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Dynamic programming using radial basis functions
1. | Center for Mathematics, Technische Universität München, 85747 Garching bei München, Germany, Germany |
References:
[1] | |
[2] |
H. Alwardi, S. Wang, L. Jennings and S. Richardson, An adaptive least-squares collocation radial basis function method for the HJB equation, J. Glob. Opt., 52 (2012), 305-322.
doi: 10.1007/s10898-011-9667-4. |
[3] |
M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Birkhäuser, Boston, Boston, MA, 1997.
doi: 10.1007/978-0-8176-4755-1. |
[4] |
R. Bellman, Dynamic Programming, Princeton University Press, Princeton, NJ, 1957. |
[5] |
D. Bertsekas, Dynamic Programming, Prentice Hall Inc., Englewood Cliffs, NJ, 1987. |
[6] |
I. Capuzzo-Dolcetta, On a discrete approximation of the Hamilton-Jacobi equation of dynamic programming, Appl. Math. Optim., 10 (1983), 367-377.
doi: 10.1007/BF01448394. |
[7] |
I. Capuzzo-Dolcetta and M. Falcone, Discrete dynamic programming and viscosity solutions of the Bellman equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 161-183. |
[8] |
E. Carlini, M. Falcone and R. Ferretti, An efficient algorithm for Hamilton-Jacobi equations in high dimension, Comput. Vis. Sci., 7 (2004), 15-29.
doi: 10.1007/s00791-004-0124-5. |
[9] |
T. Cecil, J. Qian and S. Osher, Numerical methods for high dimensional Hamilton-Jacobi equations using radial basis functions, J. Comp. Phys., 196 (2004), 327-347.
doi: 10.1016/j.jcp.2003.11.010. |
[10] |
M. Falcone, A numerical approach to the infinite horizon problem of deterministic control theory, Appl. Math. Optim., 15 (1987), 1-13.
doi: 10.1007/BF01442644. |
[11] |
M. Falcone and R. Ferretti, Discrete time high-order schemes for viscosity solutions of Hamilton-Jacobi-Bellman equations, Numer. Math., 67 (1994), 315-344.
doi: 10.1007/s002110050031. |
[12] |
M. Falcone and R. Ferretti, High-order approximations for viscosity solutions of Hamilton-Jacobi-Bellman equations, in Nonlinear variational problems and partial differential equations (Isola d'Elba, 1990), vol. 320 of Pitman Res. Notes Math. Ser., Longman Sci. Tech., Harlow, (1995), 197-209. |
[13] |
G. Fasshauer, Meshfree Approximation Methods with Matlab, World Scientific, 2007.
doi: 10.1142/6437. |
[14] |
P. Giesl, On the determination of the basin of attraction of discrete dynamical systems, J. Diff. Eq. Appl., 13 (2007), 523-546.
doi: 10.1080/10236190601135209. |
[15] |
P. Giesl, Construction of a local and global Lyapunov function for discrete dynamical systems using radial basis functions, Journal of Approximation Theory, 153 (2008), 184-211.
doi: 10.1016/j.jat.2008.01.007. |
[16] |
E. Gottzein, R. Meisinger and L. Miller, Anwendung des "Magnetischen Rades" in Hochgeschwindigkeitsmagnetschwebebahnen, ZEV-Glasers Annalen, 103. |
[17] |
L. Grüne, An adaptive grid scheme for the discrete Hamilton-Jacobi-Bellman equation, Numer. Math., 75 (1997), 319-337.
doi: 10.1007/s002110050241. |
[18] |
L. Grüne, Error estimation and adaptive discretization for the discrete stochastic Hamilton-Jacobi-Bellman equation, Numer. Math., 99 (2004), 85-112.
doi: 10.1007/s00211-004-0555-4. |
[19] |
L. Grüne and O. Junge, A set oriented approach to optimal feedback stabilization, Syst. Cont. Lett., 54 (2005), 169-180.
doi: 10.1016/j.sysconle.2004.08.005. |
[20] |
C. Huang, S. Wang, C. Chen and Z. Li, A radial basis collocation method for Hamilton-Jacobi-Bellman equations, Automatica, 42 (2006), 2201-2207.
