Ordered upwind methods for static Hamilton-Jacobi equations: Theory and algorithms

SIAM Journal on Numerical Analysis

Volumn 41, Issue 1, 2003, Pages 325-363

  Sethian, James A ^a   Vladimirsky, Alexander ^b  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b Department of Obstetrics Gynecology   (United States)

Author keywords

Anisotropic front propagation; Anisotropic optimal control; Dijkstra like methods; Dynamic programming; Fast marching methods; Ordered upwind methods; Viscosity solution

Indexed keywords

BOUNDARY VALUE PROBLEMS; CONVERGENCE OF NUMERICAL METHODS; DYNAMIC PROGRAMMING; FINITE DIFFERENCE METHOD; LAGRANGE MULTIPLIERS; NONLINEAR EQUATIONS; PARTIAL DIFFERENTIAL EQUATIONS; PROBLEM SOLVING; SEISMOLOGY;

ANISOTROPIC FRONT PROPAGATION; ANISOTROPIC OPTIMAL CONTROL; DIJKSTRA-LIKE METHODS; FAST MARCHING METHODS; ORDERED UPWIND METHODS; STATIC HAMILTON-JACOBI EQUATIONS; VISCOSITY SOLUTION;

ALGORITHMS;

EID: 1842511489     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142901392742     Document Type: Article

References (47)

