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Level set methods: An overview and some recent results. (English) Zbl 0988.65093
The basic goal of the paper is to compare and analyze the subsequent motion of \(\Gamma\) under a velocity field \(v\) (depending on \(\Gamma\)).
The paper contains the key definitions and basic level set technology, as well as a few words about the numerical implementation. Moreover, recent variants, extensions and a rather interesting selection of related fast numerical methods are given. Applications of the moving interfaces in compressible, incompressible flows, Stefano problems, kinetic crystal growth, epitaxial growth of thin fibers, vertex-dominated flows, and extensions to multiphase motion are presented.
Reviewer: V.Dolejsi (Praha)

MSC:
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
76M25 Other numerical methods (fluid mechanics) (MSC2010)
76T30 Three or more component flows
80A22 Stefan problems, phase changes, etc.
35K55 Nonlinear parabolic equations
35R35 Free boundary problems for PDEs
82C26 Dynamic and nonequilibrium phase transitions (general) in statistical mechanics
82D25 Statistical mechanics of crystals
Software:
SLIC
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References:
[1] Adalsteinsson, D.; Sethian, J.A., The fast construction of extension velocities in level set methods, J. comput. phys., 148, 2, (1999) · Zbl 0919.65074
[2] Adalsteinsson, D.; Sethian, J.A., A fast level set method for propagating interfaces, J. comput. phys., 118, 269, (1995) · Zbl 0823.65137
[3] Adalsteinsson, D.; Sethian, J.A., A level set approach to a unified model for etching, deposition, and lithography. II. three-dimensional simulations, J. comput. phys., 122, 348, (1995) · Zbl 0840.65131
[4] Alvarez, L.; Guichard, F.; Lions, P.-L.; Morel, J.-M., Axioms and fundamental equations of image processing, Arch. ration. mech. anal., 123, 199, (1993) · Zbl 0788.68153
[5] Ambrosio, L.; Soner, H.M., Level set approach to Mean curvature flow in arbitrary codimension, J. differential geom., 43, 693, (1996) · Zbl 0868.35046
[6] Bardi, M.; Evans, L.C., On Hopf’s formulas for solutions of hamilton – jacobi equations, Nonlinear anal. TMA, 8, 1373, (1984) · Zbl 0569.35011
[7] Bardi, M.; Osher, S., The nonconvex multidimensional Riemann problem for hamilton – jacobi equations, SIAM J. anal., 22, 344, (1991) · Zbl 0733.35073
[8] Barles, G., Solutions de viscosité des equations de hamilton – jacobi, (1996)
[9] G. Bellettini, M. Novaga, and, M. Paolini, An example of three dimensional fattening for linked space curves evolving by curvature, Comm. Partial Differential Equations, in press. · Zbl 0958.35065
[10] M. Bertalmio, L. T. Cheng, S. Osher, and, G. Sapiro, Variational Problems and Partial Differential Equations on Implicit Surfaces: The Framework and Examples in Image Processing and Pattern Formation, CAM Report 00-23, University of California, Los Angeles), submitted for publication. · Zbl 0991.65055
[11] Boué, M.; Dupuis, P., Markov chain approximations for deterministic control problems with affine dynamics and quadratic cost in the control, SIAM J. numer. anal., 36, 667, (1999) · Zbl 0933.65073
[12] Brackbill, J.U.; Kothe, D.B.; Zemach, C., A continuum method for modeling surface tension, J. comput. phys., 100, 335, (1992) · Zbl 0775.76110
[13] P. Burchard, L.-T. Cheng, B. Merriman, and, S. Osher, Motion of Curves in Three Spatial Dimensions Using a Level Set Approach, CAM Report 00-29, University of California, Los Angeles, submitted for publication. · Zbl 0991.65077
[14] Burton, W.K.; Cabrera, N.; Frank, F.C., The growth of crystals and the equilibrium structure of their surfaces, Philos. trans. R. soc. London ser. A, 243, (1951) · Zbl 0043.23402
[15] Caflisch, R.E.; Gyure, M.; Merriman, B.; Osher, S.; Ratsch, C.; Vvedensky, D.; Zinck, J., Island dynamics and the level set method for epitaxial growth, Appl. math. lett., 12, 13, (1999) · Zbl 0937.35191
[16] Caselles, V.; Catté, F.; Coll, T.; Dibo, F., A geometric model for active contours in image processing, Numer. math., 66, 1, (1993) · Zbl 0804.68159
[17] Caselles, V.; Kimmel, R.; Sapiro, G., Geodesic active contours, Int. J. comput. vision, 22, 61, (1997) · Zbl 0894.68131
[18] Caselles, V.; Morel, J.-M.; Sapiro, G.; Tannenbaum, A., Special issue on partial differential equations and geometry-driven diffusion in image processing and analysis, IEEE trans. image process., 7, 269, (1998)
[19] T. Chan, R. Fedkiw, M. Kang, and, L. Vese, Improvements in the Efficiency and Robustness of Active Contour Algorithms, in preparation.
