Multiscale differential Riccati equations for linear quadratic regulator problems
Målqvist, Axel ; Persson, Anna and Stillfjord, Tony LU
(2018)
In SIAM Journal on Scientific Computing
40(4).
p.2406-2426
- 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... (More)
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 L2 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.
(Less)
- author
- Målqvist, Axel
; Persson, Anna
and Stillfjord, Tony
LU
- publishing date
- 2018-08-02
- type
- Contribution to journal
- publication status
- published
- subject
- keywords
- Differential Riccati equations, Finite elements, Linear quadratic regulator problems, Localized orthogonal decomposition, Multiscale
- in
- SIAM Journal on Scientific Computing
- volume
- 40
- issue
- 4
- pages
- 2406 - 2426
- publisher
- Society for Industrial and Applied Mathematics
- external identifiers
-
- scopus:85053750253
- ISSN
- 1064-8275
- DOI
- 10.1137/17M1134500
- language
- English
- LU publication?
- no
- additional info
- Submitted to the journal’s Methods and Algorithms for Scientific Computing section June 13, 2017; accepted for publication (in revised form) May 29, 2018; published electronically August 2, 2018. http://www.siam.org/journals/sisc/40-4/M113450.html Funding: This work was supported by the Swedish Research Council under grant 2015-04964. Publisher Copyright: © 2018 Society for Industrial and Applied Mathematics.
- id
- 4a1bc310-ed80-4022-91f6-41676a52b34c
- date added to LUP
- 2024-01-23 17:35:56
- date last changed
- 2024-02-26 09:00:33
@article{4a1bc310-ed80-4022-91f6-41676a52b34c,
abstract = {{<p>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<sup>2</sup> 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.</p>}},
author = {{Målqvist, Axel and Persson, Anna and Stillfjord, Tony}},
issn = {{1064-8275}},
keywords = {{Differential Riccati equations; Finite elements; Linear quadratic regulator problems; Localized orthogonal decomposition; Multiscale}},
language = {{eng}},
month = {{08}},
number = {{4}},
pages = {{2406--2426}},
publisher = {{Society for Industrial and Applied Mathematics}},
series = {{SIAM Journal on Scientific Computing}},
title = {{Multiscale differential Riccati equations for linear quadratic regulator problems}},
url = {{http://dx.doi.org/10.1137/17M1134500}},
doi = {{10.1137/17M1134500}},
volume = {{40}},
year = {{2018}},
}