Abstract
We introduce a three-step method to construct on a simplicial complex a combinatorial dynamical system in the sense of Forman from vector field data by means of a linear minimization problem, where the solution to this problem induces an admissible matching for the dynamical system. We show that the matrix of the minimization problem is unimodular, allowing us to relax the problem from binary-valued to real-valued and permitting its resolution in polynomial time. We demonstrate the effectiveness of the method on the Lotka–Volterra model and on the Lorenz attractor model. We also describe three potential extensions to our method: how barycentric subdivision can be applied to the simplicial complex to increase the resolution and obtain a solution that better fits the underlying dynamics of the data, how to add constraints to the minimization problem to fix the number of critical simplices, and how to add constraints to obtain a solution that induces a gradient matching.
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References
Allili, M., Kaczynski, T., Landi, C., Masoni, F.: Acyclic partial matchings for multidimensional persistence: algorithm and combinatorial interpretation. J. Math. Imaging Vis. 61, 174–192 (2019). https://doi.org/10.1007/s10851-018-0843-8
Batko, B., Kaczynski, T., Mrozek, M., Wanner, T.: Linking combinatorial and classical dynamics: Conley index and Morse decompositions. Found. Comput. Math. 20, 1–46 (2020). https://doi.org/10.1007/s10208-020-09444-1
Dey, T., Mrozek, M., Slechta, R.: Persistence of the Conley index in combinatorial dynamical systems. In: 36th International Symposium on Computational Geometry (SoCG 2020) Leibniz International Proceedings in Informatics (LIPIcs), pp. 37–13717 (2020)
Dey, T., Juda, M., Kapela, T., Kubica, J., Lipiński, M., Mrozek, M.: Persistent homology of Morse decomposition in combinatorial dynamics. SIAM J. Appl. Dyn. Syst. 18, 510–530 (2019). https://doi.org/10.1137/18M1198946
Dowker, C.: Homology groups of relations. Ann. Math. 56, 84–95 (1952). https://doi.org/10.2307/1969768
Edelsbrunner, H., Harer, J.L.: Computational Topology: An Introduction. American Mathematical Society, Providence, RI (2010)
Eidi, M., Jost, J.: Floer homology: from generalized Morse-Smale dynamical systems to Forman’s combinatorial vector fields (2021). arXiv. https://doi.org/10.48550/ARXIV.2105.02567
Forman, R.: Morse theory for cell complexes. Adv. Math. 134(AI971650), 90–145 (1998). https://doi.org/10.1006/aima.1997.1650
Forman, R.: Combinatorial vector fields and dynamical systems. Math. Z. 228, 629–681 (1998). https://doi.org/10.1007/PL00004638
Forman, R.: A user’s guide to discrete Morse theory. Sém. Lothar. Combin. 48(B48c), 35 (2002)
Harary, F., Palmer, E.M.: Graphical Enumeration. Academic Press, New York (1973)
Harker, S., Mischaikow, K., Mrozek, M., Nanda, V.: Discrete Morse theoretic algorithms for computing homology of complexes and maps. Found. Comput. Math. 14, 151–184 (2014). https://doi.org/10.1007/s10208-013-9145-0
Heller, I., Tompkins, C.B.: An extension of a theorem of Dantzig’s. Linear Inequal. Relat. Syst. 38, 247–254 (1957)
Hoffman, A.J.: Total unimodularity and combinatorial theorems. Linear Algebra Appl. 13, 103–108 (1976)
Joswig, M., Pfetsch, M.E.: Computing optimal Morse matchings. SIAM J. Discret. Math. 20(1), 11–25 (2006). https://doi.org/10.1137/S0895480104445885
Kaczynski, T., Mrozek, M., Wanner, T.: Towards a formal tie between combinatorial and classical vector field dynamics. J. Comput. Dyn. 3(1), 17–50 (2016). https://doi.org/10.3934/jcd.2016002
King, H., Knudson, K., Mramor, N.: Generating discrete Morse functions from point data. Exp. Math. 14(4), 435–444 (2005). https://doi.org/10.1080/10586458.2005.10128941
Lipiński, M., Kubica, J., Mrozek, M., Wanner, T.: Conley-Morse-Forman theory for generalized combinatorial multivector fields on finite topological spaces, pp. 1–44 (2020). arXiv:1911.12698. https://doi.org/10.48550/arXiv.1911.12698
Matsumoto, Y.: An Introduction to Morse Theory. American Mathematical Society, Providence, RI (2002)
Mrozek, M.: Conley-Morse-Forman theory for combinatorial multivector fields on Lefschetz complexes. Found. Comput. Math. 17(6), 1585–1633 (2017). https://doi.org/10.1007/s10208-016-9330-z
Mrozek, M., Wanner, T.: Creating semiflows on simplicial complexes from combinatorial vector fields. J. Differ. Equ. 304, 375–434 (2021). https://doi.org/10.1016/j.jde.2021.10.001
Munkres, J.R.: Elements of Algebraic Topology. Addison-Weslay, Cambridge (1984)
Nocedal, J., Wright, S.J.: Numerical Optimization. Springer, New York (2006)
Reininghaus, J., Hotz, I.: Combinatorial 2D vector field topology. In: Topological Methods in Data Analysis and Visualization: Theory, Algorithms, and Applications, pp. 103–114. Springer, Cham (2011). https://doi.org/10.1007/978-3-642-15014-2_9
Reininghaus, J., Lowen, C., Hotz, I.: Fast combinatorial vector field topology. IEEE Trans. Vis. Comput. Graph. 17(10), 1433–1443 (2010). https://doi.org/10.1109/TVCG.2010.235
Robins, V., John Wood, P., Sheppard, A.P.: Theory and algorithms for constructing discrete Morse complexes from grayscale digital images. IEEE Trans. Pattern Anal. Mach. Intell. 33(8), 14 (2010). https://doi.org/10.1109/TPAMI.2011.95
Szymczak, A.: Morse connection graphs for piecewise constant vector fields on surfaces. Comput. Aided Geom. Des. 30(6), 529–541 (2013). https://doi.org/10.1016/j.cagd.2012.03.022
Szymczak, A., Zhang, E.: Robust Morse decompositions of piecewise constant vector fields. IEEE Trans. Vis. Comput. Graph. 18(6), 938–951 (2012). https://doi.org/10.1109/TVCG.2011.88
Veinott, A.F., Dantzig, G.B.: Short notes: integral extreme points. SIAM Rev. 10(3), 371–372 (1968)
Yen, I.E.-H., Zhong, K., Hsieh, C.-J., Ravikumar, P.K., Dhillon, I.S.: Sparse linear programming via primal and dual augmented coordinate descent. In: Advances in Neural Information Processing Systems, vol. 28 (2015)
Acknowledgements
I would like to thank the Institut des Sciences Mathématiques (ISM) for their financial support. I will also like to thank Tomasz Kaczynski for discussions beneficial for this work and Marc Ethier for editing this paper.
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Institut des Sciences Mathématiques, Université du Québec à Montréal (14540).
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Partial financial support was received from Institut des Sciences Mathématiques (ISM). The author has no conflict of interest to declare that are relevant to the content of this article.
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Desjardins Côté, D. From finite vector field data to combinatorial dynamical systems in the sense of Forman. J Appl. and Comput. Topology 8, 669–694 (2024). https://doi.org/10.1007/s41468-024-00181-w
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DOI: https://doi.org/10.1007/s41468-024-00181-w
Keywords
- Computational topology
- Simplicial complex
- Combinatorial dynamical system
- Finite vector field data
- Matching algorithm