Efficient low-rank solution of large-scale matrix equations

Autor*innen

• Patrick Kürschner

Parameter

• Lehrbücher
• Universitätslehrbücher

Mehr zum Buch

This thesis investigates numerical solutions for large-scale algebraic matrix equations, focusing on methods based on alternating directions implicit (ADI) iteration to compute low-rank approximations. These low-rank ADI iterations are particularly useful for large-scale Lyapunov and Sylvester equations. A significant portion of the work enhances the performance of these iterative methods by developing algorithmic improvements that reduce computational effort at various stages of each iteration. Key innovations include novel low-rank expressions for the residual matrix, facilitating efficient computation of the residual norm, and strategies to minimize complex arithmetic operations. Since ADI methods depend on shift parameters that affect convergence speed, new shift generation strategies are proposed to enhance the convergence rate of low-rank ADI iterations, enabling automatic and cost-effective implementation. The improved low-rank ADI methods are subsequently applied in Newton-type algorithms to find approximate solutions for quadratic matrix equations, including symmetric and nonsymmetric algebraic Riccati equations. Finally, techniques for solving large-scale Lyapunov equations are utilized for balanced truncation model order reduction in linear control systems, leading to the development of a novel and efficient algorithmic framework for frequency-limited balanced truncation.

• Sprache: Englisch
• Erscheinungsdatum: 2016