Level Set Methods: An Overview and Some Recent Results

Journal of Computational Physics

Volumn 169, Issue 2, 2001, Pages 463-502

  Osher, Stanley ^a   Fedkiw, Ronald P ^b  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b Stanford University   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

DROP BREAKUP; INCOMPRESSIBLE FLOW; LEVEL MEASUREMENT;

CONTINUOUS FUNCTIONS; LEVEL SET METHOD; LIPSCHITZ CONTINUOUS; MULTIPLY-CONNECTED REGION; SIMPLE METHOD; THREE DIMENSIONS; TWO-DIMENSIONS; VELOCITY FIELD; VERSATILE METHODS; ZERO LEVEL SET;

NUMERICAL METHODS;

EID: 0000653947     PISSN: 00219991     EISSN: None     Source Type: Journal    
DOI: 10.1006/jcph.2000.6636     Document Type: Article

References (90)

1. Adalsteinsson D., Sethian J. A. The fast construction of extension velocities in level set methods. J. Comput. Phys. 148:1999;2.
2. Adalsteinsson D., Sethian J. A. A fast level set method for propagating interfaces. J. Comput. Phys. 118:1995;269.
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. 122:1995;348.
4. Alvarez L., Guichard F., Lions P.-L., Morel J.-M. Axioms and fundamental equations of image processing. Arch. Ration. Mech. Anal. 123:1993;199.
5. Ambrosio L., Soner H. M. Level set approach to mean curvature flow in arbitrary codimension. J. Differential Geom. 43:1996;693.
6. Bardi M., Evans L. C. On Hopf's formulas for solutions of Hamilton-Jacobi equations. Nonlinear Anal. TMA. 8:1984;1373.
7. Bardi M., Osher S. The nonconvex multidimensional Riemann problem for Hamilton-Jacobi equations. SIAM J. Anal. 22:1991;344.
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.
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.
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. 36:1999;667.
12. Brackbill J. U., Kothe D. B., Zemach C. A continuum method for modeling surface tension. J. Comput. Phys. 100:1992;335.
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.
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. 1951;243.
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. 12:1999;13.
16. Caselles V., Catté F., Coll T., Dibo F. A geometric model for active contours in image processing. Numer. Math. 66:1993;1.
17. Caselles V., Kimmel R., Sapiro G. Geodesic active contours. Int. J. Comput. Vision. 22:1997;61.
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. 7:1998;269.
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.
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. 124:1996;449.
23. Chen Y. G., Giga Y., Goto S. Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations. J. Differential Geom. 33:1991;749.
24. Chen S., Merriman B., Osher S., Smereka P. A simple level set method for solving Stefan problems. J. Comput. Phys. 135:1997;8.
25. Cheng L. T., Fedkiw R. P., Gibou F., Kang M. A Symmetric Method for Implicit Time Discretization of the Stefan Problem. 2000;00-37.
26. Colella P., Majda A., Roytburd V. Theoretical and numerical structure for reacting shock waves. SIAM J. Sci. Stat. Comput. 7:1986;1059.
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. 27:1992;1.
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. 65:1992;1097.
30. Evans Y. C., Spruck J. Motion of level sets by mean curvature, I. J. Differential Geom. 33:1991;635.
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. 152:1999;457.
33. Fedkiw R., Aslam T., Xu S. The ghost fluid method for deflagration and detonation discountinuities. J. Comput. Phys. 154:1999;393.
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. 35:1918;137.
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.
37. Harabetian E., Osher S. Regularization of ill-posed problems via the level set approach. SIAM J. Appl. Math. 58:1998;1689.
38. Harabetian E., Osher S., Shu C.-W. An Eulerian approach for vortex motion using a level set approach. J. Comput. Phys. 127:1996;15.
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. 148:1999;366.
40. Helmsen J., Puckett E., Colella P., Dorr M. Two new methods for simulating photolithography development in 3D. Proc. SPIE. 2726:1996;253.
41. Hou T. Numerical solutions to free boundary problems. Acta Numer. 4:1995;335.
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. 134:1997;236.
