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April 2016
Patrick Kürschner
Efficient Low-Rank Solution of Large-Scale Matrix Equations
In this thesis, we investigate the numerical solution of large-scale, algebraic matrix equations.
The focus lies on numerical methods based on the alternating directions implicit (ADI) iteration, which can be formulated to compute approximate solutions of matrix equations in form of low-rank factorizations. These low-rank versions of the ADI iteration can be used to deal with large-scale Lyapunov and Sylvester equations.
The major part of this thesis is devoted to improving the performance of these iterative methods. At first, we develop algorithmic enhancements that aim at reducing the computational effort of certain stages in each iteration step. This includes novel low-rank expressions of the residual matrix, which allows an efficient computation of the residual norm, and approaches for the reduction of the amount of occurring complex arithmetic operations.
ADI based methods rely on shift parameters, which influence how fast the iteration generates an approximate solution. For this, we propose novel shift generation strategies which improve the convergence speed of the low-rank ADI iteration and, at the same time, can be performed in an automatic and cost efficient numerical way.
Later on, the improved low-rank ADI methods for Lyapunov and Sylvester equations are used in Newton type methods for finding approximate solutions of quadratic matrix equations in the form of symmetric and nonsymmetric algebraic Riccati equations. In the last part of this thesis, the methods for solving large-scale Lyapunov equations are applied in order to execute balanced truncation model order reduction for linear control systems in a numerically feasible way.
For frequency-limited balanced truncation, a novel and efficient algorithmic framework is developed.
The focus lies on numerical methods based on the alternating directions implicit (ADI) iteration, which can be formulated to compute approximate solutions of matrix equations in form of low-rank factorizations. These low-rank versions of the ADI iteration can be used to deal with large-scale Lyapunov and Sylvester equations.
The major part of this thesis is devoted to improving the performance of these iterative methods. At first, we develop algorithmic enhancements that aim at reducing the computational effort of certain stages in each iteration step. This includes novel low-rank expressions of the residual matrix, which allows an efficient computation of the residual norm, and approaches for the reduction of the amount of occurring complex arithmetic operations.
ADI based methods rely on shift parameters, which influence how fast the iteration generates an approximate solution. For this, we propose novel shift generation strategies which improve the convergence speed of the low-rank ADI iteration and, at the same time, can be performed in an automatic and cost efficient numerical way.
Later on, the improved low-rank ADI methods for Lyapunov and Sylvester equations are used in Newton type methods for finding approximate solutions of quadratic matrix equations in the form of symmetric and nonsymmetric algebraic Riccati equations. In the last part of this thesis, the methods for solving large-scale Lyapunov equations are applied in order to execute balanced truncation model order reduction for linear control systems in a numerically feasible way.
For frequency-limited balanced truncation, a novel and efficient algorithmic framework is developed.
Schlagwörter: Numerical Linear Algebra; Matrix Equations; Model Order Reduction
Forschungsberichte aus dem Max-Planck-Institut für Dynamik komplexer technischer Systeme
Herausgegeben von Prof. Dr. Peter Benner, Prof. Dr.-Ing. Udo Reichl, Prof. Dr.-Ing. Andreas Seidel-Morgenstern und Prof. Dr.-Ing. Kai Sundmacher, Magdeburg
Band 45
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