Skip to main content
SpringerLink
Log in

A Level Set Method for the Dirichlet k-Partition Problem

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

We propose a simple level set method for the Dirichlet k-partition problem which aims to partition an open domain into K different subdomains as to minimize the sum of the smallest eigenvalue of the Dirichlet Laplace operator in each subdomain. We first formulate the problem as a nested minimization problem of a functional of the level set function and the eigenfunction defined in each subdomain. As an approximation, we propose to simply replace the eigenfunction by the level set function so that the nested minimization can then be converted to a single minimization problem. We apply the standard gradient descent method so that the problem leads to a Hamilton–Jacobi type equation. Various numerical examples will be given to demonstrate the effectiveness of our proposed method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24

Similar content being viewed by others

References

  1. Allen, S.M., Cahn, J.W.: A microscope theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27, 1085–1095 (1979)

    Article  Google Scholar 

  2. Bao, W.: Ground states and dynamics of rotating Bose–Einstein condensates. Model. Simul. Sci. Eng. Technol. 2(9780817644895), 215–255 (2007)

    Article  MathSciNet  Google Scholar 

  3. Bao, W., Du, Q.: Computing the ground state solution of Bose–Einstein condensates by a normalized gradient flow. SIAM J. Sci. Comput. 25(5), 1674–1697 (2004)

    Article  MathSciNet  Google Scholar 

  4. Bogosel, B.: Efficient algorithm for large spectral partitions. arXiv:1705.08739 (2017)

  5. Bogosel, B.: Efficient algorithm for optimizing spectral partition. Appl. Math. Comput. 333, 61–75 (2018)

    MathSciNet  MATH  Google Scholar 

  6. Bogosel, B., Bonnaillie-Noel, V.: Minimal partitions for p-norms of eigenvalues. Interfaces Free Bound. 20(1), 129 (2018)

    Article  MathSciNet  Google Scholar 

  7. Bourdin, B., Bucur, D., Oudet, D.: Optimal Partitions for Eigenvalues. SIAM J. Sci. Comput. 31(6), 4100–4114 (2010)

    Article  MathSciNet  Google Scholar 

  8. Bozorgnia, F.: Numerical algorithm for spatial segregation of competitive systems. SIAM J. Sci. Comput. 31(5), 3946–3958 (2009)

    Article  MathSciNet  Google Scholar 

  9. Bozorgnia, F.: Optimal partitions for first eigenvalues of the Laplace operator. Numer. Methods Partial Differ. Equ. 31(3), 923–949 (2015)

    Article  MathSciNet  Google Scholar 

  10. Bucur, D., Buttazzo, G., Henrot, A.: Existence results for some optimal partition problems. Adv. Math. Sci. Appl. 8, 571–579 (1998)

    MathSciNet  MATH  Google Scholar 

  11. Buttazzo, G.: Spectral optimization problems. Rev. Mat. Complut. 24, 277–322 (2011)

    Article  MathSciNet  Google Scholar 

  12. Cafferelli, L.A., Lin, F.H.: An optimal partition problem for eigenvalues. J. Sci. Comput. 31, 1–2 (2007)

    Article  MathSciNet  Google Scholar 

  13. Cahn, J.W., Hilliard, J.E.: Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 28, 258–267 (1958)

    Article  Google Scholar 

  14. Conti, M., Terracini, S., Verzini, G.: Nehari’s problem and competing species systems. Ann. Inst. H. Poincare Anal. Non Lineaire 19(6), 871–888 (2002)

    Article  MathSciNet  Google Scholar 

  15. Conti, M., Terracini, S., Verzini, G.: An optimal partition problem related to nonlinear eigenvalues. J. Funct. Anal. 198, 160–196 (2003)

    Article  MathSciNet  Google Scholar 

  16. Conti, M., Terracini, S., Verzini, G.: On a class of optimal partition problems related to the fucik spectrum and to the monotonicity formula. Calc. Var. 22, 45–72 (2005)

    Article  MathSciNet  Google Scholar 

  17. Conti, M., Terracini, S., Verzini, G.: A variational problem for the spatial segregation of rection-diffusion systems. Indiana Univ. Math. J. 54, 779–815 (2005)

    Article  MathSciNet  Google Scholar 

  18. Cybulski, O., Babin, V., Holyst, R.: Minimization of the Renyi entropy production in the space-partitioning process. Phys. Rev. E 71, 046130 (2005)

    Article  MathSciNet  Google Scholar 

  19. Cybulski, O., Holyst, R.: Three-dimensional space partition based on the first Laplacian eigenvalues in cells. Phys. Rev. E 77, 056101 (2008)

    Article  MathSciNet  Google Scholar 

  20. Du, Q., Feng, X.: The phase field method for geometric moving interfaces and their numerical approximations. In: Geometric Partial Differential Equations, Handbook of Numerical Analysis, vol. 21(I) (2020)

  21. Du, Q., Lin, F.: Numerical approximations of a norm-preserving gradient flow and applications to an optimal partition problem. Nonlinearity 22(1), 67–83 (2009)

