Journal of Scientific Computing

, Volume 55, Issue 3, pp 575–605 | Cite as

An Adaptive Sparse Grid Semi-Lagrangian Scheme for First Order Hamilton-Jacobi Bellman Equations

  • Olivier Bokanowski
  • Jochen Garcke
  • Michael Griebel
  • Irene Klompmaker
Article

Abstract

We propose a semi-Lagrangian scheme using a spatially adaptive sparse grid to deal with non-linear time-dependent Hamilton-Jacobi Bellman equations. We focus in particular on front propagation models in higher dimensions which are related to control problems. We test the numerical efficiency of the method on several benchmark problems up to space dimension d=8, and give evidence of convergence towards the exact viscosity solution. In addition, we study how the complexity and precision scale with the dimension of the problem.

Keywords

Sparse grids Hamilton-Jacobi Bellman equation Front propagation Semi-Lagrangian scheme Adaptivity 

Notes

Acknowledgements

This work was partially supported by the EU under the 7th Framework Programme Marie Curie Initial Training Network “FP7-PEOPLE-2010-ITN”, SADCO project, GA number 264735-SADCO, and by the DFG priority programme SPP 1324 “Mathematical methods for extracting quantifiable information from complex systems”; Michael Griebel was also partially supported by the Hausdorff Research Institute for Mathematics in Bonn and by the Sonderforschungsbereich 611 Singular phenomena and scaling in mathematical models funded by the Deutsche Forschungsgemeinschaft.

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Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • Olivier Bokanowski
    • 1
    • 2
    • 3
  • Jochen Garcke
    • 4
    • 5
  • Michael Griebel
    • 4
    • 5
  • Irene Klompmaker
    • 6
  1. 1.Lab. Jacques-Louis LionsUniv. Pierre et Marie CurieParis Cedex 05France
  2. 2.UFR de Mathématiques, Site ChevaleretUniv. Paris-DiderotParis CedexFrance
  3. 3.Projet CommandsENSTA ParisTechPalaiseau CedexFrance
  4. 4.Institut für Numerische SimulationUniverstität BonnBonnGermany
  5. 5.Fraunhofer SCAISankt AugustinGermany
  6. 6.Institut für MathematikTechnische Universität BerlinBerlinGermany

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