Skip to main content
Log in

Multi-grid methods for Hamilton-Jacobi-Bellman equations

  • About Cubature Formulas with a Minimal Number of Knots
  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Summary

In this paper we develop multi-grid algorithms for the numerical solution of Hamilton-Jacobi-Bellman equations. The proposed schemes result from a combination of standard multi-grid techniques and the iterative methods used by Lions and mercier in [11]. A convergence result is given and the efficiency of the algorithms is illustrated by some numerical examples.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Bensoussan, A., Lions, J.-L.: Applications of variational inequalities in stochastic control. Amsterdam, New York, Oxford: North-Holland 1982

    Google Scholar 

  2. Bramble, J.H., Hubbard, B.E.: A theorem on error estimation for finite difference analogues of the Dirichlet problem for elliptic equations. Control. Diff. Eqn.2, 319–340 (1963)

    Google Scholar 

  3. Delebecque, F., Quadrat, J.P.: Optimal control of Markov chains admitting strong and weak interactions. Automatica17, 281–296 (1981)

    Google Scholar 

  4. Evans, L.C.: Classical solutions of the Hamilton-Jacobi-Bellman equation for uniformly elliptic operators. Trans. Am. Math. Soc.275, 245–255 (1983)

    Google Scholar 

  5. Fleming, W.H., Rishel, R.: Deterministic and stochastic optimal control Berlin, Heidelberg, New York: Springer 1975

    Google Scholar 

  6. Hackbusch, W.: Convergence of multi-grid iterations applied to difference equations. Math. Comput.34, 425–440 (1980)

    Google Scholar 

  7. Hackbusch, W.: Regularity of difference schemes, Pt. II. Regularity estimates for linear and nonlinear problems. Rep. 80-13, Math. Institute, University of Cologne (1980)

  8. Hackbusch, W.: Multi-grid convergence theory. In: Multigrid methods (W. Hackbusch, U. Trottenberg, (eds.). Lect. Notes Math. Vol. 960. Berlin, Heidelberg, New York: Springer 1982

    Google Scholar 

  9. Hackbusch, W., Mittelmann, H.D.: On multi-grid methods for variational inequalities. Numer. Math.42, 65–76 (1983)

    Google Scholar 

  10. Ladyzhenskaya, O.A., Ural'tseva, N.N.: Linear and quasilinear elliptic equations. New York, London: Academic Press 1968

    Google Scholar 

  11. Lions, P.L., Mercier, B.: Approximation numérique des équations de Hamilton-Jacobi-Bellman. R.A.I.R.O. Analyse numérique/Numer. Anal.14, 369–393 (1980)

    Google Scholar 

  12. Motzkin, T., Wasow, W.: On the approximation of linear elliptic differential equations by difference equations with positive coefficients. J. Math. Phys.31, 253–259 (1953)

    Google Scholar 

  13. Stüben, K., Trottenberg, U.: Multigrid methods: Fundamental algorithms, model problem analysis and applications. In: Multigrid methods (W. Hackbusch, U. Trottenberg, eds.). Lect. Notes Math. Vol. 960. Berlin, Heidelberg, New York: Springer 1982

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hoppe, R.H.W. Multi-grid methods for Hamilton-Jacobi-Bellman equations. Numer. Math. 49, 239–254 (1986). https://doi.org/10.1007/BF01389627

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01389627

Subject Classifications

Navigation