Patrick Kürschner

Efficient handling of complex shift parameters in the low-rank Cholesky factor ADI method

The solution of large-scale Lyapunov equations is a crucial problem for several fields of modern applied mathematics. The low-rank Cholesky factor version of the alternating directions implicit method (LRCF-ADI) is one iterative algorithm that computes approximate low-rank factors of the solution. In order to achieve fast convergence it requires adequate shift parameters, which can be complex if the matrices defining the Lyapunov equation are unsymmetric. This will require complex arithmetic computations as well as storage of complex data and thus, increase the overall complexity and memory requirements of the method. In this article we propose a novel reformulation of LRCF-ADI which generates real low-rank factors by carefully exploiting the dependencies of the iterates with respect to pairs of complex conjugate shift parameters. It significantly reduces the amount of complex arithmetic calculations and requirements for complex storage. It is hence often superior in terms of efficiency compared to other real formulations.

Self-Generating and Efficient Shift Parameters in ADI Methods for Large Lyapunov and Sylvester Equations

Low-rank versions of the alternating direction implicit (ADI) iteration are popular and well estab- lished methods for the numerical solution of large-scale Sylvester and Lyapunov equations. Probably the biggest disadvantage of these methods is their dependence on a set of shift parameters that are crucial for fast convergence. Here we firstly review existing shift generation strategies that compute a number of shifts before the actual itera- tion. These approaches come with several disadvantages such as, e.g., expensive numerical computations and the difficulty to obtain necessary spectral information or data n eeded to initially setup their generation. Secondly, we propose two novel shift selection strategies with the motivation to resolve these issues at least partially. Both ap- proaches generate shifts automatically in the course of the ADI iterations. Extensive numerical tests show that one of these new approaches, based on a Galerkin projection onto the space spanned by the current ADI data, is superior to other approaches in the majority of cases both in terms of convergence speed and required execution time.

An Improved Numerical Method for Balanced Truncation for Symmetric Second-Order Systems

We consider balanced truncation model order reduction for symmetric second-order systems. The occurring large-scale generalized and structured Lyapunov equations are solved with a specially adapted low-rank alternating directions implicit (ADI) type method. Stopping criteria for this iteration are investigated, and a new result concerning the Lyapunov residual within the low-rank ADI method is established. We also propose a goal-oriented stopping criterion which tries to incorporate the balanced truncation approach already during the ADI iteration. The model reduction approach using the ADI method with different stopping criteria is evaluated on several test systems.

Computing Real Low-Rank Solutions of Sylvester Equations by the Factored ADI Method

Abstract We investigate the factored alternating directions implicit (ADI) iteration for large and sparse Sylvester equations. A novel low-rank expression for the associated Sylvester residual is established which enables cheap computations of the residual norm along the iteration, and which yields a reformulated factored ADI iteration. The application to generalized Sylvester equations is considered as well. We also discuss the efficient handling of complex shift parameters and reveal interconnections between the ADI iterates w.r.t. those complex shifts. This yields a further modification of the factored ADI iteration which employs only an absolutely necessary amount of complex arithmetic operations and storage, and which produces low-rank solution factors consisting of entirely real data. Certain linear matrix equations, such as, e.g., cross Gramian Sylvester and Stein equations, are in fact special cases of generalized Sylvester equations and we show how specially tailored low-rank ADI iterations can be deduced from the generalized factored ADI iteration.

A Goal-Oriented Dual LRCF-ADI for Balanced Truncation

Abstract In this contribution we propose a novel dual Lyapunov solver that computes low rank factors of the reachability and observability Gramians of a control system simultaneously. This is especially helpful in balanced truncation model order reduction applications, where the singular values of the product of these Gramian factors are the basis of the truncation process. For stopping the dual iteration scheme we therefore propose a tailored stopping criterion aiming at the accurate computation of these singular values.