doi: 10.1016/j.automatica.2006.07.013. |
[21] |
O. Junge and H. M. Osinga, A set oriented approach to global optimal control, ESAIM Control Optim. Calc. Var., 10 (2004), 259-270.
doi: 10.1051/cocv:2004006. |
[22] |
C.-Y. Kao, S. Osher and Y.-H. Tsai, Fast sweeping methods for static Hamilton-Jacobi equations, SIAM J. Numer. Anal., 42 (2005), 2612-2632.
doi: 10.1137/S0036142902419600. |
[23] |
S. N. Kružkov, Generalized solutions of Hamilton-Jacobi equations of eikonal type. I, Mat. Sb. (N.S.), 98 (1975), 450-493, 496. |
[24] |
B. Lincoln and A. Rantzer, Relaxing dynamic programming, IEEE Trans. Auto. Ctrl., 51 (2006), 1249-1260.
doi: 10.1109/TAC.2006.878720. |
[25] |
W. McEneaney, Max-plus Methods for Nonlinear Control and Estimation, Birkhäuser, Boston, 2006. |
[26] |
R. Schaback and H. Wendland, Adaptive greedy techniques for approximate solution of large rbf systems, Numerical Algorithms, 24 (2000), 239-254.
doi: 10.1023/A:1019105612985. |
[27] |
R. Schaback and H. Wendland, Numerical Techniques Based on Radial Basis Functions, Technical report, DTIC Document, 2000. |
[28] |
J. A. Sethian and A. Vladimirsky, Ordered upwind methods for static Hamilton-Jacobi equations: theory and algorithms, SIAM J. Numer. Anal., 41 (2003), 325-363.
doi: 10.1137/S0036142901392742. |
[29] |
D. Shepard, A two-dimensional interpolation function for irregularly-spaced data, in Proc. 23rd ACM, ACM, (1968), 517-524.
doi: 10.1145/800186.810616. |
[30] |
H. Wendland, Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree, Adv. Comp. Math., 4 (1995), 389-396.
doi: 10.1007/BF02123482. |
show all references
References:
[1] | |
[2] |
H. Alwardi, S. Wang, L. Jennings and S. Richardson, An adaptive least-squares collocation radial basis function method for the HJB equation, J. Glob. Opt., 52 (2012), 305-322.
doi: 10.1007/s10898-011-9667-4. |
[3] |
M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Birkhäuser, Boston, Boston, MA, 1997.
doi: 10.1007/978-0-8176-4755-1. |
[4] |
R. Bellman, Dynamic Programming, Princeton University Press, Princeton, NJ, 1957. |
[5] |
D. Bertsekas, Dynamic Programming, Prentice Hall Inc., Englewood Cliffs, NJ, 1987. |
[6] |
I. Capuzzo-Dolcetta, On a discrete approximation of the Hamilton-Jacobi equation of dynamic programming, Appl. Math. Optim., 10 (1983), 367-377.
doi: 10.1007/BF01448394. |
[7] |
I. Capuzzo-Dolcetta and M. Falcone, Discrete dynamic programming and viscosity solutions of the Bellman equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 161-183. |
[8] |
E. Carlini, M. Falcone and R. Ferretti, An efficient algorithm for Hamilton-Jacobi equations in high dimension, Comput. Vis. Sci., 7 (2004), 15-29.
doi: 10.1007/s00791-004-0124-5. |
[9] |
T. Cecil, J. Qian and S. Osher, Numerical methods for high dimensional Hamilton-Jacobi equations using radial basis functions, J. Comp. Phys., 196 (2004), 327-347.
doi: 10.1016/j.jcp.2003.11.010. |
[10] |
M. Falcone, A numerical approach to the infinite horizon problem of deterministic control theory, Appl. Math. Optim., 15 (1987), 1-13.
doi: 10.1007/BF01442644. |
[11] |
M. Falcone and R. Ferretti, Discrete time high-order schemes for viscosity solutions of Hamilton-Jacobi-Bellman equations, Numer. Math., 67 (1994), 315-344.
