Low-rank updates and a divide-and-conquer method for linear matrix equations
For researchers and practitioners solving large-scale linear matrix equations in control theory and PDEs, this provides a faster and more memory-efficient method for structured problems.
This work presents a new algorithm based on tensorized Krylov subspaces for efficiently updating solutions of linear matrix equations under low-rank changes, and uses it to accelerate the Newton method for Riccati equations and to develop a divide-and-conquer approach for hierarchical low-rank matrices, achieving improvements in computational time and memory consumption.
Linear matrix equations, such as the Sylvester and Lyapunov equations, play an important role in various applications, including the stability analysis and dimensionality reduction of linear dynamical control systems and the solution of partial differential equations. In this work, we present and analyze a new algorithm, based on tensorized Krylov subspaces, for quickly updating the solution of such a matrix equation when its coefficients undergo low-rank changes. We demonstrate how our algorithm can be utilized to accelerate the Newton method for solving continuous-time algebraic Riccati equations. Our algorithm also forms the basis of a new divide-and-conquer approach for linear matrix equations with coefficients that feature hierarchical low-rank structure, such as HODLR, HSS, and banded matrices. Numerical experiments demonstrate the advantages of divide-and-conquer over existing approaches, in terms of computational time and memory consumption.