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

Authors:
Olivier Bokanowski;Jochen Garcke;Michael Griebel;Irene Klompmaker
Affiliations:
Lab. Jacques-Louis Lions, Univ. Pierre et Marie Curie, Paris Cedex 05, France 75252 and UFR de Mathématiques, Site Chevaleret, Univ. Paris-Diderot, Paris Cedex, France 75205 and Projet Comman ...;Institut für Numerische Simulation, Universtität Bonn, Bonn, Germany and Fraunhofer SCAI, Sankt Augustin, Germany;Institut für Numerische Simulation, Universtität Bonn, Bonn, Germany and Fraunhofer SCAI, Sankt Augustin, Germany;Institut für Mathematik, Technische Universität Berlin, Berlin, Germany
Venue:
Journal of Scientific Computing
Year:
2013

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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.