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A Fast Semi-Implicit Level Set Method for Curvature Dependent Flows with an Application to Limit Cycles Extraction in Dynamical Systems

Part of: Approximation methods and numerical treatment of dynamical systems Ergodic theory Basic methods in fluid mechanics

Published online by Cambridge University Press:  03 July 2015

Guoqiao You  and
Shingyu Leung
Guoqiao You
Affiliation:
The School of Science, Nanjing Audit University, Nanjing, Jiangsu Province, China
Shingyu Leung*
Affiliation:
Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong
*
*Corresponding author. Email addresses: [email protected] (G. You), [email protected] (S. Leung)
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Abstract

We propose a new semi-implicit level set approach to a class of curvature dependent flows. The method generalizes a recent algorithm proposed for the motion by mean curvature where the interface is updated by solving the Rudin-Osher-Fatemi (ROF) model for image regularization. Our proposal is general enough so that one can easily extend and apply the method to other curvature dependent motions. Since the derivation is based on a semi-implicit time discretization, this suggests that the numerical scheme is stable even using a time-step significantly larger than that of the corresponding explicit method. As an interesting application of the numerical approach, we propose a new variational approach for extracting limit cycles in dynamical systems. The resulting algorithm can automatically detect multiple limit cycles staying inside the initial guess with no condition imposed on the number nor the location of the limit cycles. Further, we also propose in this work an Eulerian approach based on the level set method to test if the limit cycles are stable or unstable.

Keywords

Numerical methods for PDEslevel set methoddynamical systemsflow visualization

MSC classification

Secondary: 37A25: Ergodicity, mixing, rates of mixing 37M25: Computational methods for ergodic theory (approximation of invariant measures, computation of Lyapunov exponents, entropy) 76M27: Visualization algorithms
Type
Research Article
Information
Communications in Computational Physics , Volume 18 , Issue 1 , July 2015 , pp. 203 - 229
Copyright
Copyright © Global-Science Press 2015 

