870 research outputs found
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
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