[20] Chan, T.; Vese, L., Active contours without edges, (1998) · Zbl 1039.68779
[21] T. Chan, and, L. Vese, An active contour model without edges, in Lecture Notes in Computer Science, edited by, M. Neilsen, P. JohansenO. F. Olsen, and J. Weickert, Springer-Verlag, Berlin/New York, 1999, Vol, 1687, p, 141.
[22] Chang, Y.C.; Hou, T.Y.; Merriman, B.; Osher, S., A level set formulation of Eulerian interface capturing methods for incompressible fluid flows, J. comput. phys., 124, 449, (1996) · Zbl 0847.76048
[23] Chen, Y.G.; Giga, Y.; Goto, S., Uniqueness and existence of viscosity solutions of generalized Mean curvature flow equations, J. differential geom., 33, 749, (1991) · Zbl 0696.35087
[24] Chen, S.; Merriman, B.; Osher, S.; Smereka, P., A simple level set method for solving Stefan problems, J. comput. phys., 135, 8, (1997) · Zbl 0889.65133
[25] Cheng, L.T.; Fedkiw, R.P.; Gibou, F.; Kang, M., A symmetric method for implicit time discretization of the Stefan problem, 00-37, (2000)
[26] Colella, P.; Majda, A.; Roytburd, V., Theoretical and numerical structure for reacting shock waves, SIAM J. sci. stat. comput., 7, 1059, (1986) · Zbl 0633.76060
[27] Crandall, M.G.; Ishii, H.; Lions, P.-L., User’s guide to viscosity solutions of second order partial differential equations, Am. math. soc. bull., 27, 1, (1992) · Zbl 0755.35015
[28] E. DeGiorgi, Barriers, boundaries, motion of manifolds, lectures presented in Pavia, Italy, 1994.
[29] Evans, Y.C.; Soner, H.M.; Souganidis, P.E., Phase transitions and generalized motion by Mean curvature, Commun. pure appl. math., 65, 1097, (1992) · Zbl 0801.35045
[30] Evans, Y.C.; Spruck, J., Motion of level sets by Mean curvature, I, J. differential geom., 33, 635, (1991) · Zbl 0726.53029
[31] R. Fedkiw, A symmetric spatial discretization for implicit time discretization of Stefan type problems, unpublished.
[32] Fedkiw, R.; Aslam, T.; Merriman, B.; Osher, S., A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method), J. comput. phys., 152, 457, (1999) · Zbl 0957.76052
[33] Fedkiw, R.; Aslam, T.; Xu, S., The ghost fluid method for deflagration and detonation discountinuities, J. comput. phys., 154, 393, (1999) · Zbl 0955.76071
[34] R. Fedkiw, and, X.-D. Liu, The ghost fluid method for viscous flows, in, Progress in Numerical Solutions of Partial Differential Equations, edited by, M. Hafez, Arcachon, 1998.
[35] Gross, R., Zur theorie des washstrums und losungsforganges kristalliner materic, Abh. math.-phys. kl. saechs. akad. wiss., 35, 137, (1918)
[36] Gyure, M.; Ratsch, C.; Merriman, B.; Caflisch, R.E.; Osher, S.; Zinck, J.; Vvedensky, D., Level set methods for the simulation of epitaxial phenomena, Phys. rev. E, 59, (1998) · Zbl 0937.35191
[37] Harabetian, E.; Osher, S., Regularization of ill-posed problems via the level set approach, SIAM J. appl. math., 58, 1689, (1998) · Zbl 0914.65098
[38] Harabetian, E.; Osher, S.; Shu, C.-W., An Eulerian approach for vortex motion using a level set approach, J. comput. phys., 127, 15, (1996) · Zbl 0859.76052
[39] Helenbrook, B.T.; Martinelli, L.; Law, C.K., A numerical method for solving incompressible flow problems with a surface of discontinuity, J. comput. phys., 148, 366, (1999) · Zbl 0931.76058
[40] Helmsen, J.; Puckett, E.; Colella, P.; Dorr, M., Two new methods for simulating photolithography development in 3D, Proc. SPIE, 2726, 253, (1996)
