis, Cambridge University Press, 1990.Google Scholar
  60. 60.
    H. Hu, T. Laurent, M. A. Porter, and A. L. Bertozzi, A method based on total variation for network modularity optimization using the MBO scheme, submitted (2013).Google Scholar
  61. 61.
    H. J. Hupkes, D. Pelinovsky, and B. Sandstede, Propagation failure in the discrete Nagumo equation, Proc. Amer. Math. Soc. 139 (2011), no. 10, 3537–3551.Google Scholar
  62. 62.
    C. Imbert, Level set approach for fractional mean curvature flows, Interfaces and Free Boundaries 11 (2009), no. 1, 153–176.Google Scholar
  63. 63.
    S. Janson, T. Łuczak, T. Turova, and T. Vallier, Bootstrap percolation on the random graph G n,p, Ann. Appl. Probab. 22 (2012), no. 5, 1989–2047.Google Scholar
  64. 64.
    R. Kannan, S. Vempala, and A. Vetta, On clusterings—good, bad and spectral, 41st Annual Symposium on Foundations of Computer Science (Redondo Beach, CA, 2000), IEEE Comput. Soc. Press, Los Alamitos, CA, 2000, 367–377.Google Scholar
  65. 65.
    R. Kannan, S. Vempala, and A. Vetta, On clusterings: good, bad and spectral, J. ACM 51 (2004), no. 3, 497–515.Google Scholar
  66. 66.
    M. A. Katsoulakis and P. E. Souganidis, Interacting particle systems and generalized evolution of fronts, Archive for rational mechanics and analysis 127 (1994), no. 2, 133–157.Google Scholar
  67. 67.
    M. A. Katsoulakis and P. E. Souganidis, Stochastic ising models and anisotropic front propagation, Journal of statistical physics 87 (1997), no. 1-2, 63–89.Google Scholar
  68. 68.
    S. Luckhaus and T. Sturzenhecker, Implicit time discretization for the mean curvature flow equation, Calc. Var. Partial Differential Equations 3 (1995), no. 2, 253–271.Google Scholar
  69. 69.
    R. Lyons, The ising model and percolation on trees and tree-like graphs, Communications in Mathematical Physics 125 (1989), no. 2, 337–353.Google Scholar
  70. 70.
    R. Lyons, Phase transitions on nonamenable graphs. (english summary), J. Math. Phys 41 (2000), no. 3, 1099–1126.Google Scholar
  71. 71.
    J. J. Manfredi, A. M. Oberman, and A. P. Sviridov, Nonlinear elliptic partial differential equations and p-harmonic functions on graphs, Preprint (2012).Google Scholar
  72. 72.
    P. Mascarenhas, Diffusion generated motion by mean curvature, UCLA Department of Mathematics CAM report CAM 92–33 (1992).Google Scholar
  73. 73.
    Meeks W.H. III, Pérez J.: The classical theory of minimal surfaces. Bull. Amer. Math. Soc. 48, 325–407 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  74. 74.
    E. Merkurjev, T. Kostić, and A. L. Bertozzi, An MBO scheme on graphs for segmentation and image processing, submitted (2012).Google Scholar
  75. 75.
    B. Merriman, J. K. Bence, and S. Osher, Diffusion generated motion by mean curvature, UCLA Department of Mathematics CAM report CAM 06–32 (1992).Google Scholar
  76. 76.
    B. Merriman, J.K. Bence, and S. Osher, Diffusion generated motion by mean curvature, AMS Selected Letters, Crystal Grower’s Workshop (1993), 73–83.Google Scholar
  77. 77.
    B. Merriman, J. K. Bence, and S. J. Osher, Motion of multiple functions: a level set approach, J. Comput. Phys. 112 (1994), no. 2, 334–363.Google Scholar
  78. 78.
    L. Modica, The gradient theory of phase transitions and the minimal interface criterion, Arch. Rational Mech. Anal. 98 (1987), no. 2, 123–142.Google Scholar
  79. 79.
    L. Modica and S. Mortola, Un esempio di Γ-convergenza, Bollettino U.M.I. 5 (1977), no. 14-B, 285–299.Google Scholar
  80. 80.
    B. Mohar, The Laplacian spectrum of graphs, Graph Theory, Combinatorics, and Applications, vol. 2, Wiley, 1991, 871–898.Google Scholar
  81. 81.
    W. W. Mullins and R. F. Sekerka, Stability of a planar interface during solidification of a dilute binary alloy, Journal of Applied Physics 35 (1964), no. 2, 444–451.Google Scholar
  82. 82.
    A. Ng, M. Jordan, and Y. Weiss, On spectral clustering: Analysis and an algorithm, Dietterich, T., Becker, S., Ghahramani, Z. (eds.) Advances in Neural Information Processing Systems 14, MIT Press, Cambridge, 2002, 849–856.Google Scholar
  83. 83.
    Oberman A. M.: A convergent monotone difference scheme for motion of level sets by mean curvature. Numer. Math. 99, 365–279 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  84. 84.
    R. Olfati-Saber, A. Fax, and R. M. Murray, Consensus and cooperation in networked multi-agent systems, Proceedings of the IEEE 95 (2007), no. 1, 215–233.Google Scholar
  85. 85.
    S. J. Osher and R. Fedkiw, Level set methods and dynamic implicit surfaces, vol. 153, Springer Verlag, 2003.Google Scholar
  86. 86.
    B. Osting, C. Brune, and S. Osher, Enhanced statistical rankings via targeted data collection, JMLR, W&CP 28 (2013), no. 1, 489–497.Google Scholar
  87. 87.
    R. L. Pego, Front migration in the nonlinear cahn-hilliard equation, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 422 (1989), no. 1863, 261–278.Google Scholar
  88. 88.