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References

[1]Almgren, F., Taylor, J.E., and Wang, L.. Curvature-driven flow: A variational approach. SIAM J. Control and Optimization, 31:387–437, 1993.Google Scholar
[2]Angenent, S., Sapiro, G., and Tannenbaum, A.. On the affine invariant heat equation for non-convex curves. Journal of the American Mathematical Society, 11:601–634, 1998.CrossRefGoogle Scholar
[3]Arrow, K.J., Hurwics, L., and Uzawa, H.. Studies in linear and non-linear programming. Stanford Mathematical Studies in the Social Sciences, Vol II, Stanford University Press, 1958.Google Scholar
[4]Aubert, G. and Kornprobst, P.. Mathematical Problems in Image Processing - Partial Differential Equations and the Calculus of Variations. Springer, 2006.Google Scholar
[5]Barles, G. and Georgelin, C.. A simple proof of convergence for an approximation scheme for computing motions by mean curvature. SIAM J. Num. Anal., 32:484–500, 1995.CrossRefGoogle Scholar
[6]Caselles, V., Kimmel, R., and Sapiro, G.. Geodesic active contours. International Journal of Computer Vision, 22(1):61–79, 1997.Google Scholar
[7]Chambolle, A.. An algorithm for mean curvature motion. Interfaces and Free Boundaries, 6:195–218, 2004.Google Scholar
[8]Chambolle, A.. An algorithm for total variation minimization and applications. Journal of Mathematical Imaging and Vision, 20:89–97, 2004.Google Scholar
[9]Chambolle, A. and Pock, T.. A first-order primal-dual algorithm for convex problems with applications to imaging. Journal of Mathematical Imaging and Vision, 40(1):120–145, 2011.Google Scholar
[10]Chan, T. and Shen, J.. Image Processing and Analysis: Variational, PDE, Wavelet and Stochastic Methods. SIAM, Pennsylvania, US., 2005.Google Scholar
[11]Esch, J. and Rogers, T.D.. On the detection of limit cycles by the variational velocity method. Acta Applicandae Mathematicae, 54:345–354, 1998.CrossRefGoogle Scholar
[12]Esedoglu, S., Ruuth, S., and Tsai, R.. Diffusion generated motion using signed distance functions. J. Comput. Phys., 229:1017–1042, 2010.Google Scholar
[13]Esser, E., Zhang, X., and Chan, T.. A general framework for a class of first order primal-dual algorithms for tv minimization. UCLA CAM 09-67, 2009.Google Scholar
[14]Goldfarb, D. and Yin, W.. Second-order cone programming methods for total variation based image restoration. SIAM J. on Scientific Computing, 27:622–645, 2005.Google Scholar
[15]Goldstein, T. and Osher, S.. The split bregman method for l1 regularized problems. SIAM J. Imaging Sciences, 2:323–343, 2008.Google Scholar
[16]Gottlieb, S. and Shu, C.-W.. Total variation diminishing Runge-Kutta schemes. Mathematics of Computation, 67:73–85, 1998.Google Scholar
[17]Hale, E.T., Yin, W., and Zhang, Y.. Fixed-point continuation for 11-minimization: Methodology and convergence. SIAM J. on Optimization, 19:1107–1130, 2008.Google Scholar
[18]Haller, G.. Distinguished material surfaces and coherent structures in three-dimensional fluid flows. Physica D, 149:248–277, 2001.Google Scholar
[19]Haller, G. and Yuan, G.. Lagrangian coherent structures and mixing in two-dimensional turbulence. Physica D, 147:352–370, 2000.Google Scholar
[20]He, Y., Liu, Y., and Tang, T.. On large time-stepping methods for the cahn-hilliard equation. Appl. Num. Math., 57:616–628, 2007.Google Scholar
[21]Hilbert, D.. Mathematical problems. Bull. Amer. Math. Soc., 8:437–479, 1902.Google Scholar
[22]Ilyashenko, Y.. Centennial history of Hilbert’s 16th problem. Bull. Amer. Math. Soc., 39(3):301–354, 2002.Google Scholar
[23]Jiang, G. S. and Peng, D.. Weighted ENO schemes for Hamilton-Jacobi equations. SIAM J. Sci. Comput., 21:2126–2143, 2000.Google Scholar
[24]Kass, M., Witkin, A., and Terzopoulos, D.. Snakes: Active contour models. International Journal of Computer Vision, 1(4):321–331, 1988.Google Scholar
[25]Kimura, M. and Notsu, H.. A level set method using the signed distance function. Japan J. Indust. Appl. Math., 19:415–446, 2002.Google Scholar
[26]Korpelevich, G.M.. Extrapolational gradient methods and their connection with modified Lagrangians. Ehkon. Mat. Metody., 19:694–703, 1983.Google Scholar
[27]Leung, S.. An Eulerian approach for computing the finite time Lyapunov exponent. J. Comput. Phys., 230:3500–3524, 2011.Google Scholar
[28]Leung, S.. A backward phase flow method for the finite time Lyapunov exponent. Chaos, 23(043132), 2013.Google Scholar
[29]Leung, S. and Qian, J.. The backward phase flow and FBI-transform-based Eulerian Gaussian beams for the Schrödinger equation. J. Comput. Phys., 229:8888–8917, 2010.Google Scholar
[30]Liu, X. D., Osher, S. J., and Chan, T.. Weighted Essentially NonOscillatory schemes. J. Comput. Phys., 115:200–212, 1994.Google Scholar
[31]Malladi, R. and Sethian, J.A.. Flows under min/max curvature flow and mean curvature: Applications in image processing. Lecture Notes in Computer Science, 1064, 1996.Google Scholar
[32]Mediana, D. and Padilla, P.. A geometric approach to invariant sets for dynamical systems. Electronic Journal of Differential Equations, Conference 18:45–56, 2010.Google Scholar
[33]Merriman, B., Bence, J. K., and Osher, S. J.. Motion of multiple junctions: a level set approach. J. Comput. Phys., 112:334–363, 1994.Google Scholar
[34]Moisan, L.. Affine plane curve evolution: A fully consistent scheme. IEEE Transactions on Image Processing, 7:411–420, 1998.Google Scholar
[35]Oberman, A., Osher, S., Takei, R., and Tsai, R.. Numerical methods for smooth and crystalline mean curvature flow. UCLA CAM 10–21, 2010.Google Scholar
[36]Osher, S., Mao, Y., Dong, B., and Yin, W.. Fast linearized bregman iteration for compressive sensing and sparse denoising. Commun. Math. Sci., 8:93–111, 2010.Google Scholar
[37]Osher, S. J. and Fedkiw, R. P.. Level Set Methods and Dynamic Implicit Surfaces. Springer-Verlag, New York, 2003.Google Scholar
[38]Osher, S. J. and Sethian, J. A.. Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys., 79:12–49, 1988.CrossRefGoogle Scholar
[39]Pock, T., Cremers, D., Bischof, H., and Chambolle, A.. An algorithm for minimizing the mumford-shah functional. ICCV Proceedings, LNCS, Springer, 2009.Google Scholar
[40]Rudin, L., Osher, S.J., and Fatemi, E.. Nonlinear total variation based noise removal algorithms. Physica D, 60:259–268, 1992.Google Scholar
[41]Sethian, J.. Curvature and the evolution of fronts. Commun. Math. Phys., 101:487–499, 1985.Google Scholar
[42]Shadden, S.C., Lekien, F., and Marsden, J.E.. Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows. Physica D, 212:271–304, 2005.CrossRefGoogle Scholar
[43]Shu, C. W.. Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. In Cockburn, B., Johnson, C., Shu, C.W., and Tadmor, E., editors, Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, volume 1697, pages 325–432. Springer, 1998. Lecture Notes in Mathematics.Google Scholar
[44]Smereka, P.. Semi-implicit level set methods for curvature and surface diffusion motion. J. Sci. Comput., 1–3:439–456, 2003.Google Scholar
[45]Tasdizen, T., Whitaker, R., Burchard, P., and Osher, S.. Geometric surface processing via anisotropic diffusion of normals. Proceedings of IEEE Visualization 2002, pages 125–132, 2002.Google Scholar
[46]Tasdizen, T., Whitaker, R., Burchard, P., and Osher, S.. Geometric surface processing via normal maps. ACM Transactions on Graphics, 22, 2003.Google Scholar
[47]You, G. and Leung, S.. An Eulerian method for computing the coherent ergodic partition of continuous dynamical systems. J. Comp. Phys., 264:112–132, 2014.Google Scholar
[48]You, G. and Leung, S.. VIALS: An Eulerian tool based on total variation and the level set method for studying dynamical systems. J. Comp. Phys., 266:139–160, 2014.Google Scholar
[49]Zhao, H.-K., Chan, T., Merriman, B., and Osher, S. J.. A variational level set approach for multiphase motion. J. Comput. Phys., 127:179–195, 1996.Google Scholar
[50]Zhu, M. and Chan, T.. An efficient primal-dual hybrid gradient algorithm for total variation image restoration. UCLA CAM 08–34, 2009.Google Scholar