arXiv subjects

Ninoslav Truhar

Publications and source records attributed to Ninoslav Truhar.

8 recordsLinked to original sources

arXiv2026

Structure-preserving deflation of critical eigenvalues in quadratic eigenvalue problems associated with damped mass-spring systems

For a quadratic matrix polynomial associated with a damped mass-spring system there are three types of critical eigenvalues, the eigenvalues $\infty$ and $0$ and the eigenvalues on the imaginary axis. All these are on the boundary of the set of (robustly) stable eigenvalues. For numerical methods, but also for (robust) stability analysis, it is desirable to deflate such eigenvalues by projecting the matrix polynomial to a lower dimensional subspace before computing the other eigenvalues and eigenvectors. We describe structure-preserving deflation strategies that deflate these eigenvalues via a trimmed structure-preserving linearization. We employ these results for the special case of hyperbolic problems. We also analyze the effect of a (possibly low rank) parametric damping matrix on purely imaginary eigenvalues.

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arXiv2026

Optimal placement and tuning of pointwise dampers for vibrating strings via a Lyapunov framework

We study the optimal placement and tuning of a small number of pointwise viscous dampers for a vibrating string. Starting from a finite element discretization of the damped wave equation, the system is transformed into a first-order phase-space formulation, which enables a unified Lyapunov trace framework. Three optimization criteria are considered: average total energy, average total displacement, and energy for a fixed initial state. For all criteria, explicit gradient formulas with respect to damper positions and viscosities are derived, requiring only one primal and one dual Lyapunov solve. Due to the strong non-convexity of the problem, a simple heuristic based on an explicit single-damper formula is proposed to generate effective initial guesses. Numerical examples illustrate the influence of spectral selection and discretization on the optimal damping configuration.

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arXiv2025

Damping optimization of discrete mechanical systems -- rod/string model

This paper investigates two optimization criteria for damping optimization in a multi-body oscillator system with arbitrary degrees of freedom ($n$), resembling string/rod free vibrations. The total average energy over all possible initial data and the total average displacement over all possible initial data. Our first result shows that both criteria are equivalent to the trace minimization of the solution of the Lyapunov equation with different right-hand sides. As the second result, we prove that in the case of damping with one damper, for the discrete system, the minimal trace for each criterion can be expressed as a linear or cubic function of the dimension $n$. Consequently, the optimal damping position is determined solely by the number of dominant eigenfrequencies and the optimal viscosity, independent of the dimension $n$, offering efficient damping optimization in discrete systems. The paper concludes with numerical examples illustrating the presented theoretical framework and results.

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arXiv2024

On Stewart's Perturbation Theorem for SVD

This paper establishes a variant of Stewart's theorem (Theorem~6.4 of Stewart, {\em SIAM Rev.}, 15:727--764, 1973) for the singular subspaces associated with the SVD of a matrix subject to perturbations. Stewart's original version uses both the Frobenius and spectral norms, whereas the new variant uses the spectral norm and any unitarily invariant norm that offer choices per convenience of particular applications and lead to sharper bounds than that straightforwardly derived from Stewart's original theorem with the help of the well-known equivalence inequalities between matrix norms. Of interest in their own right, bounds on the solution to two couple Sylvester equations are established for a few different circumstances.

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arXiv2020

Relative Perturbation Theory for Quadratic Hermitian Eigenvalue Problem

In this paper, we derive new relative perturbation bounds for eigenvectors and eigenvalues for regular quadratic eigenvalue problems of the form $λ^2 M x + λC x + K x = 0$, where $M$ and $K$ are nonsingular Hermitian matrices and $C$ is a general Hermitian matrix. We base our findings on new results for an equivalent regular Hermitian matrix pair $A-λB$. The new bounds can be applied to many interesting quadratic eigenvalue problems appearing in applications, such as mechanical models with indefinite damping. The quality of our bounds is demonstrated by several numerical experiments.

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arXiv2019

Mixed control of vibrational systems

We consider new performance measures for vibrational systems based on the $H_2$ 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.

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arXiv2012

Generalized Eigenvalue Problems with Specified Eigenvalues

We consider the distance from a (square or rectangular) matrix pencil to the nearest matrix pencil in 2-norm that has a set of specified eigenvalues. We derive a singular value optimization characterization for this problem and illustrate its usefulness for two applications. First, the characterization yields a singular value formula for determining the nearest pencil whose eigenvalues lie in a specified region in the complex plane. For instance, this enables the numerical computation of the nearest stable descriptor system in control theory. Second, the characterization partially solves the problem posed in [Boutry et al. 2005] regarding the distance from a general rectangular pencil to the nearest pencil with a complete set of eigenvalues. The involved singular value optimization problems are solved by means of BFGS and Lipschitz-based global optimization algorithms.

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arXiv2010

The Rotation of Eigenspaces of Perturbed Matrix Pairs

We revisit the relative perturbation theory for invariant subspaces of positive definite matrix pairs. As a prototype model problem for our results we consider parameter dependent families of eigenvalue problems. We show that new estimates are a natural way to obtain sharp --- as functions of the parameter indexing the family of matrix pairs --- estimates for the rotation of spectral subspaces.

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