Generalized Eigenvalue Problem Research Articles

The pseudospectrum and transient of Kaluza–Klein black holes in Einstein–Gauss–Bonnet gravity

Abstract The spectrum and dynamical instability, as well as the transient effect of the tensor perturbation for the so-called Maeda–Dadhich black hole, a type of Kaluza–Klein black hole, in Einstein–Gauss–Bonnet gravity have been investigated in framework of pseudospectrum. We cast the problem of solving quasinormal modes (QNMs) in AdS-like spacetime as the linear evolution problem of the non-normal operator in null slicing by using ingoing Eddington–Finkelstein coordinates. In terms of spectrum instability, based on the generalized eigenvalue problem, the QNM spectrum and ε-pseudospectrum has been studied, while the open structure of ε-pseudospectrum caused by the non-normality of operator indicates the spectrum instability. In terms of dynamical instability, we introduce the concept of the distance to dynamical instability, which plays a crucial role in bridging the spectrum instability and the dynamical instability. We calculate such distance, named the complex stability radius, as parameters vary. Finally, we show the behavior of the energy norm of the evolution operator, which can be roughly reflected by the three kinds of abscissas in context of pseudospectrum, and find the transient growth of the energy norm of the evolution operator.

A Finite Integral Transform-Based Generalized Eigenvalue Formulation for Buckling of Orthotropic Rectangular Plates with Rotational Restraints

Although the finite integral transform (FIT) method has been developed for buckling analysis of rectangular thin plates, the existing formulation involves solving complex nonlinear determinantal equations, leading to potentially questionable numerical results. To address this challenge, a new FIT-based formulation is established that transforms the solution of a complex nonlinear equation into that of a straightforward generalized eigenvalue problem. New analytical solutions for buckling of orthotropic rectangular thin plates with rotational restraints are obtained. The solution procedure is implemented via the following four steps: imposing the two-dimensional FIT on the governing equation; inputting deflection conditions of four edges, by which the relations between the transformed deflections and specific unknowns are provided; imposing the one-dimensional FIT on the elastic conditions to replace the unknowns with the transformed deflections, a generalized eigenvalue problem is constructed; determining the final analytical solutions by solving the generalized eigenvalue problem. Comprehensive highly accurate buckling load/mode solutions for typical rotationally restrained orthotropic plates are presented as benchmarks. The effects of boundary conditions, rotational fixity factors, and loading ratios on the buckling behaviors of orthotropic plates are expediently investigated adopting the present solutions.

Optimal weighting strategies for maximizing contrast-to-noise ratio in photon counting CT images.

Photon counting detectors (PCDs) with energy discrimination capabilities have the potential to generate grayscale CT images with improved contrast-to-noise ratio (CNR) through optimal weighting of their spectral measurements. This study evaluates the CNR performance of grayscale CT projections and images generated from spectral measurements of PCDs using three energy-weighting strategies: pre-log weighting, post-log weighting, and material decomposition (MD) weighting. This study provides the expressions of optimal weights and maximum achievable CNR of these energy-weighting strategies, which only require the knowledge of detected bin counts and do not require information of PCD energy responses or imaging techniques. We defined and solved a generalized eigenvalue problem to obtain the maximum achievable CNR in the projection domain for low-contrast tasks using three energy-weighting strategies: pre-log weighting (weighted sum of energy bin counts), post-log weighting (weighted sum of line integrals), and MD weighting (weighted sum of basis material thicknesses, which is equivalent to virtual monoenergetic images [VMIs]). These expressions only contain energy bin counts from PCD measurements. We used a realistic PCD energy response model to simulate the detected bin counts and conducted Monte Carlo simulations of different contrast tasks and phantoms to evaluate the projection- and image-domain CNR performance of these energy-weighting strategies. Additionally, the total counts method (a special case of pre-log weighting with unity weights) was included for comparison. We also conducted Gammex head and body phantom scans on an edge-on-irradiated silicon PCCT prototype to evaluate the image-domain CNR performance of these energy-weighting strategies. The results show that pre-log, post-log, and MD weighting strategies generate approximately equal projection-domain maximum achievable CNR, with a difference of less than 2%, and outperform the total counts method. These three energy-weighting strategies also generate approximately equal image-domain maximum CNR when the contrast task is located at the center of a homogeneous phantom. Pre-log weighting generates the highest image-domain CNR for an off-center contrast task location or inhomogeneous phantoms while also outperforming the total counts method. We derived the expression of projection-domain maximum achievable CNR using three energy-weighting strategies. Our results suggest that using pre-log weighting strategies enables fast grayscale CT image generation with high CNR from spectral PCD measurements for inhomogeneous phantoms and off-center region of interests (ROIs).

