2014/08/18 by Aleksandar Haber, Haber, Aleksandar, Michel Verhaegen +1 · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #FOS: Mathematics #Matrix Theory and Algorithms #Model Reduction and Neural Networks #Numerical methods for differential equations #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.1408.3898
openalex publication_date 2014/08/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the problem of computing an approximate banded solution of the\ncontinuous-time Lyapunov equation\n underlineA underlineX+ underlineX underlineAT= underlineP,\nwhere the coefficient matrices underlineA and underlineP are large,\nsymmetric banded matrices. The (sparsity) pattern of underlineA describes\nthe interconnection structure of a large-scale interconnected system. Recently,\nit has been shown that the entries of the solution underlineX are\nspatially localized or decaying away from a banded pattern. We show that the\ndecay of the entries of underlineX is faster if the condition number of\n underlineA is smaller. By exploiting the decay of entries of\n underlineX, we develop two computationally efficient methods for\napproximating underlineX by a banded matrix. For a well-conditioned and\nsparse banded underlineA, the computational and memory complexities of the\nmethods scale linearly with the state dimension. We perform extensive numerical\nexperiments that confirm this, and that demonstrate the effectiveness of the\ndeveloped methods. The methods proposed in this paper can be generalized to\n(sparsity) patterns of underlineA and underlineP that are more\ngeneral than banded matrices. The results of this paper open the possibility\nfor developing computationally efficient methods for approximating the solution\nof the large-scale Riccati equation by a sparse matrix.\n