Domain decomposition algorithms for solving hamilton-jacobi-bellman equations

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Abstract
We focus on numerically solving a typical type of Hamilton-Jacobi-Bellman equations (HJB) arising from a class of optimal controls with diffusion models and from a similar one with finite Markov chain models. For computational purpose both cases eventually deduce to a discrete (HJB) of this form max{(I – Q k )u – c k 1 ≤ k ≤ m} = 0 (DHJB) The Bellman's curse of dimensionality seems to be the main obstacle to the applicability of most numerical algorithms for solving (DHJB). In duffusion models we decompose the state domain into a finite number of subregions of manageable sizes. In the finite Markov chain case, the state space is partitioned into a finite number of groups of desirable sizes. Consequently, we are led to alternatively solving a number of subproblems similar to (DHJB). We will present some convergence results, as well as experimental ones.

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