doi: 10.1007/s002110050031. |
[12] |
M. Falcone and R. Ferretti, High-order approximations for viscosity solutions of Hamilton-Jacobi-Bellman equations, in Nonlinear variational problems and partial differential equations (Isola d'Elba, 1990), vol. 320 of Pitman Res. Notes Math. Ser., Longman Sci. Tech., Harlow, (1995), 197-209. |
[13] |
G. Fasshauer, Meshfree Approximation Methods with Matlab, World Scientific, 2007.
doi: 10.1142/6437. |
[14] |
P. Giesl, On the determination of the basin of attraction of discrete dynamical systems, J. Diff. Eq. Appl., 13 (2007), 523-546.
doi: 10.1080/10236190601135209. |
[15] |
P. Giesl, Construction of a local and global Lyapunov function for discrete dynamical systems using radial basis functions, Journal of Approximation Theory, 153 (2008), 184-211.
doi: 10.1016/j.jat.2008.01.007. |
[16] |
E. Gottzein, R. Meisinger and L. Miller, Anwendung des "Magnetischen Rades" in Hochgeschwindigkeitsmagnetschwebebahnen, ZEV-Glasers Annalen, 103. |
[17] |
L. Grüne, An adaptive grid scheme for the discrete Hamilton-Jacobi-Bellman equation, Numer. Math., 75 (1997), 319-337.
doi: 10.1007/s002110050241. |
[18] |
L. Grüne, Error estimation and adaptive discretization for the discrete stochastic Hamilton-Jacobi-Bellman equation, Numer. Math., 99 (2004), 85-112.
doi: 10.1007/s00211-004-0555-4. |
[19] |
L. Grüne and O. Junge, A set oriented approach to optimal feedback stabilization, Syst. Cont. Lett., 54 (2005), 169-180.
doi: 10.1016/j.sysconle.2004.08.005. |
[20] |
C. Huang, S. Wang, C. Chen and Z. Li, A radial basis collocation method for Hamilton-Jacobi-Bellman equations, Automatica, 42 (2006), 2201-2207.
doi: 10.1016/j.automatica.2006.07.013. |
[21] |
O. Junge and H. M. Osinga, A set oriented approach to global optimal control, ESAIM Control Optim. Calc. Var., 10 (2004), 259-270.
doi: 10.1051/cocv:2004006. |
[22] |
C.-Y. Kao, S. Osher and Y.-H. Tsai, Fast sweeping methods for static Hamilton-Jacobi equations, SIAM J. Numer. Anal., 42 (2005), 2612-2632.
doi: 10.1137/S0036142902419600. |
[23] |
S. N. Kružkov, Generalized solutions of Hamilton-Jacobi equations of eikonal type. I, Mat. Sb. (N.S.), 98 (1975), 450-493, 496. |
[24] |
B. Lincoln and A. Rantzer, Relaxing dynamic programming, IEEE Trans. Auto. Ctrl., 51 (2006), 1249-1260.
doi: 10.1109/TAC.2006.878720. |
[25] |
W. McEneaney, Max-plus Methods for Nonlinear Control and Estimation, Birkhäuser, Boston, 2006. |
[26] |
R. Schaback and H. Wendland, Adaptive greedy techniques for approximate solution of large rbf systems, Numerical Algorithms, 24 (2000), 239-254.
doi: 10.1023/A:1019105612985. |
[27] |
R. Schaback and H. Wendland, Numerical Techniques Based on Radial Basis Functions, Technical report, DTIC Document, 2000. |
[28] |
J. A. Sethian and A. Vladimirsky, Ordered upwind methods for static Hamilton-Jacobi equations: theory and algorithms, SIAM J. Numer. Anal., 41 (2003), 325-363.
doi: 10.1137/S0036142901392742. |
[29] |
D. Shepard, A two-dimensional interpolation function for irregularly-spaced data, in Proc. 23rd ACM, ACM, (1968), 517-524.
doi: 10.1145/800186.810616. |
[30] |
H. Wendland, Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree, Adv. Comp. Math., 4 (1995), 389-396.
doi: 10.1007/BF02123482. |
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