Mixing between flavor singlets in lattice gauge theories coupled to matter fields in multiple representations

We provide the first extensive, numerical study of the nontrivial problem of mixing between flavor-singlet composite states emerging in strongly coupled lattice field theories with matter field content consisting of fermions transforming in different representations of the gauge group. The theory of interest is the minimal candidate for a composite Higgs model that also accommodates a mechanism for top partial compositeness: the Sp(4) gauge theory coupled to two (Dirac) fermions transforming as the fundamental and three as the two-index antisymmetric representation of the gauge group, respectively. We apply an admixture of APE smearing and Wuppertal smearings, as well as the generalized eigenvalue problem approach, to two-point functions involving flavor-singlet mesons, for ensembles having time extent longer than the space extent. We demonstrate that, in the region of lattice parameter space accessible to this study, both masses and mixing angles can be measured effectively, despite the presence of (numerically noisy) contributions from disconnected diagrams. Published by the American Physical Society 2024

Acceleration effect on stress singularity and edge strength in piezoelectric/Piezomagnetic heterostructures

The acceleration effect on edge stress singularity and edge strength in piezoelectric/piezomagnetic heterostructures is investigated in this work. For the stress singularity analysis, we propose a stress function based iterative approach based on the Lekhnitskii stress functions with body forces and harmonic assumption of initial stress fields. The acceleration effect is introduced by adding body force terms to the equilibrium equations. The governing equations are obtained by applying variational principle in each process and solved by general eigenvalue problems to obtain homogeneous solutions, as well as to obtain particular solutions based on the forms of load and acceleration conditions. During the iterations, the stress oscillations can be gradually eliminated and the stress concentration can be predicted exactly located at the interfaces. Finally, an example of symmetrically layered heterostructure is presented under both in-plane acceleration and out-of-plane acceleration. It is found that both accelerations have significant effect on the edge normal and shear stresses which may further cause failure. The edge strength is also evaluated by calculating the edge average stress. This work may help to understand the acceleration effect on edge stresses and edge strength for heterostructures, as well as the trend of stress magnitude change caused by external accelerations.

A contour integral-based method for nonlinear eigenvalue problems for semi-infinite photonic crystals

In this study, we introduce an efficient method for determining isolated singular points of two-dimensional semi-infinite and bi-infinite photonic crystals, equipped with perfect electric conductor and quasi-periodic mixed boundary conditions. This specific problem can be modeled by a Helmholtz equation and is recast as a generalized eigenvalue problem involving an infinite-dimensional block quasi-Toeplitz matrix. Through an intelligent implementation of cyclic structure-preserving matrix transformations, the contour integral method is elegantly employed to calculate the isolated eigenvalue and to extract a component of the associated eigenvector. Moreover, a propagation formula for electromagnetic fields is derived. This formulation enables rapid computation of field distributions across the expansive semi-infinite and bi-infinite domains, thus highlighting the attributes of edge states. The preliminary MATLAB implementation is available at https://github.com/FAME-GPU/2D_Semi-infinite_PhC.

Influence of radiation on the stability of MHD micropolar fluid in a vertical channel

The objective of the present article is to explore the stability of micropolar fluid flow in a vertical channel in the presence of thermal radiation and a transverse magnetic field. The generalized eigenvalue problem is numerically solved by utilizing the Chebyshev spectral collocation method, which is obtained from the perturbed state using the normal mode technique. The numerical data were compared with previously published results for particular cases. The critical modified Grashof number (Grc′) and the associated wave numbers (αc) are calculated and displayed graphically for different values of the parameters. It is noticed that the boundaries of instability may be increased or decreased with the flow governing parameters because of the presence of a magnetic field and thermal radiation.

