Fast marching methods for stationary Hamilton-Jacobi equations with axis-aligned anisotropy

SIAM Journal on Numerical Analysis

Volumn 47, Issue 1, 2008, Pages 363-385

  Alton, Ken ^a   Mitchell, Ian M ^a  

^a UNIVERSITY OF BRITISH COLUMBIA   (Canada)

Author keywords

Anisotropic optimal control; Fast marching method; Hamilton Jacobi equation; Viscosity solution

Indexed keywords

ARBITRARY DIMENSION; DIJKSTRA'S ALGORITHMS; EFFICIENT ALGORITHM; EIKONAL EQUATION; EUCLIDEAN; FAST MARCHING METHODS; FINITE-DIFFERENCE SCHEME; HAMILTON JACOBI EQUATIONS; HAMILTON-JACOBI; INFINITY NORM; MANHATTANS; OPTIMAL CONTROLS; ORTHOGONAL GRID; PATH-PLANNING; SHORTEST PATH; VISCOSITY SOLUTIONS;

AIRCRAFT ENGINES; ALGORITHMS; ANISOTROPY; GEOMETRICAL OPTICS; MECHANICS; MOTION PLANNING; OPTIMIZATION; VISCOSITY;

DIFFERENCE EQUATIONS;

EID: 77952721815     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/070680357     Document Type: Article

References (31)

1. K. ALTON AND I. M. MITCHELL, Fast Marching Methods for a Class of Anistrophic Stationary Hamilton-Jacobi Equations, Technical report TR-2006-2027, Department of Computer Science, University of British Columbia, Vancouver, 2007.
2. K. ALTON AND I. M. MITCHELL, Fast Marching Methods for Hamilton-Jacobi Equations with Axis-aligned Anisotropy, Technical report TR-2008-2102, Department of Computer Science, University of British Columbia, Vancouver, 2008.
3. K. ALTON AND I. MITCHELL, Optimal path planning under different norms in continuous state spaces, in Proceedings of the International Conference on Robotics and Automation, IEEE, 2006, pp. 866-872.
4. G. BARLES AND P. E. SOUGANIDIS, Convergence of approximation schemes for fully nonlinear second order equations, Asymptot. Anal., 4 (1991), pp. 271-283.
5. F. BORNEMANN AND C. RASCH, Finite-element discretization of static Hamilton-Jacobi equations based on a local variational principle, Comput. Vis. Sci., 9 (2006), pp. 57-69.
6. M. BOUE AND P. DUPUIS, Markov chain approximations for deterministic control problems with affine dynamics and quadratic cost in the control, SIAM J. Numer. Anal., 36 (1999), pp. 667-695.
7. M. G. CRANDALL, H. ISHII, AND P. LIONS, User's guide to viscosity solutions of second order partial differential equations, Bull. Amcr. Math. Soc., 27 (1992), pp. 1-67.
8. P.-E. DANIELSSON, Euclidean distance mapping, Comput. Graph. Imago Process., 14 (1980), pp. 227-248.
9. E. W. DIJKSTRA, A note on two problems in connection with graphs, Numer. Math., 1 (1959), pp. 269-271.
10. S. FOMEL, On anelliptic approximations for qp velocities in vti media, Goophys. Prospecting, 52 (2004), pp. 247-259.
11. P. A. GREMAUD AND C. M. KUSTER., Computational study of fast methods for the Eikonal equation, SIAM J. Sci. Comput., 27 (2006), pp. 1803-1816.
12. S.-R. HYSING AND S. TUREK, The Eikonal equation: Numerical efficiency vs. algorithmic complexity on quadrilateral grids, in Proceedings of Algoritmy 2005, Vsako Tatry, Pobansko, Slovakia, 2005, pp. 22-31.
13. C. Y. KAO, S. OSHER, AND Y. H. TSAI, Fast sweeping methods for static Hamilton-Jacobi equations, SIAM J. Numer. Anal., 42 (2005), pp. 2612-2632.
14. R. KIMMEL AND J. A. SETHIAN, Fast marching methods on triangulated domains, Proc. Natl. Acad. Sci. USA, 95 (1998), pp. 8341-8435.
15. R. KIMMEL AND J. A. SETHIAN, Optimal algorithm for shape from shading and path planning, J. Math. Imaging Vision, 14 (2001), pp. 237-244.
16. I. M. MITCHELL AND S. SASTRY, Continuous Path Planning with Multiple Constraints, Technical report UCB/ERL M03/34, UC Berkeley Engineering Research Laboratory, Berkeley, CA, 2003.
17. S. OSHER, AND R. P. FEDKIW, Level set methods: An overview and some recent results, J. Comput. Phys., 169 (2001), pp. 463-502.
18. L. C. POLYMENAKOS, D. P. BERTSEKAS, AND J. N. TSITSIKLIs, Implementation of efficient algorithms for globally optimal trajectories, IEEE Trans. Automat. Control, 43 (1998), pp. 278-283.
19. J. QIAN, Y. ZHANG, AND H. ZHAO, A fast sweeping method for static convex Hamilton-Jacobi equations, J. Sci. Comput., 31 (2007), pp. 237-271.
20. J. A. SETHIAN AND A. VLADIMIRSKY, Fast methods for Eikonal and related Hamilton-Jacobi equations on unstructured meshes, Proc. Natl. Acad. Sci. USA, 97 (2000), pp. 5699-5703.
21. J. A. SETHIAN AND A. VLADIMIRSKY, Ordered upwind methods for static Hamilton-Jacobi equations, Proc. Natl. Acad. Sci. USA, 98 (2001), pp. 11069-11074.
22. J. A. SETHIAN AND A. VLADIMIRSKY, Ordered upwind methods for static Hamilton-Jacobi equations: Theory and algorithms, SIAM J. Numer. Anal., 41 (2003), pp. 325-363.
23. J. A. SETHIAN, A fast marching level set method for monotonically advancing fronts, Proc. Natl. Acad. Sci. USA, 93 (1996), pp. 1591-1595.
24. J. A. SETHIAN, Level Set Methods, Cambridge University Press, Cambridge, UK, 1996.
25. J. A. SETHIAN, Fast marching methods, SIAM Rev., 41 (1999), pp. 199-235.
26. J. A. SETHIAN, Level Set Methods and Fast Marching Methods, Cambridge University Press, Cambridge, UK, 1999.
27. Y.-H. R. TSAI, L.-T. CHENG, S. OSHER, AND H.-K. ZHAO, Fast sweeping algorithms for a class of Hamilton-Jacobi equations, SIAM J. Numer. Anal., 41 (2003), pp. 673-694.
28. J. N. TSITSIKLIS, Efficient algorithms for globally optimal trajectories, in Proceedings of the 33rd IEEE Conference on Decision and Control, Lake Buena Vista, FL, 1994, pp. 1368-1373.
29. J. N. TSITSIKLIS, Efficient algorithms for globally optimal trajectories, IEEE Trans. Automat. Control, 40 (1995), pp. 1528-1538.
30. L. YATZIV, A. BARTESAGHI, AND G. SAPIRO, O(n) implementation of the fast marching method, J. Comput. Phys., 212 (2006), pp. 393-399.
31. H. ZHAO, A fast sweeping method for Eikonal equations, Math. Comp., 74 (2004), pp. 603-627.