Approximated structured pseudospectra

Pseudospectra and structured pseudospectra are important tools for the analysis of matrices. Their computation, however, can be very demanding for all but small-matrices. A new approach to compute approximations of pseudospectra and structured pseudospectra, based on determining the spectra of many suitably chosen rank-one or projected rank-one perturbations of the given matrix is proposed. The choice of rank-one or projected rank-one perturbations is inspired by Wilkinson's analysis of eigenvalue sensitivity. Numerical examples illustrate that the proposed approach gives much better insight into the pseudospectra and structured pseudospectra than random or structured random rank-one perturbations with lower computational burden. The latter approach is presently commonly used for the determination of structured pseudospectra

A Support Based Algorithm for Optimization with Eigenvalue Constraints

Optimization of convex functions subject to eigenvalue constraints is intriguing because of peculiar analytical properties of eigenvalues, and is of practical interest because of wide range of applications in fields such as structural design and control theory. Here we focus on the optimization of a linear objective subject to a constraint on the smallest eigenvalue of an analytical and Hermitian matrix-valued function. We offer a quadratic support function based numerical solution. The quadratic support functions are derived utilizing the variational properties of an eigenvalue over a set of Hermitian matrices. Then we establish the local convergence of the algorithm under mild assumptions, and deduce a precise rate of convergence result by viewing the algorithm as a fixed point iteration. We illustrate its applicability in practice on the pseudospectral functions.Comment: 18 pages, 2 figure

Computing the structured pseudospectrum of a Toeplitz matrix and its extreme points

The computation of the structured pseudospectral abscissa and radius (with respect to the Frobenius norm) of a Toeplitz matrix is discussed and two algorithms based on a low rank property to construct extremal perturbations are presented. The algorithms are inspired by those considered in [SIAM J. Matrix Anal. Appl., 32 (2011), pp. 1166-1192] for the unstructured case, but their extension to structured pseudospectra and analysis presents several difficulties. Natural generalizations of the algorithms, allowing to draw significant sections of the structured pseudospectra in proximity of extremal points are also discussed. Since no algorithms are available in the literature to draw such structured pseudospectra, the approach we present seems promising to extend existing software tools (Eigtool, Seigtool) to structured pseudospectra representation for Toeplitz matrices. We discuss local convergence properties of the algorithms and show some applications to a few illustrative examples.Comment: 21 pages, 11 figure

Pseudospectra of waveform relaxation operators

Abstract--The performance of the waveform relaxation method for solving systems of ODEs depends largely on the choices that are made for splitting, size of time window, and preconditioning. Although it is known that superlinear convergence is obtained on finite time windows, the convergence may be slow in the first few iterations. We propose the use of pseudcepectra to analyze the convergence ratio of the first few iterations when waveform relaxation is applied to linear systems of ODEs. Through pseudcepectral radii, we can examine the effect of preconditioning and overlapping on the rate of convergence. We may also use this to estimate a suitable size of the time window. Numerical experiments performed on a system of ODEs arising from the discretization of an advection-diffusion equation confirm the validity of the obtained estimates. (~) 1998 Elsevier Science Ltd. All rights reserved

Approximating the Real Structured Stability Radius with Frobenius Norm Bounded Perturbations

We propose a fast method to approximate the real stability radius of a linear dynamical system with output feedback, where the perturbations are restricted to be real valued and bounded with respect to the Frobenius norm. Our work builds on a number of scalable algorithms that have been proposed in recent years, ranging from methods that approximate the complex or real pseudospectral abscissa and radius of large sparse matrices (and generalizations of these methods for pseudospectra to spectral value sets) to algorithms for approximating the complex stability radius (the reciprocal of the $H_\infty$ norm). Although our algorithm is guaranteed to find only upper bounds to the real stability radius, it seems quite effective in practice. As far as we know, this is the first algorithm that addresses the Frobenius-norm version of this problem. Because the cost mainly consists of computing the eigenvalue with maximal real part for continuous-time systems (or modulus for discrete-time systems) of a sequence of matrices, our algorithm remains very efficient for large-scale systems provided that the system matrices are sparse

