Abstract
In this paper an adaptive finite difference scheme for the solution of the discrete first order Hamilton-Jacobi-Bellman equation is presented. Local a posteriori error estimates are established and certain properties of these estimates are proved. Based on these estimates an adapting iteration for the discretization of the state space is developed. An implementation of the scheme for two-dimensional grids is given and numerical examples are discussed.
Highlights
In this paper an adaptive grid scheme for the solution of the discrete first order HamiltonJacobi-Bellman equation sup{vh(x) − βvh(Φh(x, u)) − hg(x, u)} = 0 u∈U on Ω ⊂ Rn with 0 < β < 1 is developed
The setup is related to Markov Chain Approximations of discounted continuous time stochastic control problems
The adapting iteration – based on numerically calculated approximations ei of the error estimates ei – is given for the general n-dimensional case and a refinement and coarsening method is described for two-dimensional grids
Summary
For β = 1 − δh the solution vh of this equation is the optimal value function of the discrete discounted optimal control problem. To solve this equation numerically a finite difference scheme is used, for which a discretization of Ω is necessary. In order to approximate this optimal value function it is necessary to solve (1.1) for small discount rates δ > 0. The adapting iteration – based on numerically calculated approximations ei of the error estimates ei – is given for the general n-dimensional case and a refinement and coarsening method is described for two-dimensional grids.
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