Dynamic and modal analysis of nearly incompressible structures with stabilised displacement-volumetric strain formulations

This paper presents a dynamic formulation for the simulation of nearly incompressible structures using a mixed finite element method with equal-order interpolation pairs. Specifically, the nodal unknowns are the displacement and the volumetric strain component, something that makes possible the reconstruction of the complete stain at the integration point level and thus enables the use of strain-driven constitutive laws. Furthermore, we also discuss the resulting eigenvalue problem and how it can be applied for the modal analysis of linear elastic solids. The article puts special emphasis on the stabilisation technique used, which becomes crucial in the resolution of the generalised eigenvalue problem. In particular, we prove that using a variational multiscale method assuming the sub-grid scales to lie in the finite element space orthogonal to that of the approximation, namely the Orthogonal Sub-Grid Scales (OSGS), results in a convenient linear and symmetric generalised eigenvalue problem. The correctness, convergence and performance of the method are proven by solving a series of two- and three-dimensional examples.

Match-based solution of general parametric eigenvalue problems

We describe a novel algorithm for solving general parametric (nonlinear) eigenvalue problems. Our method has two steps: first, high-accuracy solutions of non-parametric versions of the problem are gathered at some values of the parameters; these are then combined to obtain global approximations of the parametric eigenvalues. To gather the non-parametric data, we use non-intrusive contour-integration-based methods, which, however, cannot track eigenvalues that migrate into/out of the contour as the parameter changes. Special strategies are described for performing the combination-over-parameter step despite having only partial information on such migrating eigenvalues. Moreover, we dedicate a special focus to the approximation of eigenvalues that undergo bifurcations. Finally, we propose an adaptive strategy that allows one to effectively apply our method even without any a priori information on the behavior of the sought-after eigenvalues. Numerical tests are performed, showing that our algorithm can achieve remarkably high approximation accuracy.

Hydrodynamic stability of magnetic boundary layer flow of viscoelastic Walters' liquid B embedded in a porous medium

The present study investigates the linear stability of stagnation boundary layer flow of viscoelastic Walters' liquid B in the presence of magnetic field and porous medium by solving modified Orr–Sommerfeld equation numerically using the Chebyshev collocation method. The model is characterized mainly by the elasticity number (E), the magnetic number (Q), and the permeability parameter (K) in addition to the Reynolds number(Re). The Prandtl boundary layer equations derived for the present model are converted through appropriate similarity transformations, to an ordinary differential equation whose solution describes the velocity, which has oscillatory behavior. The solution of generalized eigenvalue problem governing the stability of the boundary layer has an interesting eigenspectrum. The spectra for different values of E, K, and Q are shown to be a continuation of Newtonian eigenspectrum with the instability belongs to viscoelastic wall mode for certain range of parameters. It is shown that the role of elasticity number is to destabilize the viscoelastic boundary layer flow, whereas both magnetic field and porous medium have the stabilizing effect on the flow. These interesting features are further confirmed by performing the energy budget analysis on the perturbed quantities. Region of negative production due to the Reynolds stress as well as production due to viscous dissipation and viscoelastic contributions in the positive region, and there is reduction in the growth rate of kinetic energy that causes stability. Other physical mechanisms related to flow stability are discussed in detail.

Efficient Mixed-Precision Matrix Factorization of the Inverse Overlap Matrix in Electronic Structure Calculations with AI-Hardware and GPUs.

