@Article{JCM-42-638,
author = {Oster , MathiasSallandt , Leon and Schneider , Reinhold},
title = {Approximating the Stationary Bellman Equation by Hierarchical Tensor Products},
journal = {Journal of Computational Mathematics},
year = {2024},
volume = {42},
number = {3},
pages = {638--661},
abstract = {<p style="text-align: justify;">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.</p>},
issn = {1991-7139},
doi = {https://doi.org/10.4208/jcm.2112-m2021-0084},
url = {http://global-sci.org/intro/article_detail/jcm/23030.html}
}