Network Flows That Solve Sylvester Matrix Equations

• 成果类型: Article
• 署名作者: Deng, Wen; Hong, Yiguang; Anderson, Brian D. O.; Shi, Guodong
• 署名单位: Tongji University; Tongji University; Hangzhou Dianzi University; Commonwealth Scientific & Industrial Research Organisation (CSIRO); CSIRO Data61; Australian National University; University of Sydney
• 刊物名称: IEEE TRANSACTIONS ON AUTOMATIC CONTROL
• ISSN/ISSBN: 0018-9286
• DOI: 10.1109/TAC.2021.3130877
• 发表日期: 2022
• 页码: 6731-6738
• 关键词: Convergence rate; Distributed algorithms; network flows; Sylvester equation

摘要

In this article, we study methods to solve a Sylvester equation in the form of AX + XB = C for given matrices A, B, C is an element of R-nxn, inspired by the distributed linear equation flows. The entries of A. B. and C are separately partitioned into a number of pieces (or sometimes we permit these pieces to overlap), which are allocated to nodes in a network. Nodes hold a dynamic state shared among their neighbors defined from the network structure. Natural partial or full row , column partitions and block partitions of the data A, B, and C are formulated by use of the vectorized matrix equation. We show that existing network flows for distributed linear algebraic equations can be extended to solve this special form of matrix equations over networks. A consensus + projection + symmetrization flow is also developed for equations with symmetry constraints on the matrix variables. We prove the convergence of these flows and obtain the fastest convergence rates that these flows can achieve regardless of the choices of node interaction strengths and network structures.