Year: 2024
Author: Mathias Oster, Leon Sallandt, Reinhold Schneider
Journal of Computational Mathematics, Vol. 42 (2024), Iss. 3 : pp. 638–661
Abstract
We treat infinite horizon optimal control problems by solving the associated stationary Bellman equation numerically to compute the value function and an optimal feedback law. The dynamical systems under consideration are spatial discretizations of non linear parabolic partial differential equations (PDE), which means that the Bellman equation suffers from the curse of dimensionality. Its non linearity is handled by the Policy Iteration algorithm, where the problem is reduced to a sequence of linear equations, which remain the computational bottleneck due to their high dimensions. We reformulate the linearized Bellman equations via the Koopman operator into an operator equation, that is solved using a minimal residual method. Using the Koopman operator we identify a preconditioner for operator equation, which deems essential in our numerical tests. To overcome computational infeasibility we use low rank hierarchical tensor product approximation/tree-based tensor formats, in particular tensor trains (TT tensors) and multi-polynomials, together with high-dimensional quadrature, e.g. Monte-Carlo. By controlling a destabilized version of viscous Burgers and a diffusion equation with unstable reaction term numerical evidence is given.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jcm.2112-m2021-0084
Journal of Computational Mathematics, Vol. 42 (2024), Iss. 3 : pp. 638–661
Published online: 2024-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 24
Keywords: Feedback control Dynamic programming Hamilton-Jacobi-Bellman Tensor product approximation Variational Monte-Carlo.