Convolution-thresholding methods for interface motion. (English) Zbl 0988.65094
The paper deals with numerical methods for convolution thresholding of interface motion. This approach generalizes Huygens’ principle, threshold growth cellular automata and reaction-diffusion equations. The paper summarizes the relation of convolution-thresholding schemes to previous methods and reviews the theoretical and algorithmic development of this approach. The paper is more oriented for practical computations.
Reviewer: V.Dolejsi (Praha)
MSC:
| 65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |
| 35K57 | Reaction-diffusion equations |
| 35R35 | Free boundary problems for PDEs |
Keywords:
convolution-thresholding methods; spectral method; interface motion; Huygens’ principle; cellular automata; reaction-diffusion equationsReferences:
| [1] | Barles, G.; Georgelin, C., A simple proof of convergence for an approximation scheme for computing motions by mean curvature, SIAM J. Numer. Anal., 32, 484 (1995) · Zbl 0831.65138 |
| [2] | Beylkin, G., On the fast Fourier transform of functions with singularities, Appl. Comput. Harmonic Anal., 2, 363 (1995) · Zbl 0838.65142 |
| [3] | Bronsard, L.; Stoth, B., Volume Preserving Mean Curvature Flow as a Limit of a Nonlocal Ginzburg-Landau Equation (1994) |
| [4] | Chopp, D. L., Computing minimal surfaces via level set curvature flow, J. Comput. Phys., 106, 77 (1993) · Zbl 0786.65015 |
| [5] | Ermentrout, G. B.; Edelstein-Keshet, L., Cellular automata approaches to biological modeling, J. Theor. Biol., 160, 97 (1993) |
| [6] | Evans, L. C., Convergence of an algorithm for mean curvature motion, Indiana Univ. Math. J., 42, 553 (1993) · Zbl 0802.65098 |
| [7] | Evans, L. C.; Soner, H. M.; Souganidis, P. E., Phase transitions and generalized motion by mean curvature, Commun. Pure Appl. Math., 45, 1097 (1992) · Zbl 0801.35045 |
| [8] | Fast, V. G.; Efimov, I. G., Stability of vortex rotation in an excitable cellular medium, Physica D, 49, 75 (1991) |
| [9] | Gerhardt, M.; Schuster, H.; Tyson, J. J., A cellular automata model of excitable media. II. Curvature, dispersion, rotating waves and meandering waves, Physica D, 46, 392 (1990) · Zbl 0800.92028 |
| [10] | Gerhardt, M.; Schuster, H.; Tyson, J. J., A cellular automata model of excitable media. III. Fitting the Belousov-Zhabotinsky reaction, Physica D, 46 (1990) · Zbl 0800.92029 |
| [11] | Gerhardt, M.; Schuster, H.; Tyson, J. J., A cellular automata model of excitable media including curvature and dispersion, Science, 247, 416 (1990) · Zbl 0800.92029 |
| [12] | Gravner, J.; Griffeath, D., Threshold growth dynamics, Trans. Am. Math. Soc., 340, 837 (1993) · Zbl 0791.58053 |
| [13] | Gravner, J.; Griffeath, D., Cellular automata growth on \(z^2\), Adv. Appl. Math., 21, 241 (1998) · Zbl 0919.68090 |
| [14] | Gutowitz, H. A., Introduction, Cellular Automata Theory and Experiment, vii (1991) |
| [15] | Henze, C.; Tyson, J., Cellular automaton model of three-dimensional excitable media, J. Chem. Soc. Faraday Trans., 92, 2883 (1996) |
| [16] | C. Herring, The use of classical macroscopic concepts in surface energy problems, in, Structure and Pro- perities of Solid Surfaces, edited by, R. Gomer and C. S. Smith, University of Chicago, Chicago, 1952, p, 5.; C. Herring, The use of classical macroscopic concepts in surface energy problems, in, Structure and Pro- perities of Solid Surfaces, edited by, R. Gomer and C. S. Smith, University of Chicago, Chicago, 1952, p, 5. |
| [17] | H. Ishii, A generalization of the Bence, Merriman and Osher algorithm for motion by mean curvature, in, Curvature Flows and Related Topics, edited by, A. Damlamian, J. Spruck, and A. Visintin, Gakkôtosho, Tokyo, 1995, p, 111.; H. Ishii, A generalization of the Bence, Merriman and Osher algorithm for motion by mean curvature, in, Curvature Flows and Related Topics, edited by, A. Damlamian, J. Spruck, and A. Visintin, Gakkôtosho, Tokyo, 1995, p, 111. · Zbl 0844.35043 |
