Multiscale differential Riccati equations for linear quadratic regulator problems
For researchers and engineers solving optimal control problems with multiscale dynamics, this work provides a computationally efficient method that overcomes the need for fine-scale discretizations.
The paper tackles the computational challenges of solving differential Riccati equations for linear quadratic regulator problems with multiscale operators. It introduces a localized orthogonal decomposition method that achieves second-order convergence in the L2 operator norm and first-order in energy norm, independent of multiscale variations, while drastically reducing computational cost.
We consider approximations to the solutions of differential Riccati equations in the context of linear quadratic regulator problems, where the state equation is governed by a multiscale operator. Similarly to elliptic and parabolic problems, standard finite element discretizations perform poorly in this setting unless the grid resolves the fine-scale features of the problem. This results in unfeasible amounts of computation and high memory requirements. In this paper, we demonstrate how the localized orthogonal decomposition method may be used to acquire accurate results also for coarse discretizations, at the low cost of solving a series of small, localized elliptic problems. We prove second-order convergence (except for a logarithmic factor) in the $L^2$ operator norm, and first-order convergence in the corresponding energy norm. These results are both independent of the multiscale variations in the state equation. In addition, we provide a detailed derivation of the fully discrete matrix-valued equations, and show how they can be handled in a low-rank setting for large-scale computations. In connection to this, we also show how to efficiently compute the relevant operator-norm errors. Finally, our theoretical results are validated by several numerical experiments.