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An adaptive least-squares collocation radial basis function method for the HJB equation

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Abstract

We present a novel numerical method for the Hamilton–Jacobi–Bellman equation governing a class of optimal feedback control problems. The spatial discretization is based on a least-squares collocation Radial Basis Function method and the time discretization is the backward Euler finite difference. A stability analysis is performed for the discretization method. An adaptive algorithm is proposed so that at each time step, the approximate solution can be constructed recursively and optimally. Numerical results are presented to demonstrate the efficiency and accuracy of the method.

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References

  1. Bardi M., Capuzzo-Dolcetta I.: Optimal Control and Viscosity Solutions of Hamilton–Jacobi–Bellman equations. Birlhauser, Boston (1997)

    Book  Google Scholar 

  2. Behrens J., Iske A.: Grid-free adaptive semi-Lagrangian advection using radial basis functions. Comput. Math. Appl. 43(3–5), 319–327 (2002)

    Article  Google Scholar 

  3. Behrens J., Iske A., Kaser M.: Adaptive meshfree method of backward characteristics for nonlinear transport equations, Meshfree methods for partial differential equations (Bonn, 2001). Lect. Notes Comput. Sci. Eng. 26, 21–36 (2003)

    Article  Google Scholar 

  4. Buhman M.D.: Radial Basis Functions: Theory and Implementations, Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge (2003)

    Book  Google Scholar 

  5. Carlson R.E., Foley T.A.: The parameter \({\mathbb{R}^{2}}\) in multiquadric interpolation. Comput. Math. Appl. 21, 29–42 (1991)

    Article  Google Scholar 

  6. Falcone M., Giorgo T.: An approximation scheme for evolutive Hamilton–Jacobi equations. In: McEneancy, W.M., Yin, G., Zhang, Q. (eds) Stochastic Analysis, Control, Optimization and Applications, pp. 289–303. Birkhuser, Boston (1999)

    Chapter  Google Scholar 

  7. Foley T.A.: Near optimal parameter selection for multiquadric interpolation. J. Appl. Sci. Comput. 1, 54–69 (1991)

    Google Scholar 

  8. Franke C., Schaback R.: Solving partial differential equations by collocation using radial basis functions. Appl. Math. Comput. 93, 73–82 (1998)

    Article  Google Scholar 

  9. Golub G.H., Van Loan C.F.: Matrix Computation. 3rd edn . The Johns Hopkins University Press, Baltimore (1996)

    Google Scholar 

  10. Holmstrom K.: An adaptive radial basis algorithm (ARBF) for expensive black-box global optimization. J. Glob. Optim. 41, 17–184 (2008)

    Article  Google Scholar 

  11. Hon Y.C.: Multiquadric collocation method with adaptive technique for problems with boundary layer. Int. J. Appl. Sci. Comput. 6, 447–464 (1999)

    Google Scholar 

  12. Horst R., Pardalos P.M.: Handbook of Global Optimization. Kluwer, Dordrecht (1995)

    Google Scholar 

  13. Huang C.-S., Wang S., Teo K.L.: Solving Hamilton–Jacobi–Bellman equations by a Modified Method of Characteristics. Nonlinear Anal. TMA 40, 279–293 (2000)

    Article  Google Scholar 

  14. Huang C.-S., Wang S., Teo K.L.: On application of an alternating direction method to Hamilton–Jacobi–Bellman equations. J. Comp. Appl. Math. 166, 153–166 (2004)

    Article  Google Scholar 

  15. Huang C.-S., Wang S., Chen C.S., Li Z.-C.: A radial basis collocation method for Hamilton–Jacobi–Bellman equations. Automatica 42, 2201–2207 (2006)

    Article  Google Scholar 

  16. Larsson E., Fornberg B.: A numerical study of some radial basis function based solution methods for elliptic PDEs. Comput. Math. Appl. 46, 891–902 (2003)

    Article  Google Scholar 

  17. Ling L., Trummer M.R.: Adaptive multiquadric collocation for boundary layer problem. J. Comput. Appl. Math. 188, 265–282 (2006)

    Article  Google Scholar 

  18. Lorentz R.A., Narcowich F.J., Ward J.D.: Collocation discretizations of the transport equation with radial basis functions. Appl. Math. Comput. 145, 97–116 (2003)

    Article  Google Scholar 

  19. Pereyra V., Scherer G.: Least squares collocation solution of elliptic problems in general regions. Math. Comput. Simul. 73, 226–230 (2006)

    Article  Google Scholar 

  20. Richardson S., Wang S.: Numerical solution of Hamilton–Jacobi–Bellman equations by an exponentially fitted finite volume method. Optimization 55, 121–140 (2006)

    Article  Google Scholar 

  21. Richardson S., Wang S.: The viscosity approximation to the Hamilton–Jacobi–Bellman equation in optimal feedback control: upper bounds for extended domains. J. Ind. Manag. Optim. 6, 161–175 (2010)

    Article  Google Scholar 

  22. Rippa S.: An algorithm for selecting a good value for the parameter c in radial basis function interpolation. Adv. Comput. Math. 11, 193–210 (1999)

    Article  Google Scholar 

  23. Sarra S.A.: Adaptive radial basis function methods for time dependent partial differential equations. Appl. Numer. Math. 54, 79–94 (2005)

    Article  Google Scholar 

  24. Schittkowski K.: NLPQL: a Fortran subroutines for solving constrained nonlinear programming problems. Ann. Oper. Res. 5, 485–500 (1986)

    Google Scholar 

  25. Wang S., Gao F., Teo K.L.: An upwind finite difference method for the approximation of viscosity solutions to Hamilton–Jacobi–Bellman equations. IMA J. Math. Control Inf. 17, 169–178 (2000)

    Article  Google Scholar 

  26. Wang S., Jennings L.S., Teo K.L.: Numerical solution of Hamilton–Jacobi–Bellman equations by an upwind finite volume method. J. Glob. Optim. 27, 177–192 (2003)

    Article  Google Scholar 

  27. Zilinskas A.: On similarities between two models of global optimization: statistical models and radial basis functions. J. Glob. Optim. 48, 173–182 (2010)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Nizwa College of Applied Sciences, PO Box 699, Nizwa, 611, Sultanate of Oman

    H. Alwardi

  2. School of Mathematics and Statistics, The University of Western Australia, 35 Stirling Hwy, Crawley, WA, 6009, Australia

    S. Wang & L. S. Jennings

  3. School of Engineering, Edith Cowan University, 270 Joondalup Drive, Joondalup, WA, 6027, Australia

    S. Richardson

Authors
  1. H. Alwardi
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  2. S. Wang
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  3. L. S. Jennings
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  4. S. Richardson
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Correspondence to S. Wang.

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Alwardi, H., Wang, S., Jennings, L.S. et al. An adaptive least-squares collocation radial basis function method for the HJB equation. J Glob Optim 52, 305–322 (2012). https://doi.org/10.1007/s10898-011-9667-4

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  • DOI: https://doi.org/10.1007/s10898-011-9667-4

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