In recent years, a new kind of accelerated hardware has gained popularity in the artificial intelligence (AI) community which enables extremely high-performance tensor contractions in reduced precision for deep neural network calculations. In this article, we exploit Nvidia Tensor cores, a prototypical example of such AI-hardware, to develop a mixed precision approach for computing a dense matrix factorization of the inverse overlap matrix in electronic structure theory, S-1. This factorization of S-1, written as ZZT = S-1, is used to transform the general matrix eigenvalue problem into a standard matrix eigenvalue problem. Here we present a mixed precision iterative refinement algorithm where Z is given recursively using matrix-matrix multiplications and can be computed with high performance on Tensor cores. To understand the performance and accuracy of Tensor cores, comparisons are made to GPU-only implementations in single and double precision. Additionally, we propose a nonparametric stopping criteria which is robust in the face of lower precision floating point operations. The algorithm is particularly useful when we have a good initial guess to Z, for example, from previous time steps in quantum-mechanical molecular dynamics simulations or from a previous iteration in a geometry optimization.

A computationally efficient [formula omitted]-spectral form for partial dispersion analyses within the wave finite element framework

This paper addresses the computation of frequency-dependent dispersion curves (i.e., k(ω)) and wave modes within the framework of the Wave Finite Element Method (WFEM) and in the context of high-dimensional periodic unit cell models. Numerous applications, ranging from phononics to vibroacoustics, now rely on dispersion analyses or wave expansion over a subset of eigensolutions – complex wavenumbers and Bloch waves – resulting from the resolution of an eigenvalue problem with a T-palindromic quadratic structure (T-PQEP). To exploit the structure of finite element models, various structure-preserving linearizations such as the Zhong-Williams and the (S+S−1)-transform have already been developed to achieve partial wave resolution of large T-PQEP, primarily targeting the dominating (least decaying) waves. In this paper we derive an alternative linearization of the T-PQEP for the k(ω) problem, which leads to enhanced targeting of the eigenvalues around the unit circle and reduces the inaccuracies induced by root multiplicity. A specific form of the problem is then proposed as an optimal compromise between ease of implementation, numerical stability, convergence and accuracy enhancement. The performance of our proposed linearization is compared against existing ones across various iterative eigensolvers, since the generalized eigenvalue problems involve complex non-hermitian matrices, which are not extensively included in eigensolvers. Results indicate that the proposed linearization should be favored for the WFEM, as it provides numerical enhancements in dispersion and wave vectors computation for large eigenvalue problems, as well as for further wave expansion applications.

Fluid-structure coupling analysis in liquid-filled containers using scaled boundary finite element method

In this study, a semi-analytical model is developed to investigate the fluid-structure coupling characteristics of liquid sloshing in an elastic rectangular container subjected to horizontal external excitation based on the scaled boundary finite element method (SBFEM) for the first time. The fluid inside the container is assumed to be incompressible, inviscid and irrotational, with the hydrodynamic pressure chosen as independent nodal variable in the governing equations. The container walls are considered as cantilever beams. The coupled fluid-structure system is initially divided into the structural domain and fluid domain, after which the SBFEM is employed to obtain the governing equations for each sub-domain. In the framework of the SBFEM, only the boundary of each sub-domain, rather than the entire computational domain, needs to be meshed and discretized. This method reduces the spatial dimension of the problem by one and offers an efficient approach to model the computational domain, while allowing for analytical formulations to be derived for the internal of the domain, resulting in an accurate description of the field variables. The fundamental equation of the entire coupled fluid-structure system is then assembled by performing the equilibrium condition and compatibility condition to ensure the balance of interaction forces at the interface between container walls and the liquid. The free vibrations analysis of the fluid-structure coupling system is solved by utilizing the generalized eigenvalue problem, and the transient dynamic response is determined using the synchronous solution algorithm in conjunction with the implicit-implicit scheme of the Newmark method. To validate the excellent accuracy and stability of the proposed formulation, several numerical examples are presented to investigate the free vibration and transient dynamic characteristics for the fluid-structure coupling problem. The obtained results show good agreement with reference solutions available in the literature. Additionally, the effects of geometrical and material parameters on the system responses are examined and discussed.

