# Multiscale differential Riccati equations for linear quadratic regulator problems

## TL;DR

This paper introduces a multiscale approach using localized orthogonal decomposition to efficiently approximate solutions to differential Riccati equations in linear quadratic regulator problems with multiscale operators, achieving high accuracy with low computational cost.

## Contribution

It develops a novel multiscale method for Riccati equations that maintains accuracy independent of multiscale variations and provides detailed analysis and validation.

## Key findings

- Second-order convergence in $L^2$ operator norm
- First-order convergence in energy norm
- Effective low-rank computations for large-scale problems

## Abstract

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.