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.

• Författare: Eskil Hansen; Tony Stillfjord
• Avdelning/ar: Matematik LTH; Partial differential equations; Numerical Analysis
• Publiceringsår: 2014
• Språk: Engelska
• Sidor: 3128-3139
• Publikation/Tidskrift/Serie: SIAM Journal on Numerical Analysis
• Volym: 52
• Issue: 6
• Fulltext: - 363 kB; Download statistics
• Länkar: http://epubs.siam.org/doi/abs/10.1137/130935501; http://dx.doi.org/10.1137/130935501
• Dokumenttyp: Artikel i tidskrift
• Förlag: Society for Industrial and Applied Mathematics
• Ämne: Mathematics
• Nyckelord: Abstract differential Riccati equation; convergence order; splitting; low-rank approximation; Hilbert-Schmidt operators
• Status: Published
• Forskningsgrupp: Partial differential equations; Numerical Analysis
• ISBN/ISSN/Övrigt: ISSN: 0036-1429