Rapid computation of Hopf bifurcation points of continuous and discrete systems through minimization

For an autonomous nonlinear system, the Hopf bifurcation point along the equilibrium path is a critical feature that indicates whether the values of the parameters change from exhibiting fixed-point behavior to having a periodic orbit. To solve these problems, we developed a method of transforming an eigenvalue problem based on the Jacobian matrix at equilibrium into a minimization problem, enabling the rapid identification of a solution. Specifically, this generalized eigenvalue problem is solved by identifying the vector variable after reducing the number of eigenequations by one in the nonhomogeneous linear system. This can be achieved by normalizing the value of a selected nonzero component of the eigenvector and then moving the column containing this component to the other side of the equation. An appropriate merit function was established in terms of the Euclidean norm of the eigenequation, and this merit function was minimized using the golden section search algorithm to determine the eigenparameters of the bifurcation point. The accuracy of the method for identifying the parameter values and the corresponding imaginary eigenvalues at the Hopf bifurcation points was evaluated for numerous examples for both the continuous and discrete systems. The method was both fast and accurate. Moreover, its stability in the presence of noise was investigated, and the method was robust.

Analysis of eigenvalue condition numbers for a class of randomized numerical methods for singular matrix pencils

The numerical solution of the generalized eigenvalue problem for a singular matrix pencil is challenging due to the discontinuity of its eigenvalues. Classically, such problems are addressed by first extracting the regular part through the staircase form and then applying a standard solver, such as the QZ algorithm, to that regular part. Recently, several novel approaches have been proposed to transform the singular pencil into a regular pencil by relatively simple randomized modifications. In this work, we analyze three such methods by Hochstenbach, Mehl, and Plestenjak that modify, project, or augment the pencil using random matrices. All three methods rely on the normal rank and do not alter the finite eigenvalues of the original pencil. We show that the eigenvalue condition numbers of the transformed pencils are unlikely to be much larger than the δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta $$\\end{document}-weak eigenvalue condition numbers, introduced by Lotz and Noferini, of the original pencil. This not only indicates favorable numerical stability but also reconfirms that these condition numbers are a reliable criterion for detecting simple finite eigenvalues. We also provide evidence that, from a numerical stability perspective, the use of complex instead of real random matrices is preferable even for real singular matrix pencils and real eigenvalues. As a side result, we provide sharp left tail bounds for a product of two independent random variables distributed with the generalized beta distribution of the first kind or Kumaraswamy distribution.

Intuitionistic fuzzy generalized eigenvalue proximal support vector machine

Generalized eigenvalue proximal support vector machine (GEPSVM) has attracted widespread attention due to its simple architecture, rapid execution, and commendable performance. GEPSVM gives equal significance to all samples, thereby diminishing its robustness and efficacy when confronted with real-world datasets containing noise and outliers. In order to reduce the impact of noises and outliers, we propose a novel intuitionistic fuzzy generalized eigenvalue proximal support vector machine (IF-GEPSVM). The proposed IF-GEPSVM assigns the intuitionistic fuzzy score to each training sample based on its location and surroundings in the high-dimensional feature space by using a kernel function. The solution of the IF-GEPSVM optimization problem is obtained by solving a generalized eigenvalue problem. Further, we propose an intuitionistic fuzzy improved generalized eigenvalue proximal support vector machine (IF-IGEPSVM) by solving standard eigenvalue decomposition resulting in simpler optimization problems with less computation cost which leads to an efficient intuitionistic fuzzy-based model. We conduct a comprehensive evaluation of the proposed IF-GEPSVM and IF-IGEPSVM models on UCI and KEEL benchmark datasets. Moreover, to evaluate the robustness of the proposed IF-GEPSVM and IF-IGEPSVM models, label noise is introduced into some UCI and KEEL datasets. The experimental findings showcase the superior generalization performance of the proposed IF-GEPSVM and IF-IGEPSVM models when compared to the existing baseline models, both with and without label noise. Our experimental results, supported by rigorous statistical analyses, confirm the superior generalization abilities of the proposed IF-GEPSVM and IF-IGEPSVM models over the baseline models. Furthermore, we implement the proposed IF-GEPSVM and IF-IGEPSVM models on the USPS recognition dataset, yielding promising results that underscore the models’ effectiveness in practical and real-world applications. The source code of the proposed IF-GEPSVM and IF-IGEPSVM models are available at https://github.com/mtanveer1/IF-GEPSVM.

