Approximating the Stationary Bellman Equation by Hierarchical Tensor Products (Journal of Computational Mathematics, 2024)
We treat infinite horizon optimal control problems by solving the associated stationary Bellman equation numerically to compute the value function and an optimal feedbacklaw. The dynamical systems under consideration are spatial discretizations of non linearparabolic partial differential equations (PDE), which means that the Bellman equation suffers from the curse of dimensionality. Its non linearity is handled by the Policy Iterationalgorithm, where the problem is reduced to a sequence of linear equations, which remainthe computational bottleneck due to their high dimensions. We reformulate the linearizedBellman equations via the Koopman operator into an operator equation, that is solved using a minimal residual method. Using the Koopman operator we identify a preconditionerfor operator equation, which deems essential in our numerical tests. To overcome computational infeasibility we use low rank hierarchical tensor product approximation/tree-basedtensor formats, in particular tensor trains (TT tensors) and multi-polynomials, togetherwith high-dimensional quadrature, e.g. Monte-Carlo. By controlling a destabilized versionof viscous Burgers and a diffusion equation with unstable reaction term numerical evidenceis given.
Keywords: Feedback control; Dynamic programming; Hamilton-Jacobi-Bellman; Tensor product approximation; Variational Monte-Carlo.