This paper introduces a new algorithm for solving large-scale continuous-time algebraic Riccati equations (CARE). The advantage of the new algorithm is in its immediate and efficient low-rank formulation, which is a generalization of the Cholesky-factored variant of the Lyapunov ADI method. We discuss important implementation aspects of the algorithm, such as reducing the use of complex arithmetic and shift selection strategies. We show that there is a very tight relation between the new algorithm and three other algorithms for CARE previously known in the literature—all of these seemingly different methods in fact produce exactly the same iterates when used with the same parameters: they are algorithmically different descriptions of the same approximation sequence to the Riccati solution.

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.

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 timelimited 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.

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. Algorithm 1.1. Square-root balanced truncation (low-rank version).Input : System matrices A, B, C defining an asymptotically stable dynamical system (1.1). Output: MatricesÃ,B,C of the reduced system. 1 Compute low-rank solution factors Z P , Z Q of the solutions of (1.2), such that P ≈ Z P Z T P , Q ≈ Z Q Z T Q . 2 Compute and partition a (thin) singular value decompositionwhere Σ 1 = diag(σ 1 , . . . , σ r ) contains the largest r (approximate) Hankel singular values.1 . 4 Generate reduced order model 1.2. Goals and overview of this article. By looking at the error bound (1.4), the above BT framework generates reduced order models that are accurate for all values ω ∈ R, which, from an application oriented view, are typically considered as Downloaded 03/03/16 to 193.175.53.21. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.phpCopyright © by SIAM. Unauthorized reproduction of this article is prohibited.FLBT WITH LOW-RANK APPROXIMATIONS A473 frequencies. In several applications, however, the underlying physical or technical system operates only in a small frequency interval [ω 1 , ω 2 ] of interest. Restricting the BT procedure to this frequency interval has lead to frequency-limited balanced truncation (FLBT), which was proposed in [19]. One motivation for FLBT is that, compared to ordinary BT, by restricting to a small interval [ω 1 , ω 2 ], we hope to obtain higher accuracies with reduced order models of the same dimension, or to achieve a comparable accuracy with smaller reduced order models inside the interval, while allowing for larger errors outside.The main purpose of this article is to provide a numerically efficient framework for carrying out FLBT for high-dimensional systems. We start by reviewing the concept of frequency-limited Gramians and show how to formulate a procedure similar to the square root approach in Algorithm 1.1. In section 3, we investigate the eigenvalue decay of the frequency-limited Gramians. For the occurring CALEs, we will, as in the unlimited case, employ low-rank approximations of the solu...

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 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.