2018/04/06 by Tony Stillfjord, Stillfjord, Tony
Computer Science · Mathematics · Physics and Astronomy · #47A11 #47A62 #49N10 #FOS: Mathematics #Matrix Theory and Algorithms #Model Reduction and Neural Networks #Numerical Analysis (math.NA) #Numerical methods for differential equations #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.1804.02197
openalex publication_date 2018/04/06 · openalex created_date 2023/02/16 · openalex updated_date 2026/07/28
We consider operator-valued differential Lyapunov and Riccati equations,\nwhere the operators B and C may be relatively unbounded with respect to A\n(in the standard notation). In this setting, we prove that the singular values\nof the solutions decay fast under certain conditions. In fact, the decay is\nexponential in the negative square root if A generates an analytic semigroup\nand the range of C has finite dimension. This extends previous similar\nresults for algebraic equations to the differential case. When the initial\ncondition is zero, we also show that the singular values converge to zero as\ntime goes to zero, with a certain rate that depends on the degree of\nunboundedness of C. A fast decay of the singular values corresponds to a low\nnumerical rank, which is a critical feature in large-scale applications. The\nresults reported here provide a theoretical foundation for the observation\nthat, in practice, a low-rank factorization usually exists.\n