The effect of uncertainty in MEG-to-MRI coregistrations on MEG inverse problems

For high precision in source estimates of magnetoencephalography (MEG) data, high accuracy of the coregistration of sources and sensors is mandatory. Usually, the source space is derived from magnetic resonance imaging (MRI). Sensor-to-MRI coregistrations are the focus of this thesis. The quality of coregistrations is assessed and the effect of their uncertainties on source estimates is analyzed. Both topics, the quality assessment and the propagation of uncertainties to source estimates are treated separately. In this thesis, the target registration error (TRE) is proposed as criterion for the quality of sensor-to-MRI coregistrations. TRE measures the effect of uncertainty in coregistrations at all points of interest. In total, 5544 data sets with sensor-to-head and 128 head-to-MRI coregistrations, from a single MEG laboratory, were analyzed. An adaptive Metropolis algorithm was used to estimate the optimal coregistration and to sample the coregistration parameters (rotation and translation). I found an average TRE between 1.3 and 2.3 mm at the head surface. A mean absolute difference in coregistration parameters between the Metropolis and iterative closest point algorithm of (1.9 ± 1.5)° and (1.1 ± 0.9) mm was found. A paired sample t-test indicated a significant improvement in goal function minimization by using the Metropolis algorithm. The sampled parameters allowed computation of TRE on the entire grid of the MRI volume. Hence, I recommend the Metropolis algorithm for head-to-MRI coregistrations. The propagation of coregistration uncertainty to source estimates was performed by using pseudospectral approximations of beamformer and standardized low resolution tomography (sLORETA). This approach was tested for auditory, visual and somatosensory brain activity with different signal to noise ratios and source orientation constraints on datasets of 20 subjects. By using pseudospectral approximations as efficient surrogates, the spatial distribution of the source estimate maximum was sampled for 50000 coregistrations. From the results, it can be concluded that it is possible to apply stochastic spectral methods to MEG source estimation with high accuracy. The investigated effects of coregistration uncertainties on source estimates are small, typically the maximum location varied within a range of 5 mm, which is in the range of the localization errors. Pseudospectral approximations of the source estimates reduced computation times considerably by a factor of approximately 10000 for beamformer and 50000 for sLORETA compared to the exact original computations.Für eine hohe Präzision in der Schätzung von Gehirnaktivität, ausgehend von Daten der Magnetoenzephalographie (MEG), ist eine sehr genaue Koregistrierung der Quellen und Sensoren notwendig. Üblicherweise werden hierbei die Quellorte der Gehirnaktivität bezüglich zu Koordinaten der Magnetresonanztomographie (MRI) angegeben. Die Sensor-zu-MRI Koregistrierungen sind der Schwerpunkt dieser Arbeit. Die Qualität von Koregistrierungen wird bewertet und der Effekt ihrer Unsicherheiten auf Schätzungen der Gehirnaktivität beziehungsweise auf Quellschätzungen wird untersucht. Beide Themen, die Qualitätsbewertung und die Übertragung der Unsicherheiten auf Quellschätzungen werden separat behandelt. In dieser Arbeit wird vorgeschlagen, den target registration error (TRE) als Qualitätskriterium für Sensor-zu-MRI Koregistrierungen zu verwenden. Der TRE kann den Effekt von Koregistrierungsunsicherheiten an beliebigen Punkten messen. Insgesamt wurden 5544 Datensätze mit Sensor-zu-Kopf und 128 Datensätze mit Kopf-zu-MRI Koregistrierungen aus einem Labor analysiert. Ein adaptiver Metropolis-Algorithmus wurde genutzt um optimale Koregistrierungen zu schätzen und um Stichproben ihrer Parameter (Rotation und Translation) zu ziehen. Es wurde ein TRE von 1.3 und 2.3 mm an der Kopfoberfläche gefunden. Weiter wurde eine mittlere absolute Differenz der Koregistrierungsparameter zwischen Metropolis-Algorithmus und dem etablierten iterative closest point-Algorithmus von (1.9 ± 1.5)° und (1.1 ± 0.9) mm gefunden. Ein Zweistichproben-t-Test zeigte eine signifikante Verbesserung in der Optimierung der Zielfunktion durch den Metropolis-Algorithmus. Die Übertragung der Koregistrierungsunsicherheit auf Quellschätzungen erfolgte unter Verwendung von speziellen Polynom-Entwicklungen des Beamformers und der standardized low resolution tomography (sLORETA). Dieser Ansatz wurde für auditorische, visuelle und somatosensorische Hirnaktivität mit verschiedenen Signal-Rausch-Verhältnissen und Beschränkungen der Quellorientierung auf Datensätzen von 20 Probanden getestet. Durch die Verwendung von Polynom-Entwicklungen als effiziente Surrogate wurde die örtliche Verteilung des Quellschätzungs-Maximums für 50000 Koregistrierungen ermittelt. Aus den Ergebnissen lässt sich schließen, dass es möglich ist, Polynom-Entwicklungen mit hoher Genauigkeit auf MEG-Quellschätzungen anzuwenden. Polynom-Entwicklungen der Quellschätzungen reduzierten die Berechnungszeiten erheblich um den Faktor von etwa 10000 für Beamformer und 50000 für sLORETA im Vergleich zu den exakten Originalrechnungen