    Article  MathSciNet  Google Scholar 

  22. Elliott, C.M., Ranner, T.: A computational approach to an optimal partition problem on surfaces. Interfaces Free Bound. 17(3), 353 (2015)

    Article  MathSciNet  Google Scholar 

  23. Helffer, B.: On spectral minimal partitions: a survey. Milan J. Math. 78, 575–590 (2010)

    Article  MathSciNet  Google Scholar 

  24. Helffer, B., Hoffmann-Ostenhof, T., Terracini, S.: Nodel domains and spectral minimal partition. Ann. Inst. H. Poincare (Anal. Non Lineaire) 26, 101–138 (2009)

    Article  Google Scholar 

  25. Jiang, G.S., Peng, D.: Weighted ENO schemes for Hamilton–Jacobi equations. SIAM J. Sci. Comput. 21, 2126–2143 (2000)

    Article  MathSciNet  Google Scholar 

  26. Kao, C.-Y., Osher, S., Yablonovitch, E.: Maximizing band gaps in two dimensional photonic crystals by using level set methods. Appl. Phys. B Lasers Optics 81, 235–244 (2005)

    Article  Google Scholar 

  27. Luminita, A.V., Chan, T.F.: A new multiphase level set framework for image segmentation via the Mumford and Shah model. Int. J. Comput. Vis. 50, 271–293 (2002)

    Article  Google Scholar 

  28. Merriman, B., Bence, J.K., Osher, S.: Diffusion generated motion by mean curvature. UCLA CAM Rep. 92, 18 (1992)

    Google Scholar 

  29. Merriman, B., Bence, J.K., Osher, S.: Motion of multiple junctions: a level set approach. J. Comput. Phys. 112(2), 334–363 (1994)

    Article  MathSciNet  Google Scholar 

  30. Min, C., Gibou, F.: A second order accurate level set method on non-graded adaptive Cartesian grids. J. Comput. Phys. 225, 300–321 (2007)

    Article  MathSciNet  Google Scholar 

  31. Osher, S.J., Fedkiw, R.P.: Level Set Methods and Dynamic Implicit Surfaces. Springer, New York (2003)

    Book  Google Scholar 

  32. Osher, S.J., Sethian, J.A.: Fronts propagating with curvature dependent speed: algorithms based on Hamilton–Jacobi formulations. J. Comput. Phys. 79, 12–49 (1988)

    Article  MathSciNet  Google Scholar 

  33. Osting, B., Reeb, T.H.: Consistency of Dirichlet partitions. SIAM J. Math. Anal. 49(5), 4251–4274 (2017)

    Article  MathSciNet  Google Scholar 

  34. Oudet, E., Osting, B., White, C.: Minimal Dirichlet energy partitions for graphs. SIAM J. Sci. Comput. 36(4), A1635–A1651 (2014)

    Article  MathSciNet  Google Scholar 

  35. Peng, D., Merriman, B., Osher, S., Zhao, H.K., Kang, M.: A PDE-based fast local level set method. J. Comput. Phys. 155, 410–438 (1999)

    Article  MathSciNet  Google Scholar 

  36. Sethian, J.A.: Level Set Methods, 2nd edn. Cambridge University Press, Cambridge (1999)

    MATH  Google Scholar 

  37. Shu, C.W., Osher, S.J.: Efficient implementation of essentially non-oscillatory shock capturing schemes. J. Comput. Phys. 77, 439–471 (1988)

    Article  MathSciNet  Google Scholar 

  38. Ungar, G., Liu, Y., Zheng, X., Percec, V., Cho, W.D.: Giant supermolecular liquid crystal lattice. Science 299, 1208–11 (2003)

    Article  Google Scholar 

  39. Wang, D., Osting, B.: A diffusion generated method for computing Dirichlet partitions. J. Comput. Appl. Math. 351, 302–316 (2019)

    Article  MathSciNet  Google Scholar 

  40. Zhao, H.K., Chan, T., Merriman, B., Osher, S.: A variational level set approach to multiphase motion. J. Comput. Phys. 127(1), 179–195 (1996)

    Article  MathSciNet  Google Scholar 

  41. Ziherl, P., Kamien, R.D.: Soup froths and crystal structures. Phys. Rev. Lett. 85(16), 3528 (2000)

    Article  Google Scholar 

  42. Zosso, D., Osting, B.: A minimal surface criterion for graph partitioning. Inverse Prob. Imaging 10(4), 1149–1180 (2016)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The work of Leung was supported in part by the Hong Kong RGC Grant 16302819.

Author information

Authors and Affiliations

  1. Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong

    Kwunlun Chu & Shingyu Leung

Authors
  1. Kwunlun Chu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shingyu Leung
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shingyu Leung.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chu, K., Leung, S. A Level Set Method for the Dirichlet k-Partition Problem. J Sci Comput 86, 11 (2021). https://doi.org/10.1007/s10915-020-01368-w

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-020-01368-w

Keywords

Search

Navigation