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Convergence analysis for splitting of the abstract differential Riccati equation

Hansen, Eskil LU and Stillfjord, Tony LU (2014) In SIAM Journal on Numerical Analysis 52(6). p.3128-3139
Abstract
We consider a splitting-based approximation of the abstract differential Riccati equation in the setting of Hilbert--Schmidt operators. The Riccati equation arises in many different areas and is important within the field of optimal control. In this paper we conduct a temporal error analysis and prove that the splitting method converges with the same order as the implicit Euler scheme, under the same low regularity requirements on the initial values.

For a subsequent spatial discretization, the abstract setting also yields uniform temporal error bounds with respect to the spatial discretization parameter.

The spatial discretizations commonly lead to large-scale problems, where the use of structural properties of the... (More)
We consider a splitting-based approximation of the abstract differential Riccati equation in the setting of Hilbert--Schmidt operators. The Riccati equation arises in many different areas and is important within the field of optimal control. In this paper we conduct a temporal error analysis and prove that the splitting method converges with the same order as the implicit Euler scheme, under the same low regularity requirements on the initial values.

For a subsequent spatial discretization, the abstract setting also yields uniform temporal error bounds with respect to the spatial discretization parameter.

The spatial discretizations commonly lead to large-scale problems, where the use of structural properties of the solution is essential. We therefore conclude by proving that the splitting method preserves low-rank structure in the matrix-valued case. Numerical results demonstrate the validity of the convergence analysis. (Less)
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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Abstract differential Riccati equation, convergence order, splitting, low-rank approximation, Hilbert-Schmidt operators
in
SIAM Journal on Numerical Analysis
volume
52
issue
6
pages
3128 - 3139
publisher
SIAM Publications
external identifiers
  • wos:000346842100023
  • scopus:84919660473
ISSN
0036-1429
DOI
10.1137/130935501
language
English
LU publication?
yes
id
45005153-59d2-473f-af1a-7321b197a63f (old id 4730126)
alternative location
http://epubs.siam.org/doi/abs/10.1137/130935501
date added to LUP
2014-10-28 09:59:48
date last changed
2017-08-20 03:10:27
@article{45005153-59d2-473f-af1a-7321b197a63f,
  abstract     = {We consider a splitting-based approximation of the abstract differential Riccati equation in the setting of Hilbert--Schmidt operators. The Riccati equation arises in many different areas and is important within the field of optimal control. In this paper we conduct a temporal error analysis and prove that the splitting method converges with the same order as the implicit Euler scheme, under the same low regularity requirements on the initial values. <br/><br>
For a subsequent spatial discretization, the abstract setting also yields uniform temporal error bounds with respect to the spatial discretization parameter. <br/><br>
The spatial discretizations commonly lead to large-scale problems, where the use of structural properties of the solution is essential. We therefore conclude by proving that the splitting method preserves low-rank structure in the matrix-valued case. Numerical results demonstrate the validity of the convergence analysis.},
  author       = {Hansen, Eskil and Stillfjord, Tony},
  issn         = {0036-1429},
  keyword      = {Abstract differential Riccati equation,convergence order,splitting,low-rank approximation,Hilbert-Schmidt operators},
  language     = {eng},
  number       = {6},
  pages        = {3128--3139},
  publisher    = {SIAM Publications},
  series       = {SIAM Journal on Numerical Analysis},
  title        = {Convergence analysis for splitting of the abstract differential Riccati equation},
  url          = {http://dx.doi.org/10.1137/130935501},
  volume       = {52},
  year         = {2014},
}