    N. Peters, Turbulent combustion, Cambridge university press, 2000.Google Scholar
  89. 89.
    M. Reed and B. Simon, Methods of Modern Mathematical Physics, I: Functional Analysis, Academic Press, 1979.Google Scholar
  90. 90.
    L. I. Rudin, S. Osher, and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D: Nonlinear Phenomena, 60 (1992), no. 1–4, 259–268.Google Scholar
  91. 91.
    S. J. Ruuth, Efficient algorithms for diffusion-generated motion by mean curvature, ProQuest LLC, Ann Arbor, MI, 1996, Thesis (Ph.D.)–The University of British Columbia (Canada).Google Scholar
  92. 92.
    S. J. Ruuth, A diffusion-generated approach to multiphase motion, Journal of Computational Physics 145 (1998), no. 1, 166–192.Google Scholar
  93. 93.
    S. J. Ruuth, Efficient algorithms for diffusion-generated motion by mean curvature, Journal of Computational Physics 144 (1998), no. 2, 603–625.Google Scholar
  94. 94.
    D. Sha, Gossip algorithms, Foundations and Trends in Networking 3 (2008), no. 1, 1–125.Google Scholar
  95. 95.
    J. Shi and J. Malik, Normalized cuts and image segmentation, IEEE Transactions on Pattern Analysis and Machine Intelligence 22 (2000), no. 8, 888–905.Google Scholar
  96. 96.
    L. M. Smith, K. Lerman, C. Garcia-Cardona, A. G. Percus, and R. Ghosh, Spectral clustering with epidemic diffusion, arXiv preprint 1303.2663 (2013).Google Scholar
  97. 97.
    H. M. Soner and N. Touzi, A stochastic representation for mean curvature type geometric flows, The Annals of probability 31 (2003), no. 3, 1145–1165.Google Scholar
  98. 98.
    D. A. Spielman and S.-H. Teng, Spectral partitioning works: planar graphs and finite element meshes, 37th Annual Symposium on Foundations of Computer Science (Burlington, VT, 1996), IEEE Comput. Soc. Press, Los Alamitos, CA, 1996, 96–105.Google Scholar
  99. 99.
    D. A. Spielman and S.-H. Teng, Spectral partitioning works: planar graphs and finite element meshes, Linear Algebra Appl. 421 (2007), no. 2-3, 284–305.Google Scholar
  100. 100.
    J. Sun, S. Boyd, L. Xiao, and P. Diaconis, The fastest mixing Markov process on a graph and a connection to a maximum variance unfolding problem, SIAM Review 48 (2004), 2006.Google Scholar
  101. 101.
    J. Szarski, Differential inequalities, PWN Warsaw, 1965.Google Scholar
  102. 102.
    A. Szlam and X. Bresson, Total variation and cheeger cuts, Proceedings of the 27th International Conference on Machine Learning (ICML-10) (Haifa, Israel) (Johannes Fürnkranz and Thorsten Joachims, eds.), Omnipress, June 2010, 1039–1046.Google Scholar
  103. 103.
    V.-T. Ta, A. Elmoataz, and O. Lézoray, Partial difference equations over graphs: Morphological processing of arbitrary discrete data, Computer Vision–ECCV 2008, Springer, 2008, 668–680.Google Scholar
  104. 104.
    V.-T. Ta, A. Elmoataz, and O. Lézoray, Nonlocal PDEs-based morphology on weighted graphs for image and data processing, Image Processing, IEEE Transactions on 20 (2011), no. 6, 1504–1516.Google Scholar
  105. 105.
    J. E. Taylor, II—mean curvature and weighted mean curvature, Acta Metallurgica et Materialia 40 (1992), no. 7, 1475–1485.Google Scholar
  106. 106.
    R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, second ed., Applied Mathematical Sciences, vol. 68, Springer-Verlag, New York, 1997.Google Scholar
  107. 107.
    Y. van Gennip and A. L. Bertozzi, Γ-convergence of graph Ginzburg-Landau functionals, Adv. Differential Equations 17 (2012), no. 11–12, 1115–1180.Google Scholar
  108. 108.
    U. von Luxburg, A tutorial on spectral clustering, Statistics and Computing 17 (2007), no. 4, 395–416.Google Scholar
  109. 109.
    D. Wagner and F. Wagner, Between min cut and graph bisection, Mathematical foundations of computer science 1993 (Gdańsk, 1993), Lecture Notes in Comput. Sci., vol. 711, Springer, Berlin, 1993, 744–750.Google Scholar
  110. 110.
    D. J. Watts and S. H. Strogatz, Collective dynamics of ’small-world’ networks., Nature 393 (1998), no. 6684, 440–442.Google Scholar
  111. 111.
    J. Xin and Y. Yu, Analysis and comparison of large time front speeds in turbulent combustion models, arXiv preprint 1105.5607 (2011).Google Scholar
  112. 112.
    B. A Zalesky, Network flow optimization for restoration of images, Journal of Applied Mathematics 2 (2002), no. 4, 199–218.Google Scholar
  113. 113.
    J. Zhang and Q. Du, Numerical studies of discrete approximations to the Allen-Cahn equation in the sharp interface limit, SIAM Journal on Scientific Computing 31 (2009), no. 4, 3042–3063.Google Scholar

Copyright information

© The Author(s) 2014

Open AccessThis article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

Authors and Affiliations

  • Yves van Gennip
    • 1
  • Nestor Guillen
    • 2
  • Braxton Osting
    • 2
  • Andrea L. Bertozzi
    • 2
    Email author
  1. 1.School of Mathematical SciencesThe University of NottinghamNottinghamUK
  2. 2.Department of MathematicsUniversity of California, Los AngelesLos AngelesUSA

Personalised recommendations