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106
A new iterative method for solving largescale Lyapunov matrix equations
 SIAM J. Sci. Comput
"... Abstract. In this paper we propose a new projection method to solve largescale continuoustime Lyapunov matrix equations. The new method projects the problem onto a much smaller approximation space, generated as a combination of Krylov subspaces in A and A −1. The reduced problem is then solved by ..."
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Cited by 55 (6 self)
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Abstract. In this paper we propose a new projection method to solve largescale continuoustime Lyapunov matrix equations. The new method projects the problem onto a much smaller approximation space, generated as a combination of Krylov subspaces in A and A −1. The reduced problem is then solved by means of a direct Lyapunov scheme based on matrix factorizations. The reported numerical results show the competitiveness of the new method, compared to a stateoftheart approach based on the factorized Alternating Direction Implicit (ADI) iteration. 1. Introduction. Given
Solving LargeScale Control Problems
, 2004
"... Sparsity and parallel algorithms: two approaches to beat the curse of dimensionality. By Peter Benner I n this article we discuss sparse matrix algorithms and parallel algorithms, as well as their application to largescale systems. For illustration, we solve the linearquadratic regulator (LQR) pro ..."
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Cited by 54 (26 self)
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Sparsity and parallel algorithms: two approaches to beat the curse of dimensionality. By Peter Benner I n this article we discuss sparse matrix algorithms and parallel algorithms, as well as their application to largescale systems. For illustration, we solve the linearquadratic regulator (LQR) problem and apply balanced truncation model reduction using either parallel computing or sparse matrix algorithms. We conclude that modern tools from numerical linear algebra, along with careful investigation and exploitation of the problem structure, can be used to derive algorithms capable of solving large control problems. Since these approaches are implemented in productionquality software, control engineers can employ complex models and use computational tools to analyze and design feedback control laws. Background
StateSpace Truncation Methods for Parallel Model Reduction of LargeScale Systems
, 2003
"... We discuss a parallel library of efficient algorithms for model reduction of largescale systems with statespace dimension up to O(10⁴). We survey the numerical algorithms underlying the implementation of the chosen model reduction methods. The approach considered here is based on statespace trunc ..."
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Cited by 27 (22 self)
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We discuss a parallel library of efficient algorithms for model reduction of largescale systems with statespace dimension up to O(10⁴). We survey the numerical algorithms underlying the implementation of the chosen model reduction methods. The approach considered here is based on statespace truncation of the system matrices and includes absolute and relative error methods for both stable and unstable systems. In contrast to serial implementations of these methods, we employ Newtontype iterative algorithms for the solution of the major computational tasks. Experimental results report the numerical accuracy and the parallel performance of our approach on a cluster of Intel Pentium II processors.
A Riemannian optimization approach for computing lowrank solutions of Lyapunov equations
, 2009
"... We propose a new framework based on optimization on manifolds to approximate the solution of a Lyapunov matrix equation by a lowrank matrix. The method minimizes the error on the Riemannian manifold of symmetric positive semidefinite matrices of fixed rank. We detail how objects from differential ..."
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Cited by 26 (4 self)
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We propose a new framework based on optimization on manifolds to approximate the solution of a Lyapunov matrix equation by a lowrank matrix. The method minimizes the error on the Riemannian manifold of symmetric positive semidefinite matrices of fixed rank. We detail how objects from differential geometry, like the Riemannian gradient and Hessian, can be efficiently computed for this manifold. As minimization algorithm we use the Riemannian TrustRegion method of [Found. Comput. Math., 7 (2007), pp. 303–330] based on a secondorder model of the objective function on the manifold. Together with an efficient preconditioner this method can find lowrank solutions with very little memory. We illustrate our results with numerical examples.
On the ADI Method for Sylvester Equations
"... This paper is concerned with the numerical solution of large scale Sylvester equations AX − XB = C, Lyapunov equations as a special case in particular included, with C having very small rank. For stable Lyapunov equations, Penzl (2000) and Li and White (2002) demonstrated that the so called Cholesky ..."
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Cited by 24 (17 self)
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This paper is concerned with the numerical solution of large scale Sylvester equations AX − XB = C, Lyapunov equations as a special case in particular included, with C having very small rank. For stable Lyapunov equations, Penzl (2000) and Li and White (2002) demonstrated that the so called Cholesky factor ADI method with decent shift parameters can be very effective. In this paper we present a generalization of the Cholesky factor ADI method for Sylvester equations. We also demonstrate that often much more accurate solutions than ADI solutions can be obtained by performing Galerkin projection via the column space and row space of the computed approximate solutions.
