We address the important field of large scale matrix based algorithms in control and model order reduction. Many important tools from theory and applications in systems theory have been widely ignored during the recent decades in the context of PDE constraint optimal control problems and simulation of electric circuits. Often this is due to the fact that large scale matrices are suspected to be unsolvable in large scale applications. Since around 2000 efficient low rank theory for matrix equation solvers exists for sparse and also data sparse systems. Unfortunately upto now only incomplete or experimental Matlab implementations of most of these solvers have existed. Here we aim on the implementation of these algorithms in a higher programming language (in our case C) that allows for a high performance solver for many matrix equations arising in the context of large scale standard and generalized state space systems. We especially focus on efficient memory saving data structures and implementation techniques as well as the shared memory parallelization of the underlying algorithms.

We employ two efficient parallel approaches to reduce a model arising from a semi-discretization 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.

We investigate the numerical solution of the stable generalized Lyapunov equation via the sign function method. This approach has already been proposed to solve standard Lyapunov equations in several publications. The extension to the generalized case is straightforward. We consider some modifications and discuss how to solve generalized Lyapunov equations with semidefinite constant term for the Cholesky factor. The basic computational tools of the method are basic linear algebra operations that can be implemented efficiently on modern computer architectures and in particular on parallel computers. Hence, a considerable speed-up as compared to the Bartels-Stewart and Hammarling's methods is to be expected. We compare the algorithms by performing a variety of numerical tests.

We present a multi-grid method for a class of structured generalized Lyapunov matrix equations. Such equations need to be solved in each step of the Newton method for algebraic Riccati equations, which arise from linear-quadratic optimal control problems governed by partial differential equations. We prove the rate of convergence of the two-grid method to be bounded independent of the dimension of the problem under certain assumptions. The multi-grid method is based on matrix-matrix multiplications and thus it offers a great potential for a parallelization. The efficiency of the method is demonstrated by numerical experiments.

We investigate model reduction of large-scale linear time-invariant systems in generalized state-space form. We consider sparse state matrix pencils, including pencils with banded structure. The balancing-based methods employed here are composed of well-known linear algebra operations and have been recently shown to be applicable to large models by exploiting the structure of the matrices defining the dynamics of the system. In this paper we propose a modification of the LR-ADI iteration to solve large-scale generalized Lyapunov equations together with a practical convergence criterion, and several other implementation refinements. Using kernels from several serial and parallel linear algebra libraries, we have developed a parallel package for model reduction, SpaRed, extending the applicability of balanced truncation to sparse systems with up to $O(10^5)$ states. Experiments on an SMP parallel architecture consisting of Intel Itanium 2 processors illustrate the numerical performance of this approach and the potential of the parallel algorithms for model reduction of large-scale sparse systems.

Model reduction is a common theme within the simulation, control and optimization of complex dynamical systems. For instance, in control problems for partial differential equations, the associated large-scale systems have to be solved very often. To attack these problems in reasonable time it is absolutely necessary to reduce the dimension of the underlying system. We focus on model reduction by balanced truncation where a system theoretical background provides some desirable properties of the reduced-order system. The major computational task in balanced truncation is the solution of large-scale Lyapunov equations, thus the method is of limited use for really large-scale applications. We develop an effective implementation of balancing-related model reduction methods in exploiting the structure of the underlying problem. This is done by a data-sparse approximation of the large-scale state matrix A using the hierarchical matrix format. Furthermore, we integrate the corresponding formatted arithmetic in the sign function method for computing approximate solution factors of the Lyapunov equations. This approach is well-suited for a class of practical relevant problems and allows the application of balanced truncation and related methods to systems coming from 2D and 3D FEM and BEM discretizations.

Arbeitsbericht zum Projekt Integrierte Simulation des Systems "Werkzeugmaschine - Antriebe - Zerspanprozess" auf der Grundlage ordnungsreduzierter FEM-Strukturmodelle. Inhalt: Modellreduktion für lineare Systeme, Balanciertes Abschneiden, Singuläre Systeme 2. Ordnung, Offene Fragen und weiteres Vorgehen.

We discuss the numerical solution of large-scale sparse projected discrete-time periodic Lyapunov equations in lifted form which arise in model reduction of periodic descriptor systems. We extend the alternating direction implicit method and the Smith method to such equations. Low-rank versions of these methods are also presented, which can be used to compute low-rank approximations to the solutions of projected periodic Lyapunov equations in lifted form with low-rank right-hand side. Moreover, we consider an application of the Lyapunov solvers to balanced truncation model reduction of periodic discrete-time descriptor systems. Numerical results are given to illustrate the efficiency and accuracy of the proposed methods.:1 Introduction 2 Periodic descriptor systems 3 ADI method for causal lifted Lyapunov equations 4 Smith method for noncausal lifted Lyapunov equations 5 Application to model order reduction 6 Numerical results 7 Conclusions

A practical procedure based on implicit time integration methods applied to the differential Lyapunov equations arising in the square root balanced truncation method is presented. The application of high order time integrators results in indefinite right-hand sides of the algebraic Lyapunov equations that have to be solved within every time step. Therefore, classical methods exploiting the inherent low-rank structure often observed for practical applications end up in complex data and arithmetic. Avoiding the additional effort treating complex quantities, a symmetric indefinite factorization of both the right-hand side and the solution of the differential Lyapunov equations is applied.

We employ two efficient parallel approaches to reduce a model arising from a semi-discretization 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.