Study of circulating liquid fuel in a 1D critical system with thermal feedback

This research focuses on the description and modeling of a one-dimensional molten salt reactor (MSR), in the presence of thermal feedback. Following the example of previous works, a simple one-dimensional system is proposed, describing a molten salt reactor with a main neutron-multiplying zone called core and a recirculation loop where the salt cools down. Specific attention is paid to the precursors’ drift by modifying the neutron balance equation. Liquid nuclear fuels are characterized by a high volumetric expansion coefficient in comparison to customary solid fuels. Therefore, a strong coupling between neutronics and thermal-hydraulics is expected. As a consequence, a highly negative density coefficient characterizes the thermal feedback on the neutron reactivity. The precursor equation is here inverted analytically and combined with the neutron balance equation to obtain a generalized eigenvalue problem with the neutron flux distribution as the unknown. The balance equations are derived by finite volume integration over a discretized mesh, and the coupling between the two physical models is treated by Picard iterations. The numerical solution is finally extended to time-dependent calculations and compared to an analytical work for a one-dimensional circulating fuel reactor already existing in the literature.

Finding eigenvectors with a quantum variational algorithm

This paper presents a hybrid variational quantum algorithm that finds a random eigenvector of a unitary matrix with a known quantum circuit. The algorithm is based on the SWAP test on trial states generated by a parametrized quantum circuit. The eigenvector is described by a compact set of classical parameters that can be used to reproduce the found approximation to the eigenstate on demand. This variational eigenvector finder can be adapted to solve the generalized eigenvalue problem, to find the eigenvectors of normal matrices and to perform quantum principal component analysis on unknown input mixed states. These algorithms can all be run with low-depth quantum circuits, suitable for an efficient implementation on noisy intermediate-scale quantum computers and, with some restrictions, on linear optical systems. In full-scale quantum computers, where there might be optimization problems due to barren plateaus in larger systems, the proposed algorithms can be used as a primitive to boost known quantum algorithms. Limitations and potential applications are discussed.

Vibration of embedded restrained composite tube shafts with nonlocal and strain gradient effects

Torsional vibration response of a circular nanoshaft, which is restrained by the means of elastic springs at both ends, is a matter of great concern in the field of nano-/micromechanics. Hence, the complexities arising from the deformable boundary conditions present a formidable obstacle to the attainment of closed-form solutions. In this study, a general method is presented to calculate the torsional vibration frequencies of functionally graded porous tube nanoshafts under both deformable and rigid boundary conditions. Classical continuum theory, upgraded with nonlocal strain gradient elasticity theory, is employed to reformulate the partial differential equation of the nanoshaft. First, torsional vibration equation based on the nonlocal strain gradient theory is derived for functionally graded porous nanoshaft embedded in an elastic media via Hamilton’s principle. The ordinary differential equation is found by discretizing the partial differential equation with the separation of variables method. Then, Fourier sine series is used as the rotation function. The necessary Stokes' transformation is applied to establish the general eigenvalue problem including the different parameters. For the first time in the literature, a solution that can analyze the torsional vibration frequencies of functionally graded porous tube shafts embedded in an elastic media under general (elastic and rigid) boundary conditions on the basis of nonlocal strain gradient theory is presented in this study. The results obtained show that while the increase in the material length scale parameter, elastic media and spring stiffnesses increase the frequencies of nanoshafts, the increase in the nonlocal parameter and functionally grading index values decreases the frequencies of nanoshafts. The detailed effects of these parameters are discussed in the article.

R I biorthogonal polynomials of Hahn type

ABSTRACT A finite family of R I polynomials is introduced and studied. It consists in a set of polynomials of 3 F 2 form whose biorthogonality to an ensemble of rational functions is spelled out. These polynomials are shown to satisfy two generalized eigenvalue problems: in addition to their recurrence relation of R I type, they are also found to obey a difference equation. Underscoring this bispectrality is a triplet of operators with tridiagonal actions. The algebra associated to these operators is provided.