Numerical solution of largescale Lyapunov equations, Riccati equations, and linearquadratic optimal control problems
 NUMER. LINEAR ALGEBRA APPL. 2008; 15:1–23
, 2008
"... We study largescale, continuoustime linear timeinvariant control systems with a sparse or structured state matrix and a relatively small number of inputs and outputs. The main contributions of this paper are numerical algorithms for the solution of large algebraic Lyapunov and Riccati equations a ..."
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Cited by 18 (2 self)
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We study largescale, continuoustime linear timeinvariant control systems with a sparse or structured state matrix and a relatively small number of inputs and outputs. The main contributions of this paper are numerical algorithms for the solution of large algebraic Lyapunov and Riccati equations and linearquadratic optimal control problems, which arise from such systems. First, we review an ADI based method to compute approximate lowrank Cholesky factors of the solution matrix of largescale Lyapunov equations, and we propose a refined version of this algorithm. Second, a combination of this method with a variant of Newton’s method (in this context also called Kleinman iteration) results in an algorithm for the solution of largescale Riccati equations. Third, we describe an implicit version of this algorithm for the solution of linearquadratic optimal control problems which computes the feedback directly without solving the underlying algebraic Riccati equation explicitly. Our algorithms are efficient with respect to both memory and computation. In particular, they can be applied to problems of very large scale, where square, dense matrices of the system order cannot be stored in the computer memory. We study the performance of our algorithms in numerical experiments.
A GalerkinNewtonADI Method for Solving LargeScale Algebraic Riccati Equations
, 2010
"... The alternating directions implicit (ADI) iteration has been proven to be a highly efficient method for solving stable large scale Lyapunov equations when applied in order to compute low rank factors of the solution. Employing NewtonADI or NewtonKleinmanADI methods for solving algebraic Riccati ..."
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Cited by 15 (11 self)
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The alternating directions implicit (ADI) iteration has been proven to be a highly efficient method for solving stable large scale Lyapunov equations when applied in order to compute low rank factors of the solution. Employing NewtonADI or NewtonKleinmanADI methods for solving algebraic Riccati equations (AREs), one has to solve a stable large scale Lyapunov equation in every Newton step. It has been shown, that the sparse plus low rank structure of the Lyapunov equation in the Newton step can easily be incorporated in the low rank ADI iteration. Still, the ADI iterations convergence speed is strongly depending on certain shift parameters. In this paper we will discuss a hybrid GalerkinADI approach that can drastically accelerate the ADI iteration when good shifts are unknown or hard to compute. The same ideas can be applied to accelerate the inexact Newton iteration resulting from the approximative/iterative solution of the Lyapunov equations.
Direct methods and ADIpreconditioned Krylov subspace methods for generalized Lyapunov equations
 Numer. Lin. Alg. Appl
"... Prepared using nlaauth.cls [Version: 2002/09/18 v1.02] ..."
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Cited by 14 (1 self)
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Prepared using nlaauth.cls [Version: 2002/09/18 v1.02]
Factorized solution of Lyapunov equations based on hierarchical matrix arithmetic
, 2006
"... We investigate the numerical solution of largescale Lyapunov equations with the sign function method. Replacing the usual matrix inversion, addition, and multiplication by formatted arithmetic for hierarchical matrices, we obtain an implementation that has linearpolylogarithmic complexity and memor ..."
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Cited by 13 (6 self)
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We investigate the numerical solution of largescale Lyapunov equations with the sign function method. Replacing the usual matrix inversion, addition, and multiplication by formatted arithmetic for hierarchical matrices, we obtain an implementation that has linearpolylogarithmic complexity and memory requirements. The method is well suited for Lyapunov operators arising from FEM and BEM approximations to elliptic differential operators. With the sign function method it is possible to obtain a lowrank approximation to a fullrank factor of the solution directly. The task of computing such a factored solution arises, e.g., in model reduction based on balanced truncation. The basis of our method is a partitioned Newton iteration for computing the sign function of a suitable matrix, where one part of the iteration uses formatted arithmetic while the other part directly yields approximations to the fullrank factor of the solution. We discuss some variations of our method and its application to generalized Lyapunov equations. Numerical experiments show that the method can be applied to problems of order up to O(10 5) on workstations.
Parallel Order Reduction via Balanced Truncation for Optimal Cooling of Steel Profiles
, 2005
"... Abstract. We employ two efficient parallel approaches to reduce a model arising from a semidiscretization of a controlled heat transfer process for optimal cooling of a steel profile. Both algorithms are based on balanced truncation but differ in the numerical method that is used to solve two dual ..."
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Cited by 13 (4 self)
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Abstract. We employ two efficient parallel approaches to reduce a model arising from a semidiscretization of a controlled heat transfer process for optimal cooling of a steel profile. Both algorithms are based on balanced truncation but differ in the numerical method that is used to solve two dual generalized Lyapunov equations, which is the major computational task. Experimental results on a cluster of Intel Xeon processors compare the efficacy of the parallel model reduction algorithms.