Model reduction using a frequency-limited -cost

Abstract. The most popular method for computing the matrix logarithm is the inverse scaling and squaring method, which is the basis of the recent algorithm of [A. H. Al-Mohy and N. J. Higham, Improved inverse scaling and squaring algorithms for the matrix logarithm, SIAM J. Sci. Comput., 34 (2012), pp. C152-C169]. We show that by differentiating the latter algorithm a backward stable algorithm for computing the Fréchet derivative of the matrix logarithm is obtained. This algorithm requires complex arithmetic, but we also develop a version that uses only real arithmetic when A is real; as a special case we obtain a new algorithm for computing the logarithm of a real matrix in real arithmetic. We show experimentally that our two algorithms are more accurate and efficient than existing algorithms for computing the Fréchet derivative. We also show how the algorithms can be used to produce reliable estimates of the condition number of the matrix logarithm.Key words. matrix logarithm, principal logarithm, inverse scaling and squaring method, Fréchet derivative, condition number, Padé approximation, backward error analysis, matrix exponential, matrix square root, MATLAB, logm.

show abstract

“…The Fréchet derivative of the matrix logarithm has recently been used in nonlinear optimization techniques for model reduction [32].…”

Section: Introduction a Logarithm Of A ∈ Cmentioning

confidence: 99%

Computing the Fréchet Derivative of the Matrix Logarithm and Estimating the Condition Number

Al-Mohy¹,

Higham

Relton

2013

SIAM J. Sci. Comput.

View full text Add to dashboard Cite

show abstract

“…In [21], a nonlinear optimization-based MOR algorithm is presented to achieve the local optimality for the problem…”

Section: Frequency-limited H 2 -Optimal Mormentioning

confidence: 99%

“…MOR is applied to this extended realization, and the ROM is obtained via an inverse transformation. In [21], the optimal frequency-limited H 2 -MOR problem is considered, and an algorithm is proposed, which generates an optimal ROM. The algorithm requires the solution of Lyapunov equations and linear matrix inequalities (LMIs) to find the optimal ROM, which is not feasible in a large-scale setting.…”

Section: Introductionmentioning

confidence: 99%

Frequency-limited pseudo-optimal rational Krylov algorithm for power system reduction

Zulfiqar

Sreeram

2020

International Journal of Electrical Power & Energy Systems

View full text Add to dashboard Cite

In this paper, a computationally efficient frequencylimited model reduction algorithm is presented for large-scale interconnected power systems. The algorithm generates a reduced order model which not only preserves the electromechanical modes of the original power system but also satisfies a subset of the first-order optimality conditions for H2,ω model reduction problem within the desired frequency interval. The reduced order model accurately captures the oscillatory behavior of the original power system and provides a good time-and frequency-domain accuracy. The proposed algorithm enables fast simulation, analysis, and damping controller design for the original large-scale power system. The efficacy of the proposed algorithm is validated on benchmark power system examples.

show abstract

“…Increasingly the Fréchet derivative is also required, with recent examples including computation of correlated choice probabilities [1], registration of MRI images [6], Markov models applied to cancer data [14], matrix geometric mean computation [24], and model reduction [33], [34]. Higher order Fréchet derivatives have been used to solve nonlinear equations on Banach spaces by generalizing the Halley method [4, sec.…”

Section: Introduction Matrix Functions F : Cmentioning

confidence: 99%

Higher Order Fréchet Derivatives of Matrix Functions and the Level-2 Condition Number

Higham¹,

Relton²

2014

SIAM J. Matrix Anal. & Appl.

View full text Add to dashboard Cite

Abstract. The Fréchet derivative L f of a matrix function f : C n×n → C n×n controls the sensitivity of the function to small perturbations in the matrix. While much is known about the properties of L f and how to compute it, little attention has been given to higher order Fréchet derivatives. We derive sufficient conditions for the kth Fréchet derivative to exist and be continuous in its arguments and we develop algorithms for computing the kth derivative and its Kronecker form. We analyze the level-2 absolute condition number of a matrix function ("the condition number of the condition number") and bound it in terms of the second Fréchet derivative. For normal matrices and the exponential we show that in the 2-norm the level-1 and level-2 absolute condition numbers are equal and that the relative condition numbers are within a small constant factor of each other. We also obtain an exact relationship between the level-1 and level-2 absolute condition numbers for the matrix inverse and arbitrary nonsingular matrices, as well as a weaker connection for Hermitian matrices for a class of functions that includes the logarithm and square root. Finally, the relation between the level-1 and level-2 condition numbers is investigated more generally through numerical experiments.Key words. matrix function, Fréchet derivative, Gâteaux derivative, higher order derivative, matrix exponential, matrix logarithm, matrix square root, matrix inverse, matrix calculus, partial derivative, Kronecker form, level-2 condition number, expm, logm, sqrtm, MATLAB AMS subject classifications. 65F30, 65F60DOI. 10.1137/130945259 Introduction. Matrix functions f : Cn×n → C n×n such as the matrix exponential, the matrix logarithm, and matrix powers A t for t ∈ R are being used within a growing number of applications including model reduction [5] The Fréchet derivative of f at A ∈ C n×n is the unique function L f (A, · ) that is linear in its second argument and for all E ∈ C n×n satisfies

show abstract

Model reduction using a frequency-limited -cost

Cited by 39 publications

References 23 publications

Computing the Fréchet Derivative of the Matrix Logarithm and Estimating the Condition Number

Computing the Fréchet Derivative of the Matrix Logarithm and Estimating the Condition Number

Frequency-limited pseudo-optimal rational Krylov algorithm for power system reduction

Higher Order Fréchet Derivatives of Matrix Functions and the Level-2 Condition Number

Contact Info

Product

Resources

About