Numerical methods for high dimensional Hamilton-Jacobi equations using radial basis functions

Journal of Computational Physics

Volumn 196, Issue 1, 2004, Pages 327-347

  Cecil, Tom ^a   Qian, Jianliang ^a   Osher, Stanley ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

HEAT CONDUCTION; IMAGE SEGMENTATION; RADIAL BASIS FUNCTION NETWORKS;

ARBITRARY DIMENSION; BASE FUNCTION; DATA SET; HAMILTON - JACOBI EQUATIONS; HIGH-DIMENSIONAL; HIGHER-DIMENSIONAL; MESH-LESS DISCRETIZATION; NUMERICAL SCHEME; RADIAL BASIS; UNSTRUCTURED DATA;

NUMERICAL METHODS;

EID: 1942539307     PISSN: 00219991     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.jcp.2003.11.010     Document Type: Article

References (31)

1. R. Abgrall, On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation, J. Comput. Phys. 114 (1) (1994) 45-58.
2. R. Abgrall, Numerical discretization of the first-order Hamilton-Jacobi equation on triangular meshes, Commun. Pure Appl. Math. 49 (12) (1996) 1339-1373.
3. S. Albert, B. Cockburn, D.A. French, T.E. Peterson, A posteriori error estimates for general numerical methods for Hamilton-Jacobi equations. I. The steady state case, Math. Comput. 71 (237) (2002) 49-76 (electronic).
4. S. Augoula, R. Abgrall, High order numerical discretization for Hamilton-Jacobi equations on triangular meshes, J. Sci. Comput. 15 (2) (2000) 197-229.
5. T.J. Barth, J.A. Sethian, Numerical schemes for the Hamilton-Jacobi and level set equations on triangulated domains, J. Comput. Phys. 145 (1) (1998) 1-40.
6. T. Belytschko, Y. Krongauz, D. Organ, M. Fleming, P. Krysl, Meshless methods: an overview and recent developments, Comput. Methods Appl. Mech. Eng. 139 (1996) 3-47.
7. P. Burchard, Li-Tien Cheng, B. Merriman, S. Osher, Motion of curves in three spatial dimensions using a level set approach, J. Comput. Phys. 170 (2) (2001) 720-741.
8. 2 in multiquadric interpolation, Comput. Math. Appl. 21 (9) (1991) 29-42.
9. B. Cockburn, J. Qian, Continuous dependence results for Hamilton-Jacobi equations, in: D. Estep, S. Tavener (Eds.), Collected Lectures on the Preservation of Stability under Discretization, SIAM, Philadelphia, PA, 2002.
10. M.G. Crandall, P.-L. Lions, Two approximations of solutions of Hamilton-Jacobi equations, Math. Comput. 43 (167) (1984) 1-19.
11. M.G. Crandall, A. Majda, Monotone difference approximations for scalar conservation laws, Math. Comput. 34 (149) (1980) 1-21.
12. O. Friedrich, Weighted essentially non-oscillatory schemes for the interpolation of mean values on unstructured grids, J. Comput. Phys. 144 (1) (1998) 194-212.
13. R.L. Hardy, Multiquadric equations of topography and other irregular surfaces, J. Geophys. Res. 76 (1971) 1905-1915, Radial basis functions and their applications.
14. A. Harten, S. Chakravarthy, Multi-dimensional ENO schemes for general geometries, ICASE Report 91-76, 1991.
15. A. Harten, J.M. Hyman, P.D. Lax, On finite-difference approximations and entropy conditions for shocks, Commun. Pure Appl. Math. 29 (3) (1976) 297-322, With an appendix by B. Keyfitz.
16. A. Harten, S. Osher, B. Engquist, S.R. Chakravarthy, Some results on uniformly high-order accurate essentially nonoscillatory schemes, Appl. Numer. Math. 2 (3-5) (1986) 347-377.
17. Changqing Hu, Chi-Wang Shu, Weighted essentially non-oscillatory schemes on triangular meshes, J. Comput. Phys. 150 (1) (1999) 97-127.
18. A. Iske, T. Sonar, On the structure of function spaces in optimal recovery of point functionals for ENO-schemes by radial basis functions, Numer. Math. 74 (2) (1996) 177-201.
19. G. Kossioris, Ch. Makridakis, P.E. Souganidis, Finite volume schemes for Hamilton-Jacobi equations, Numer. Math. 83 (3) (1999) 427-442.
20. C.A. Micchelli, Interpolation and scattered data: distance matrices and conditionally positive definite functions, Constr. Approx. 2 (1986) 11-22.
21. S.J. Osher, J.A. Sethian, Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys. 79 (1988) 12-49.
22. S. Osher, Chi-Wang Shu, High-order essentially nonoscillatory schemes for Hamilton-Jacobi equations, SIAM J. Numer. Anal. 28 (4) (1991) 907-922.
23. Shmuel Rippa, An algorithm for selecting a good value for the parameter c in radial basis function interpolation, Adv. Comput. Math. 11 (2-3) (1999) 193-210, Radial basis functions and their applications.
24. R. Schaback, Error estimates and condition numbers for radial basis function interpolation, Adv. Comput. Math. 3 (3) (1995) 251-264.
25. R. Schaback, Reconstruction of Multivariate Functions from Scattered Data. Electronic publication, available from: .
26. Chi-Wang Shu, S. Osher, Efficient implementation of essentially nonoscillatory shock-capturing schemes. II, J. Comput. Phys. 83 (1) (1989) 32-78.
27. T. Sonar, Optimal recovery using thin plate splines in finite volume methods for the numerical solution of hyperbolic conservation laws, IMA J. Numer. Anal. 16 (4) (1996) 549-581.
28. T. Sonar, On families of pointwise optimal finite volume ENO approximations, SIAM J. Numer. Anal. 35 (6) (1998) 2350-2369.
29. J. Strain, Tree methods for moving interfaces, J. Comput. Phys. 151 (2) (1999) 616-648.
30. Z.-M. Wu, Multivariate compactly supported positive definite radial basis functions, Adv. Comput. Math. 4 (1995) 389-396.
31. Yong-Tao Zhang, Chi-Wang Shu, High-order WENO schemes for Hamilton-Jacobi equations on triangular meshes, SIAM J. Sci. Comput. 24 (3) (2002) 1005-1030 (electronic).