43. Jiang G.-S., Peng D. Weighted ENO schemes for Hamilton-Jacobi equations. SIAM J. Sci. Comput. 21:2000;2126.
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.
45. Karni S. Hybrid multifluid algorithms. SIAM J. Sci. Comput. 17:1996;1019.
46. Karni S. Multicomponent flow calculations by a consistent primitive algorithm. J. Comput. Phys. 112:1994;31.
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. 63:1993;410.
49. Liu X.-D., Fedkiw R. P., Kang M. A boundary condition capturing method for Poisson's equation on irregular domains. J. Comput. Phys. 160:2000;151.
50. Markstein G. H. Nonsteady Flame Propagation. 1964.
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. 112:1994;334.
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. 100:1992;209.
56. Mumford D., Shah J. Optimal approximation by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math. 42:1989;577.
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.
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. 12:1991;1207.
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.
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. 1:1997;560.
63. Osher S. A level set formulation for the solution of the Dirichlet problem for Hamilton-Jacobi equations. SIAM J. Anal. 24:1993;1145.
64. Osher S., Sethian J. A. Fronts propagating with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79:1988;12.
65. Osher S., Shu C. W. High order essentially non-oscillatory schemes for Hamilton-Jacobi equations. SIAM J. Numer. Anal. 28:1991;907.
66. Peng D., Merriman B., Osher S., Zhao H.-K., Kang M. A PDE-based fast local level set method. J. Comput. Phys. 155:1999;410.
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.
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. 126:1996;837.
69. Rouy E., Tourin A. A viscosity solutions approach to shape-from-shading. SIAM J. Numer. Anal. 29:1992;867.
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. 60:1992;259.
73. Ruuth S., Merriman B., Osher S. A Fixed Grid Method for Capturing the Motion of Self-Interesecting Interfaces and Related PDEs. 163:2000;21.
74. Ruuth S., Merriman B., Xin J., Osher S. Diffusion-Generated Motion for Mean Curvature of Filaments. 1998.
75. Sethian J. A. Algorithm for tracking interfaces in CFD and materials science. Annu. Rev. Comput. Fluid Mech. 1995.
76. Sethian J. A. Fast marching level set methods for three dimensional photolithography development. Proc. SPIE. 2726:1996;261.
77. Sethian J. A. Fast marching methods. SIAM Rev. 41:1999;199.
78. Sethian J. A., Strain J. Crystal growth and dendritic solidification. J. Comput. Phys. 98:1992;231.
79. Schwarz K. W. Simulations of dislocations on the mesoscopic scale. I. Methods and examples. J. Appl. Phys. 85:1999;108.
80. Schwarz K. W. Simulations of dislocations on the mesoscopic scale. II. Application to strained-layer relaxation. J. Appl. Phys. 85:1999;120.
81. Soravia P. Generalized motion of a front propagating along its normal direction: A differential games approach. Nonlinear Anal. TMA. 22:1994;1247.
82. Steinhoff J., Fan M., Wang L. A new Eulerian method for the computation of propagating short acoustic and electromagnetic pulses. J. Comput. Phys. 157:2000;683.
83. Sussman M., Fatemi E., Smereka P., Osher S. An improved level set method for incompressible two-phase flow. Comput. and Fluids. 27:1998;663.
84. Sussman M., Smereka P., Osher S. A level set approach for computing solutions to incompressible two-phase flow. J. Comput. Phys. 114:1994;146.
85. R. Tsai, H.-K. Zhao, and, S. Osher, Fast sweeping algorithms for a class of Hamilton-Jacobi equations, in preparation.
86. Tsitsiklis J. N. Efficient algorithms for globally optimal trajectories. IEEE Trans. Autom. Control. 40:1995;1528.
87. Unverdi S. O., Tryggvason G. A. front-tracking method for viscous, incompressible, multi-fluid flows. J. Comput. Phys. 100:1992;25.
88. Zhao H.-K., Chan T., Merriman B., Osher S. A variational level set approach to multiphase motion. J. Comput. Phys. 127:1996;179.
89. Zhao H.-K., Merriman B., Osher S., Kang M. Implicit Nonparametric Shape Reconstruction from Unorganized Points Using a Variational Level Set Method. 80:2000;295.
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. 143:1998;495.