• Corpus ID: 17778728

# A Rational Krylov Iteration for Optimal H 2 Model Reduction

@inproceedings{Gugercin2006ARK,
title={A Rational Krylov Iteration for Optimal H 2 Model Reduction},
author={Serkan Gugercin and Athanasios C. Antoulas and Christopher A. Beattie},
year={2006}
}
• Published 2006
• Computer Science, Mathematics
In the sequel, we will construct the reduced order models Gr(s) through Krylov projection methods. Toward this end, we construct matrices V ∈ R and Z ∈ R that span certain Krylov subspaces with the property that Z V = Ir. The reduced order model Gr(s) will then be obtained as Ar = Z T AV, Br = Z T B, and Cr = CV. (2) The corresponding oblique projection is given by V Z . Many researchers have worked on the problem (1); see [18], [16], [7], [5], [3], [17], [4] and references therein. Since…
54 Citations

## Figures from this paper

### Realization-independent H 2-approximation

• Computer Science, Mathematics
• 2012
By exploiting the Loewner-matrix approach for interpolation, a new formulation of IRKA is developed that only requires transfer function evaluations without access to any particular realization, which extends IRKA to H2 approximation of irrational, infinite-dimensional dynamical systems.

### Realization-independent ℌ2-approximation

• Computer Science, Mathematics
2012 IEEE 51st IEEE Conference on Decision and Control (CDC)
• 2012
A Loewner-matrix approach for interpolation is employed, and a new formulation of IRKA is developed that only uses transfer function evaluations, without requiring any particular realization, to be extended to H2 approximation of irrational, infinite-dimensional dynamical systems.

### On the ADI method for the Sylvester Equation and the optimal-H 2 points

• Computer Science, Mathematics
• 2012
This paper shows that the ADI and rational Krylov approximations are in fact equivalent when a special choice of shifts are employed in both methods, and calls these shifts pseudo H2-optimal shifts, optimal in the sense that for the Lyapunov equation, they yield a residual which is orthogonal to therational Krylov projection subspace.

### Rational Krylov decompositions : theory and applications

The new concept of continuation pairs gives rise to a near-optimal parallelization strategy that allows to control the growth of the condition number of this nonorthogonal basis, and as a consequence a more accurate and reliable parallel rational Arnoldi algorithm is obtained.

### Control and Cybernetics Iterative-interpolation Algorithms for L 2 Model Reduction *

Two convergent algorithms that draw on previous procedures presented by the same authors, are suggested: one refers to s–domain representations, the other to time–domain state–space representations.

### Méthodes de type Lanczos rationnel pour la réduction de modèles

This dissertations focuses on projection methods to efficiently construct reduced order models for large linear dynamical systems by projection onto unions of Krylov subspaces which lead to a class of reduced orders models known as rational interpolation.

### $\mathcal{L}_2$-optimal Reduced-order Modeling Using Parameter-separable Forms

• Mathematics, Computer Science
• 2022
Using parameter-separable forms of the reduced-model quantities, the gradients of the L 2 cost function are derived with respect to the reduced matrices, which allows a non-intrusive, data-driven, gradient-based descent algorithm to construct the optimal approximant using only output samples.

### Comparison of Model Reduction Methods with Applications to Circuit Simulation

• Mathematics
• 2007
Summary. We compare different model reduction methods applied to the dynamical system of a coupled transmission line: balanced truncation (BT), truncation by balancing one gramian (or PMTBR - poor

### The RKFIT Algorithm for Nonlinear Rational Approximation

• Computer Science
SIAM J. Sci. Comput.
• 2017
This paper derives a strategy for the degree reduction of the approximants, as well as methods for their conversion to partial fraction form, for the efficient evaluation, and root-finding, and puts RKFIT into a general framework.

## References

SHOWING 1-10 OF 23 REFERENCES

### Krylov Projection Methods for Model Reduction

The cornerstone of this dissertation is a collection of theory relating Krylov projection to rational interpolation, based on which three algorithms for model reduction are proposed, which are suited for parallel or approximate computations.

### The optimal projection equations for model reduction and the relationships among the methods of Wilson, Skelton, and Moore

• Mathematics
• 1985
First-order necessary conditions for quadratically optimal reduced-order modeling of linear time-invariant systems are derived in the form of a pair of modified Lyapunov equations coupled by an

### Second-order algorithm for optimal model order reduction

• Mathematics
• 1990
A second-order algorithm is given for optimal model order reduction of continuous single-input/single-output systems. Given a transfer function of order n, it finds the transfer function of a model

### Rational L/sub 2/ approximation: a non-gradient algorithm

• Computer Science
[1991] Proceedings of the 30th IEEE Conference on Decision and Control
• 1991
An efficient algorithm is presented that makes it possible to find a (local) minimum without evaluating derivatives based on a reformulation of the necessary conditions for optimality in terms of interpolation constraints.

### Solving large-scale control problems

• P. Benner
• Computer Science
IEEE Control Systems
• 2004
It is concluded that modern tools from numerical linear algebra, along with careful investigation and exploitation of the problem structure, can be used to derive algorithms capable of solving large control problems.

### Efficient numerical solution of the LQR‐problem for the heat equation

• Mathematics, Chemistry
• 2004
We discuss how the theory developed by Banks and Kunisch can be applied to a modified version of a controlled heat transfer model introduced by Tröltzsch and Unger. In the numerical implementation we

### A comparative study of 7 algorithms for model reduction

• Computer Science
Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187)
• 2000
Compares seven model reduction algorithms by applying them to four different dynamical systems and concludes that the rational Krylov algorithm gives the best results.

### Approximation of linear constant systems

• Mathematics
IEEE Transactions on Automatic Control
• 1967
In this paper two problems are considered, the problem of modeling a given constant linear system by a constant linear system of fixed lower order, and the problem of finding a filter of fixed order

### A rational Lanczos algorithm for model reduction

• Computer Science
Numerical Algorithms
• 2005
A variant of the nonsymmetric Lanczos method, rational Lanczos, is shown to yield a rational interpolant (multi-point Padé approximant) for the large-scale system.