On Error Estimation for Reduced-order Modeling of Linear Non-parametric and Parametric Systems

Motivated by a recently proposed error estimator for the transfer function of the reduced-order model of a given linear dynamical system, we further develop more theoretical results in this work. Furthermore, we propose several variants of the error estimator, and compare those variants with the existing ones both theoretically and numerically. It has been shown that some of the proposed error estimators perform better than or equally well as the existing ones. All the error estimators considered can be easily extended to estimate output error of reduced-order modeling for steady linear parametric systems.Comment: 34 pages, 12 figure

Peer Methods for the Solution of Large-Scale Differential Matrix Equations

We consider the application of implicit and linearly implicit (Rosenbrock-type) peer methods to matrix-valued ordinary differential equations. In particular the differential Riccati equation (DRE) is investigated. For the Rosenbrock-type schemes, a reformulation capable of avoiding a number of Jacobian applications is developed that, in the autonomous case, reduces the computational complexity of the algorithms. Dealing with large-scale problems, an efficient implementation based on low-rank symmetric indefinite factorizations is presented. The performance of both peer approaches up to order 4 is compared to existing implicit time integration schemes for matrix-valued differential equations.Comment: 29 pages, 2 figures (including 6 subfigures each), 3 tables, Corrected typo

Convergence Analysis of Extended LOBPCG for Computing Extreme Eigenvalues

This paper is concerned with the convergence analysis of an extended variation of the locally optimal preconditioned conjugate gradient method (LOBPCG) for the extreme eigenvalue of a Hermitian matrix polynomial which admits some extended form of Rayleigh quotient. This work is a generalization of the analysis by Ovtchinnikov (SIAM J. Numer. Anal., 46(5):2567-2592, 2008). As instances, the algorithms for definite matrix pairs and hyperbolic quadratic matrix polynomials are shown to be globally convergent and to have an asymptotically local convergence rate. Also, numerical examples are given to illustrate the convergence.Comment: 21 pages, 2 figure