The flexible, extensible and efficient toolbox of level set methods

Journal of Scientific Computing

Volumn 35, Issue 2-3, 2008, Pages 300-329

  Mitchell, Ian M ^a  

^a UNIVERSITY OF BRITISH COLUMBIA   (Canada)

Author keywords

Dynamic implicit surfaces; Hamilton Jacobi equations; Level set methods; Numerical software; Reproducible research

Indexed keywords

ARBITRARY DIMENSION; CARTESIAN GRIDS; CODING PATTERNS; DYNAMIC IMPLICIT; DYNAMIC IMPLICIT SURFACES; HAMILTON-JACOBI; HAMILTON-JACOBI EQUATIONS; IMPLEMENTATION COMPLEXITY; LEVEL SET METHODS; LEVEL SETS; LEVEL-SET ALGORITHMS; MATLAB ROUTINES; NEW RESULTS; NUMERICAL ALGORITHMS; NUMERICAL SOFTWARE; RECTANGULAR DOMAINS; REPRODUCIBLE RESEARCH;

AIRCRAFT ENGINES; ALGORITHMS; CHLORINE COMPOUNDS; CODES (STANDARDS); CODES (SYMBOLS); DROP BREAKUP; LEVEL MEASUREMENT; MATLAB; PARTIAL DIFFERENTIAL EQUATIONS; PROGRAMMING THEORY;

SET THEORY;

EID: 49749109557     PISSN: 08857474     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10915-007-9174-4     Document Type: Article

References (42)