[41] Hou, T., Numerical solutions to free boundary problems, Acta numer., 4, 335, (1995) · Zbl 0831.65137
[42] Hou, T.; Li, Z.; Osher, S.; Zhao, H.-K., A hybrid method for moving interface problems with application to the hele – shaw flow, J. comput. phys., 134, 236, (1997) · Zbl 0888.76067
[43] Jiang, G.-S.; Peng, D., Weighted ENO schemes for hamilton – jacobi equations, SIAM J. sci. comput., 21, 2126, (2000) · Zbl 0957.35014
[44] M. Kang, R. Fedkiw, and, X.-D. Liu, A Boundary Condition Capturing Method for Multiphase Incompressible Flow, CAM Report 99-21, University of California, Los Angeles, submitted for publication. · Zbl 1049.76046
[45] Karni, S., Hybrid multifluid algorithms, SIAM J. sci. comput., 17, 1019, (1996) · Zbl 0860.76056
[46] Karni, S., Multicomponent flow calculations by a consistent primitive algorithm, J. comput. phys., 112, 31, (1994) · Zbl 0811.76044
[47] Kim, Y.-T.; Goldenfeld, N.; Dantzig, J., Computation of dendritic microstructures using a level set method, Phys. rev. E, 62, (2000)
[48] Kobayashi, R., Modeling and numerical simulations of dendritic crystal growth, Physica D, 63, 410, (1993) · Zbl 0797.35175
[49] Liu, X.-D.; Fedkiw, R.P.; Kang, M., A boundary condition capturing method for Poisson’s equation on irregular domains, J. comput. phys., 160, 151, (2000) · Zbl 0958.65105
[50] Markstein, G.H., Nonsteady flame propagation, (1964) · Zbl 0077.19201
[51] Mascarenhas, P., Diffusion generated motion by Mean curvature, (1992)
[52] B. Merriman, J. Bence, and, S. Osher, Diffusion generated motion by mean curvature, in, AMS Select Lectures in Mathematics: The Computational Crystal Grower’s Workshop, edited by, J. Taylor, Am. Math. Soc. Providence, 1993, p, 73.
[53] Merriman, B.; Bence, J.; Osher, S., Motion of multiple junctions: A level set approach, J. comput. phys., 112, 334, (1994)
[54] B. Merriman, R. Cafiisch, and, S. Osher, Level set methods with an application to modelling the growth of thin flims, in, Free Boundary Value Problems, Theory and Applications, edited by, I. Athanasopoulos, G. Makrikis, and J. F. Rodriguez, CRC Press, Boca Raton, FL, 1999, p, 51.
[55] Mulder, W.; Osher, S.; Sethian, J.A., Computing interface motion in compressible gas dynamics, J. comput. phys., 100, 209, (1992) · Zbl 0758.76044
[56] Mumford, D.; Shah, J., Optimal approximation by piecewise smooth functions and associated variational problems, Commun. pure appl. math., 42, 577, (1989) · Zbl 0691.49036
[57] D. Nguyen, R. P. Fedkiw, and, M. Kang, A Boundary Condition Capturing Method for Incompressible Flame Discontinuities, CAM Report 00-19, University of California, Los Angeles, submitted for publication. · Zbl 1065.76575
[58] Nielsen, M.; Johansen, P.; Olsen, O.F.; Weickert, J., Scale space theories in computer vision, 1682, (1999)
[59] Nochetto, R.H.; Paolini, M.; Verdi, C., An adaptive finite element method for two phase Stefan problems in two space dimensions. II. implementation and numerical experiments, SIAM J. sci. comput., 12, 1207, (1991) · Zbl 0733.65088
[60] W. F. Noh, and, P. R. Woodward, SLIC (Simple Line Interface Construction), Lecture Notes in Physics, edited by, A. van de Vooren and P. J. Zandbergen, Springer-Verlag, Berlin, 1976, Vol, 59, p, 330. · Zbl 0382.76084
[61] S. Osher, and, J. Helmsen, A generalized fast algorithm with applications to ion etching, in progress.