1. D. Adalsteinsson and J.A. Sethian, A fast level set method for propagating interfaces, J. Comput. Phys., 118 (1995), pp. 269-277.
2. D. Adalsteinsson and J.A. Sethian, The fast construction of extension velocities in level set methods, J. Comput. Phys., 148 (1999), pp. 2-22.
3. R.K. Ahuja, T.L. Magnanti, and J.B. Orlin, Network Flows: Theory, Algorithms, and Applications, Prentice-Hall, Englewood Cliffs, NJ, 1993.
4. M. Bardi, M. Falcone, and P. Soravia, Fully discrete schemes for the value function of pursuit-evasion games, in Advances in Dynamic Games and Applications (Geneva, 1992), Ann. Internat. Soc. Dynam. Games 1, Birkhäuser-Boston, Cambridge, MA, 1994, pp. 89-105.
5. R. Bellman, Dynamic Programming, Princeton University Press, Princeton, NJ, 1957.
6. R. Bellman, Introduction to the Mathematical Theory of Control Processes, Academic Press, New York, 1967.
7. I. Capuzzo Dolcetta and M. Falcone, Discrete dynamic programming and viscosity solutions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), pp. 161-183.
8. I. Capuzzo Dolcetta, On a discrete approximation of the Hamilton-Jacobi equation of dynamic programming, Appl. Math. Optim., 10 (1983), pp. 367-377.
9. M.G. Crandall, L.C. Evans, and P.-L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 282 (1984), pp. 487-502.
10. M.G. Crandall and P.-L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 277 (1983), pp. 1-43.
11. J.A. Dellinger, Anisotropic Seismic Wave Propagation, Ph.D. Dissertation, Department of Geophysics, Stanford University, Stanford, CA, 1991.
12. E.W. Dijkstra, A note on two problems in connection with graphs, Numer. Math., 1 (1959), pp. 269-271.
13. L.C. Evans, Partial Differential Equations, American Mathematical Society, Providence, RI, 1998.
14. L.C. Evans and P.E. Souganidis, Differential and representation formulas for solutions of Hamilton-Jacobi-Isaacs equations, Indiana Univ. Math. J., 33 (1984), pp. 773-797.
15. M. Falcone, The minimum time problem and its applications to front propagation, in Motion by Mean Curvature and Related Topics, Proceedings of the International Conference at Trento, 1992, Walter de Gruyter, New York, 1994, pp. 70-88.
16. M. Falcone, A numerical approach to the infinite horizon problem of deterministic control theory, Appl. Math. Optim., 15 (1987), pp. 1-13
17. M. Falcone, A numerical approach to the infinite horizon problem of deterministic control theory, Appl. Math. Optim., corrigenda, 23 (1991), pp. 213-214.
18. M. Falcone and R. Ferretti, Discrete time high-order schemes for viscosity solutions of Hamilton-Jacobi-Bellman equations, Numer. Math., 67 (1994), pp. 315-344.
19. R. Gonzales and E. Rofman, On deterministic control problems: An approximate procedure for the optimal cost I. The stationary problem, SIAM J. Control Optim., 23 (1985), pp. 242-266.
20. J. Helmsen, E.G. Puckett, P. Colella, and M. Dorr, Two new methods for simulating photolithography development in three dimensions, in Proceedings of the SPIE 1996 International Symposium on Microlithography, Santa Clara, CA, SPIE 2726, SPIE, Bellingham, WA, 1996, pp. 253-261.
21. A. Jain, Fundamentals of Digital Image Processing, Prentice-Hall, Englewood Cliffs, NJ, 1989.
22. R. Kimmel and J. A. Sethian, Fast marching methods on triangulated domains, Proc. Nat. Acad. Sci. USA, 95 (1998), pp. 8341-8435.
23. R. Kimmel and J.A. Sethian, Fast Marching Methods for Robotic Navigation with Constraints, Center for Pure and Applied Mathematics Report, University of California, Berkeley, 1996.
24. R. Kimmel and J.A. Sethian, Fast Voronoi diagrams and offsets of triangulated surfaces, in Curve and Surface Design, P.J. Laurent, P. Sablonniere, and L.L. Schumaker, eds., Vanderbilt University Press, Nashville, TN, 2000, pp. 193-202.
25. S.N. Kruzhkov, Generalized solutions of the Hamilton-Jacobi equations of the Eikonal type, Math. USSR-Sb., 27 (1975), pp. 406-445.
26. H.J. Kushner, Probability Methods for Approximation in Stochastic Control and for Elliptic Equations, Springer-Verlag, New York, 1977.
27. H.J. Kushner and P.G. Dupuis, Numerical Methods for Stochastic Control Problems in Continuous Time, Academic Press, New York, 1992.
28. R. Malladi and J.A. Sethian, An O(NlogN) algorithm for shape modeling, Proc. Nat. Acad. Sci. USA, 93 (1996), 9389-9392.
29. J. McGulagh, Geometrical propositions applied to the wave theory of light, Trans. Royal Irish Acad., 17 (1837), pp. 241-263.
30. S. Osher and R.P. Fedkiw, Level set methods: An overview and some recent results, J. Comput. Phys., 169 (2001), pp. 463-502.
31. D. Peng, S. Osher, B. Merriman, and H.-K. Zhao, The geometry of Wulff crystal shapes and its relations with Riemann problems, in Nonlinear Partial Differential Equations, Contemp. Math. 238, G.-Q. Chen and E. di Benedetto, eds., AMS, Providence, RI, 1999, pp. 251-303.
32. G.W. Postma, Wave propagation in a stratified medium, Geophysics, 20 (1995), pp. 780-806.
33. J.A. Sethian, Fast Marching Methods for Computing Seismic Travel Times, manuscript.
34. J.A. Sethian, A fast marching level set method for monotonically advancing fronts, Proc. Nat. Acad. Sci. USA, 93 (1996), pp. 1591-1595.
35. J.A. Sethian, Fast marching level set methods for three-dimensional photolithography development, in Proceedings of the SPIE 1996 International Symposium on Microlithography, Santa Clara, CA, SPIE 2726, SPIE, Bellingham, WA, 1996, pp. 262-272.
36. J.A. Sethian, Fast marching methods, SIAM Rev., 41 (1999), pp. 199-235.
37. J.A. Sethian, Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision and Materials Sciences, Cambridge University Press, Cambridge, UK, 1996.
38. J.A. Sethian and M. Popovici, Three dimensional traveltimes computation using the fast marching method, Geophysics, 64 (1999), pp. 516-523.
39. J.A. Sethian and A. Vladimirsky, Fast methods for the Eikonal and related Hamilton-Jacobi equations on unstructured meshes, Proc. Nat. Acad. Sci. USA, 97 (2000), pp. 5699-5703.
40. J.A. Sethian and A. Vladimirsky, Ordered upwind methods for static Hamilton-Jacobi equations, Proc. Nat. Acad. Sci. USA, 98 (2001), pp. 11069-11074.
41. T.P. Schulze and R.V. Kohn, A geometric model for coarsening during spiral-mode growth of thin films, Phys. D, 132 (1999), pp. 520-542.
42. P. Soravia, Generalized motion of a front propagating along its normal direction: A differential games approach, Nonlinear Anal., 22 (1994), pp. 1247-1262.
43. W. Press, B. Flannery, S. Teukolsky, and W. Vetterling, Numerical Recipes in C, Cambridge University Press, Cambridge, UK, 1988.
44. J.E. Taylor, Crystalline variational problems, Bull. Amer. Math. Soc., 84 (1978), pp. 568-588.
45. J.N. Tsitsiklis, Efficient algorithms for globally optimal trajectories, in Proceedings of the IEEE 33rd Conference on Decision and Control, Lake Buena Vista, FL, 1994, pp. 1368-1373.
46. J.N. Tsitsiklis, Efficient algorithms for globally optimal trajectories, IEEE Trans. Automat. Control, 40 (1995), pp. 1528-1538.
47. A. Vladimirsky, Fast Methods for Static Hamilton-Jacobi Partial Differential Equations, Ph.D. Dissertation, Department of Mathematics, University of California, Berkeley, CA, 2001.