Frequency-Limited Balanced Truncation with Low-Rank Approximations

In this article we investigate model order reduction of large-scale systems using frequency-limited balanced truncation, which restricts the well known balanced truncation framework to prescribed frequency regions. The main emphasis is put on the efficient numerical realization of this model reduction approach. We discuss numerical methods to take care of the involved matrix-valued functions. The occurring large-scale Lyapunov equations are solved for low-rank approximations for which we also establish results regarding the eigenvalues of their solutions. These results, and also numerical experiments, will show that the eigenvalues of the Lyapunov solutions in frequency-limited balanced truncation often decay faster than those in standard balanced truncation. Moreover, we show in further numerical examples that frequency-limited balanced truncation generates reduced order models which are significantly more accurate in the considered frequency region.

Low-Rank Newton-ADI Methods for Large Nonsymmetric Algebraic Riccati Equations

Abstract The numerical treatment of large-scale, nonsymmetric algebraic Riccati equations (NAREs) by a low-rank variant of Newton׳s method is considered. We discuss a method to compute approximations to the solution of the NARE in a factorized form of low rank. The occurring large-scale Sylvester equations are dealt with using the factored alternating direction implicit iteration (fADI). Several performance enhancing strategies available for the factored ADI as well as the related Newton-ADI for symmetric algebraic Riccati equations are generalized to this combination. This includes the efficient computation of the norm of the residual matrix, adapted shift parameter strategies for fADI, and an acceleration of Newton׳s scheme by means of a Galerkin projection. Numerical experiments illustrate the capabilities of the proposed method to solve high-dimensional NAREs.

Model order reduction of large-scale dynamical systems with Jacobi-Davidson style eigensolvers

Many applications concerning physical and technical processes employ dynamical systems for simulation purposes. The increasing demand for a more accurate and detailed description of realistic phenomena leads to high dimensional dynamical systems and hence, simulation often yields an increased computational effort. An approximation, e.g. with model order reduction techniques, of these large-scale systems becomes therefore crucial for a cost efficient simulation. This paper focuses on a model order reduction method for linear time in-variant (LTI) systems based on modal approximation via dominant poles. There, the original large-scale LTI system is projected onto the left and right eigenspaces corresponding to a specific subset of the eigenvalues of the system matrices, namely the dominant poles of the systems transfer function. Since these dominant poles can lie anywhere in the spectrum, specialized eigenvalue algorithms that can compute eigentriplets of large and sparse matrices are required. The Jacobi-Davidson method has proven to be a suitable and competitive candidate for the solution of various eigenvalue problems and hence, we discuss how it can be incorporated into this modal truncation approach. Generalizations of the reduction technique and the application of the algorithms to second-order systems are also investigated. The computed reduced order models obtained with this modal approximation can be combined with the ones constructed with Krylov subspace or balanced truncation based model order reduction methods to get even higher accuracies.

Balanced Truncation Model Order Reduction in Limited Time Intervals for Large Systems

In this article we investigate model order reduction of large-scale systems using time-limited balanced truncation, which restricts the well known balanced truncation framework to prescribed finite time intervals. The main emphasis is on the efficient numerical realization of this model reduction approach in case of large system dimensions. We discuss numerical methods to deal with the resulting matrix exponential functions and Lyapunov equations which are solved for low-rank approximations. Our main tool for this purpose are rational Krylov subspace methods. We also discuss the eigenvalue decay and numerical rank of the solutions of the Lyapunov equations. These results, and also numerical experiments, will show that depending on the final time horizon, the numerical rank of the Lyapunov solutions in time-limited balanced truncation can be smaller compared to standard balanced truncation. In numerical experiments we test the approaches for computing low-rank factors of the involved Lyapunov solutions and illustrate that time-limited balanced truncation can generate reduced order models having a higher accuracy in the considered time region.

Improved Second-Order Balanced Truncation for Symmetric Systems

Abstract We investigate second order balanced truncation model order reduction for symmetric linear time invariant second order control systems. This special structure decreases the required computational effort significantly. Moreover, we show how stability of the original model can be preserved for such systems. We briefly discuss the numerical solution of the occurring large-scale Lyapunov equations with a modified low-rank ADI method. The approach is tested on finite element models of mechanical structures.