[62] Osher, S.; Merriman, B., The Wulff shape as the asymptotic limit of a growing crystalline interface, Asian J. math., 1, 560, (1997) · Zbl 0891.49023
[63] Osher, S., A level set formulation for the solution of the Dirichlet problem for hamilton – jacobi equations, SIAM J. anal., 24, 1145, (1993) · Zbl 0804.35021
[64] Osher, S.; Sethian, J.A., Fronts propagating with curvature dependent speed: algorithms based on hamilton – jacobi formulations, J. comput. phys., 79, 12, (1988) · Zbl 0659.65132
[65] Osher, S.; Shu, C.W., High order essentially non-oscillatory schemes for hamilton – jacobi equations, SIAM J. numer. anal., 28, 907, (1991) · Zbl 0736.65066
[66] Peng, D.; Merriman, B.; Osher, S.; Zhao, H.-K.; Kang, M., A PDE-based fast local level set method, J. comput. phys., 155, 410, (1999) · Zbl 0964.76069
[67] D. Peng, S. Osher, B. Merriman, and, H.-K. Zhao, The geometry of Wulff crystal shapes and its relations with Riemann problems, in Contemporary Mathematics, edited by, G. Q. Chen and E. DeBenedetto, Am. Math. Soc. Providence, RI, 1999, Vol, 238, p, 251. · Zbl 0942.35113
[68] Reitich, F.; Soner, H.M., Three phase boundary motions under constant velocities. I. the vanishing surface tension limit, Proc. R. soc. Edinburgh ser. A, 126, 837, (1996) · Zbl 0861.35122
[69] Rouy, E.; Tourin, A., A viscosity solutions approach to shape-from-shading, SIAM J. numer. anal., 29, 867, (1992) · Zbl 0754.65069
[70] Rudin, L.I., Images, numerical analysis of singularities, and shock filters, (1987)
[71] L. I. Rudin, and, S. Osher, Total variation based restoration with free local constraints, in, Proceedings of the ICIP, IEEE International Conference on Image Processing, Austin, TX, 1994, p, 31.
[72] Rudin, L.I.; Osher, S.; Fatemi, E., Nonlinear total variation based noise removal algorithms, Physica D, 60, 259, (1992) · Zbl 0780.49028
[73] Ruuth, S.; Merriman, B.; Osher, S., A fixed grid method for capturing the motion of self-interesecting interfaces and related pdes, 163, 21, (2000) · Zbl 0963.65097
[74] Ruuth, S.; Merriman, B.; Xin, J.; Osher, S., Diffusion-generated motion for Mean curvature of filaments, (1998) · Zbl 1037.35033
[75] Sethian, J.A., Algorithm for tracking interfaces in CFD and materials science, Annu. rev. comput. fluid mech., (1995) · Zbl 0875.76397
[76] Sethian, J.A., Fast marching level set methods for three dimensional photolithography development, Proc. SPIE, 2726, 261, (1996)
[77] Sethian, J.A., Fast marching methods, SIAM rev., 41, 199, (1999) · Zbl 0926.65106
[78] Sethian, J.A.; Strain, J., Crystal growth and dendritic solidification, J. comput. phys., 98, 231, (1992) · Zbl 0752.65088
[79] Schwarz, K.W., Simulations of dislocations on the mesoscopic scale. I. methods and examples, J. appl. phys., 85, 108, (1999)
[80] Schwarz, K.W., Simulations of dislocations on the mesoscopic scale. II. application to strained-layer relaxation, J. appl. phys., 85, 120, (1999)
[81] Soravia, P., Generalized motion of a front propagating along its normal direction: A differential games approach, Nonlinear anal. TMA, 22, 1247, (1994) · Zbl 0814.35140
[82] Steinhoff, J.; Fan, M.; Wang, L., A new Eulerian method for the computation of propagating short acoustic and electromagnetic pulses, J. comput. phys., 157, 683, (2000) · Zbl 1043.78556
[83] Sussman, M.; Fatemi, E.; Smereka, P.; Osher, S., An improved level set method for incompressible two-phase flow, Comput. and fluids, 27, 663, (1998) · Zbl 0967.76078
[84] Sussman, M.; Smereka, P.; Osher, S., A level set approach for computing solutions to incompressible two-phase flow, J. comput. phys., 114, 146, (1994) · Zbl 0808.76077
[85] R. Tsai, H.-K. Zhao, and, S. Osher, Fast sweeping algorithms for a class of Hamilton-Jacobi equations, in preparation. · Zbl 1049.35020
[86] Tsitsiklis, J.N., Efficient algorithms for globally optimal trajectories, IEEE trans. autom. control, 40, 1528, (1995) · Zbl 0831.93028
[87] Unverdi, S.O.; Tryggvason, G., A. front-tracking method for viscous, incompressible, multi-fluid flows, J. comput. phys., 100, 25, (1992) · Zbl 0758.76047
[88] Zhao, H.-K.; Chan, T.; Merriman, B.; Osher, S., A variational level set approach to multiphase motion, J. comput. phys., 127, 179, (1996) · Zbl 0860.65050
[89] Zhao, H.-K.; Merriman, B.; Osher, S.; Kang, M., Implicit nonparametric shape reconstruction from unorganized points using a variational level set method, 80, 295, (2000) · Zbl 1011.68538
[90] Zhao, H.-K.; Merriman, B.; Osher, S.; Wang, L., Capturing the behavior of bubbles and drops using the variational level set approach, J. comput. phys., 143, 495, (1998) · Zbl 0936.76065
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