1. Adalsteinsson, D., Sethian, J.A.: The fast construction of extension velocities in level set methods. J. Comput. Phys. 148, 2-22 (1999)
2. Barles, G., Souganidis, P.E.: Convergence of approximation schemes for fully nonlinear second order equations. Asymptot. Anal. 4(3), 271-283 (1991)
3. Cardaliaguet, P., Quincampoix, M., Saint-Pierre, P.: Set-valued numerical analysis for optimal control and differential games. In: Bardi, M., Raghavan, T.E.S., Parthasarathy, T. (eds.) Stochastic and Differential Games: Theory and Numerical Methods. Annals of International Society of Dynamic Games, vol. 4, pp. 177-247. Birkhäuser, Basel (1999)
4. Chopp, D.: Computing minimal surfaces via level set curvature flow. J. Comput. Phys. 106, 77-91 (1993)
5. Chu, K.T., Prodanovic, M.: Level set method library (LSMLIB). [Online]. Available: http://www.princeton.edu/~ktchu/software/lsmlib/
6. Crandall, M.G., Lions, P.-L.: Viscosity solutions of Hamilton-Jacobi equations. Trans. Am. Math. Soc. 277(1), 1-42 (1983)
7. Crandall, M.G., Lions, P.-L.: Two approximations of solutions of Hamilton-Jacobi equations. Math. Comput. 43(167), 1-19 (1984)
8. Crandall, M.G., Ishii, H., Lions, P.-L.: User's guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. 27(1), 1-67 (1992)
9. [Online]. Available: http://creativecommons.org
10. 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, 457-492 (1999)
11. Jiang, G.-S., Peng, D.: Weighted ENO schemes for Hamilton-Jacobi equations. SIAM J. Sci. Comput. 21, 2126-2143 (2000)
12. J. Sci. Comput. 19 (1-3) (2003)
13. Kao, C.-Y., Osher, S., Qian, J.: Lax-Friedrichs sweeping schemes for static Hamilton-Jacobi equations. J. Comput. Phys. 196, 367-391 (2004)
14. Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differential Equations, 3rd edn. Applications of Mathematics. Springer, New York (1999)
15. Kurzhanski, A.B., Mitchell, I.M., Varaiya, P.: Control synthesis for state constrained systems and obstacle problems. In: Proceedings of the IFAC Workshop on Nonlinear Control (NOLCOS). Vienna, Austria (2004)
16. Mallet, V.: Multivac C++ library. [Online]. Available: http://vivienmallet.net/fronts/index.php
17. [Online]. Available: http://www.cs.ubc.ca/~mitchell/ToolboxLS
18. Mitchell, I.M.: A toolbox of level set methods (version 1.1). Department of Computer Science, University of British Columbia, Vancouver, BC, Canada, Tech. Rep. TR-2007-11, June 2007. [Online]. Available: http://www.cs.ubc.ca/ ~mitchell/ToolboxLS/toolboxLS.pdf
19. Mitchell, I.M., Templeton, J.A.: A toolbox of Hamilton-Jacobi solvers for analysis of nondeterministic continuous and hybrid systems. In: Morari, M., Thiele, L. (eds.) Hybrid Systems: Computation and Control. Lecture Notes in Computer Science, vol. 3414, pp. 480-494. Springer, New York (2005)
20. Mitchell, I., Tomlin, C.: Level set methods for computation in hybrid systems. In: Krogh, B., Lynch, N. (eds.) Hybrid Systems: Computation and Control. Lecture Notes in Computer Science, vol. 1790, pp. 310-323. Springer, New York (2000)
21. Mitchell, I.M., Bayen, A.M., Tomlin, C.J.: A time-dependent Hamilton-Jacobi formulation of reachable sets for continuous dynamic games. IEEE Trans. Autom. Control 50(7), 947-957 (2005)
22. Oberman, A.: A convergent upwind difference scheme for motion of level sets by mean curvature. Numer. Math. 99(2), 365-379 (2004)
23. Oberman, A.: Convergent difference schemes for degenerate elliptic and parabolic equations: Hamilton-Jacobi equations and free boundary problems. SIAM J. Numer. Anal. 44(2), 879-895 (2006)
24. Øksendal, B.: Stochastic Differential Equations: An Introduction with Applications, 6th edn. Springer, New York (2003)
25. Osher, S.: A level set formulation for the solution of the Dirichlet problem for Hamilton-Jacobi equations. SIAM J. Math. Anal. 24(5), 1145-1152 (1993)
26. Osher, S., Fedkiw, R.: Level set methods: an overview and some recent results. J. Comput. Phys. 169, 463-502 (2001)
27. Osher, S., Fedkiw, R.: Level Set Methods and Dynamic Implicit Surfaces. Springer, New York (2002)
28. Osher, S., Paragios, N. (eds.): Geometric Level Set Methods in Imaging, Vision and Graphics. Springer, New York (2003)
29. Osher, S., Sethian, J.A.: Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79(1), 12-49 (1988)
30. Osher, S., Shu, C.-W.: High-order essentially nonoscillatory schemes for Hamilton-Jacobi equations. SIAM J. Numer. Anal. 28(4), 907-922 (1991)
31. Peng, D., Merriman, B., Osher, S., Zhao, H., Kang, M.: A PDE based fast local level set method. J. Comput. Phys. 165, 410-438 (1999)
32. Peyré, G.: Toolbox fast marching. [Online]. Available: http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=6110
33. Russo, G., Smereka, P.: A remark on computing distance functions. J. Comput. Phys. 163, 51-67 (2000)
34. Sethian, J.A.: Level Set Methods and Fast Marching Methods. Cambridge University Press, New York (1999)
35. Sethian, J.A.: Evolution, implementation, and application of level set and fast marching methods for advancing fronts. J. Comput. Phys. 169, 503-555 (2001)
36. Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77, 439-471 (1988)
37. Smereka, P.: Spiral crystal growth. Physica D 138, 282-301 (2000)
38. Spiteri, R.J., Ruuth, S.J.: A new class of optimal high-order strong-stability-preserving time discretization methods. SIAM J. Numer. Anal. 40(2), 469-491 (2002)
39. Sumengen, B.: A Matlab toolbox implementing level set methods. [Online]. Available: http://barissumengen.com/level_set_methods/index.html
40. Sussman, M., Smereka, P., Osher, S.: An improved level set method for incompressible two-phase flow. J. Comput. Phys. 114, 145-159 (1994)
41. Takei, R.: Modern theory of numerical methods for motion by mean curvature. Master's thesis, Department of Mathematics, Simon Fraser University (August 2007)
42. Zalesak, S.T.: Fully multidimensional flux-corrected transport algorithms for fluids. J. Comput. Phys. 31(3), 335-362 (1979)