6 research outputs found
Large-Scale Computation of -Norms by a Greedy Subspace Method
We are concerned with the computation of the -norm for
an -function of the form ,
where the middle factor is the inverse of a meromorphic matrix-valued function,
and are meromorphic functions mapping to short-and-fat and
tall-and-skinny matrices, respectively. For instance, transfer functions of
descriptor systems and delay systems fall into this family. We focus on the
case where the middle factor is large-scale. We propose a subspace projection
method to obtain approximations of the function where the middle factor is
of much smaller dimension. The -norms are computed for the
resulting reduced functions, then the subspaces are refined by means of the
optimal points on the imaginary axis where the -norm of
the reduced function is attained. The subspace method is designed so that
certain Hermite interpolation properties hold between the largest singular
values of the original and reduced functions. This leads to a locally
superlinearly convergent algorithm with respect to the subspace dimension,
which we prove and illustrate on various numerical examples.Comment: 23 pages, 3 figure
Measuring the stability of spectral clustering
As an indicator of the stability of spectral clustering of an undirected
weighted graph into clusters, the th spectral gap of the graph Laplacian
is often considered. The th spectral gap is characterized in this paper as
an unstructured distance to ambiguity, namely as the minimal distance of the
Laplacian to arbitrary symmetric matrices with vanishing th spectral gap. As
a conceptually more appropriate measure of stability, the structured distance
to ambiguity of the -clustering is introduced as the minimal distance of the
Laplacian to Laplacians of graphs with the same vertices and edges but with
weights that are perturbed such that the th spectral gap vanishes. To
compute a solution to this matrix nearness problem, a two-level iterative
algorithm is proposed that uses a constrained gradient system of matrix
differential equations in the inner iteration and a one-dimensional
optimization of the perturbation size in the outer iteration. The structured
and unstructured distances to ambiguity are compared on some example graphs.
The numerical experiments show, in particular, that selecting the number of
clusters according to the criterion of maximal stability can lead to different
results for the structured and unstructured stability indicators
Mixed control of vibrational systems
We consider new performance measures for vibrational systems based on the
norm of linear time invariant systems. New measures will be used as an
optimization criterion for the optimal damping of vibrational systems. We
consider both theoretical and concrete cases in order to show how new measures
stack up against the standard measures. The quality and advantages of new
measures as well as the behaviour of optimal damping positions and
corresponding damping viscosities are illustrated in numerical experiments
Semi-active damping optimization by adaptive interpolation
In this work we consider the problem of semi-active damping optimization of
mechanical systems with fixed damper positions. Our goal is to compute a
damping that is locally optimal with respect to the -norm
of the transfer function from the exogenous inputs to the performance outputs.
We make use of a new greedy method for computing the -norm
of a transfer function based on rational interpolation. In this paper, this
approach is adapted to parameter-dependent transfer functions. The
interpolation leads to parametric reduced-order models that can be optimized
more efficiently. At the optimizers we then take new interpolation points to
refine the reduced-order model and to obtain updated optimizers. In our
numerical examples we show that this approach normally converges fast and thus
can highly accelerate the optimization procedure. Another contribution of this
work are heuristics for choosing initial interpolation points
Extended and improved criss-cross algorithms for computing the spectral value set abscissa and radius
In this paper, we extend the original criss-cross algorithms for computing
the -pseudospectral abscissa and radius to general spectral value
sets. By proposing new root-finding-based strategies for the horizontal/radial
search subphases, we significantly reduce the number of expensive Hamiltonian
eigenvalue decompositions incurred, which typically translates to meaningful
speedups in overall computation times. Furthermore, and partly necessitated by
our root-finding approach, we develop a new way of handling the singular
pencils or problematic interior searches that can arise when computing the
-spectral value set radius. Compared to would-be direct extensions
of the original algorithms, that is, without our additional modifications, our
improved criss-cross algorithms are not only noticeably faster but also more
robust and numerically accurate, for both spectral value set and pseudospectral
problems.Comment: Final versio
www.mpi-magdeburg.mpg.de/preprints A STRUCTURED PSEUDOSPECTRAL METHOD FOR H∞-NORM COMPUTATION OF LARGE-SCALE DESCRIPTOR SYSTEMS
Abstract. In this paper we discuss the problem of computing the H∞-norms of transfer functions associated to large-scale descriptor systems. We exploit the relationship between the H∞-norm and the structured complex stability radius of the corresponding system pencil. To compute the structured stability radius we consider so-called structured pseudospectra. Namely, we have to find the pseudospectrum touching the imaginary axis. Therefore, we set up an iteration over the real part of the rightmost pseudoeigenvalue. For that we use a new fast iterative scheme which is based on certain rank-1 perturbations of the system pencil. Finally, we analyze the performance of our algorithm by using real-world examples. Key words. Descriptor systems, H ∞ control, iterative methods, pseudospectra, sparse matrices, stability of linear systems. 1. Introduction and Preliminaries. In this paper we consider linear timeinvariant descriptor systems of the form E ˙x(t) = Ax(t) + Bu(t), y(t) = Cx(t), (1.1