Convergence analysis for splitting of the abstract differential Riccati equation

Summary, in English

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.

• Author: Eskil Hansen; Tony Stillfjord
• Department/s: Mathematics (Faculty of Engineering); Partial differential equations; Numerical Analysis
• Publishing year: 2014
• Language: English
• Pages: 3128-3139
• Publication/Series: SIAM Journal on Numerical Analysis
• Volume: 52
• Issue: 6
• Full text: - 363 kB; Download statistics
• Links: http://epubs.siam.org/doi/abs/10.1137/130935501; http://dx.doi.org/10.1137/130935501
• Document type: Journal article
• Publisher: Society for Industrial and Applied Mathematics
• Topic: Mathematics
• Keywords: Abstract differential Riccati equation; convergence order; splitting; low-rank approximation; Hilbert-Schmidt operators
• Status: Published
• Research group: Partial differential equations; Numerical Analysis
• ISBN/ISSN/Other: ISSN: 0036-1429