| [18] | Ishii, H.; Pires, G. E.; Souganidis, P. E., TMU Math. Preprint Ser., 4 (1996) |
| [19] | MacLennan, B. J., Continuous Spatial Automata (1990) |
| [20] | Markus, M.; Hess, B., Isotropic cellular automaton for modeling excitable media, Nature, 347, 56 (1990) |
| [21] | Mascarenhas, P., Diffusion Generated Motion by Mean Curvature (1992) |
| [22] | B. Merriman, J. Bence, and, S. Osher, Diffusion generated motion by mean curvature, in, Computational Crystal Growers Workshop, edited by, J. E. Taylor, American Mathematical Society, Providence, Rhode Island, 1992, p, 73. Also available as UCLA CAM Report 92-18, April 1992.; B. Merriman, J. Bence, and, S. Osher, Diffusion generated motion by mean curvature, in, Computational Crystal Growers Workshop, edited by, J. E. Taylor, American Mathematical Society, Providence, Rhode Island, 1992, p, 73. Also available as UCLA CAM Report 92-18, April 1992. |
| [23] | Merriman, B.; Bence, J.; Osher, S., J. Comput. Phys., 112, 334 (1994) |
| [24] | Osher, S.; Merriman, B., Asian J. Math., 1, 560 (1997) · Zbl 0891.49023 |
| [25] | Packard, N.; Wolfram, S., J. Stat. Phys., 38, 901 (1985) · Zbl 0625.68038 |
| [26] | Perona, P., IEEE Trans. Image Process., 7, 457 (1998) |
| [27] | Ringach, D.; Sapiro, G.; Shapley, R., Vision Res., 37, 2455 (1997) |
| [28] | Rubinstein, J., Self-induced motion of line defects, Q. Appl. Math., 49, 1 (1991) · Zbl 0728.35118 |
| [29] | Rubinstein, J.; Sternberg, P., IMA J. Appl. Math., 48, 248 (1992) |
| [30] | Ruuth, S. J., Efficient Algorithms for Diffusion-Generated Motion by Mean Curvature (1996) |
| [31] | Ruuth, S. J., A diffusion-generated approach to multiphase motion, J. Comput. Phys., 145, 166 (1998) · Zbl 0929.76101 |
| [32] | Ruuth, S. J., Efficient algorithms for diffusion-generated motion by mean curvature, J. Comput. Phys., 144, 603 (1998) · Zbl 0946.65093 |
| [33] | Ruuth, S. J.; Merriman, B., Convolution generated motion and generalized Huygens’ principles for interface motion, SIAM J. Appl. Math., 60, 868 (2000) · Zbl 0958.65021 |
| [34] | Ruuth, S. J.; Merriman, B.; Osher, S., Convolution generated motion as a link between cellular automata and continuum pattern dynamics, J. Comput. Phys., 151, 836 (1999) · Zbl 0938.68058 |
| [35] | Ruuth, S. J.; Merriman, B.; Xin, J.; Osher, S., Diffusion-Generated Motion by Mean Curvature for Filaments (1998) · Zbl 1037.35033 |
| [36] | Schönfisch, B., Anisotropy in cellular automata, Biosystems, 41, 29 (1997) |
| [37] | Swindale, N. V., A model for the formation of ocular dominance stripes, Proc. R. Soc. London B, 208, 243 (1980) |
| [38] | Taylor, J. E., Mean curvature and weighted mean curvature, Acta Metall. Meter., 40, 1475 (1992) |
| [39] | Weimar, J. R.; Tyson, J. J.; Watson, L. T., Diffusion and wave propagation in cellular automata models of excitable media, Physica D, 55, 309 (1992) · Zbl 0744.92002 |
| [40] | Weimar, J. R.; Tyson, J. J.; Watson, L. T., Physica D, 55, 328 (1992) · Zbl 0744.92003 |
| [41] | Wiener, N.; Rosenblueth, A., The mathematical formulation of the problem of conduction of impulses in a network of connected excitable elements, specifically in cardiac muscle, Arch. Inst. Cardiol. Mexico, 16, 205 (1946) · Zbl 0134.37904 |
| [42] | Winfree, A. T., When Time Breaks Down (1987) |
| [43] | Winfree, A. T., Stable particle-like solutions to the nonlinear wave equations of three-dimensional excitable media, SIAM Rev., 32, 1 (1990) · Zbl 0711.35067 |
| [44] | Wolfram, S., Universality and complexity in cellular automata, Physica D, 10, 1 (1984) · Zbl 0562.68040 |
| [45] | Young, D. A., A local activator-inhibitor model of vertebrate skin patterns, Math. Biosci., 72, 51 (1984) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
