Generalized Eigenvalue Problem Research Articles

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.

New vector transport operators extending a Riemannian CG algorithm to generalized Stiefel manifold with low-rank applications

This paper proposes two innovative vector transport operators, leveraging the Cayley transform, for the generalized Stiefel manifold embedded with a non-standard metric. Specifically, it introduces the differentiated retraction and an approximation of the Cayley transform to the differentiated matrix exponential. These vector transports are demonstrated to satisfy the Ring–Wirth non-expansive condition under non-standard metrics, and one of them is also isometric. Building upon the novel vector transport operators, we extend the modified Polak–Ribière–Polyak (PRP) conjugate gradient method to the generalized Stiefel manifold. Under a non-monotone line search condition, we prove our algorithm globally converges to a stationary point. The efficiency of the proposed vector transport operators is empirically validated through numerical experiments involving generalized eigenvalue problems and canonical correlation analysis.

Conical-Shaped Shells of Non-Uniform Thickness Vibration Analysis Using Higher-Order Shear Deformation Theory

Conical-Shaped Shells of Non-Uniform Thickness Vibration Analysis Using Higher-Order Shear Deformation Theory

The pseudospectra of black holes in AdS

We study the stability of quasinormal modes (QNMs) in electrically charged black brane spacetimes that asymptote to AdS by means of the pseudospectrum. Methodologically, we adopt ingoing Eddington-Finkelstein coordinates to cast QNMs in terms of a generalised eigenvalue problem involving a non-selfadjoint operator; this simplifies the computation significantly in comparison with previous results in the literature. Our analysis reveals spectral instability for (neutral) scalar as well as gravitoelectric perturbations. This indicates that the equilibration process of perturbed black branes is sensitive to external perturbations. Particular attention is given on the hydrodynamic modes, which are found to be the least unstable. In contrast with computations in hyperboloidal coordinates, we find that the pseudospectral contour lines cross to the upper half plane. This indicates the existence of pseudo-resonances as well as the possibility of transient instabilities. We also investigate the asymptotic structure of pseudospectral contour levels and we find remarkable universality across all sectors, persistent in the extremal limit.

On the magnetohydrodynamic stability of channel flow of Navier–Stokes–Voigt fluid

The linear stability of a pressure-driven channel flow of an electrically conducting Navier–Stokes–Voigt type of the viscoelastic fluid subject to a transverse magnetic field is investigated. The validity of Squire's theorem is proved, and the generalized eigenvalue problem for two-dimensional modes is obtained by adopting the Galerkin method, which is subsequently solved using the QZ-algorithm. Although the base flow retains its Newtonian fluid characteristics, the noticeable influence of the Kelvin–Voigt parameter in conjunction with the Hartmann number on the stability of fluid flow is perceived. Instability is exclusively identified within a specific range of the Kelvin–Voigt parameter, markedly affected by the Hartmann number. Furthermore, closed neutral stability curves arise, indicating the requirement of two values of the Reynolds number to completely assess the linear stability criteria, in contrast to the typical single value observed in the case of Newtonian fluids.

Tensor networks for solving the time-independent Boltzmann neutron transport equation

Tensor network techniques, known for their low-rank approximation ability that breaks the curse of dimensionality, are emerging as a foundation of new mathematical methods for ultra-fast numerical solutions of high-dimensional Partial Differential Equations (PDEs). Here, we present a mixed Tensor Train (TT)/Quantized Tensor Train (QTT) approach for the numerical solution of time-independent Boltzmann Neutron Transport equations (BNTEs) in Cartesian geometry. Discretizing a realistic three-dimensional (3D) BNTE by (i) diamond differencing in space, (ii) multigroup-in-energy, and (iii) discrete ordinates collocation in angle leads to large generalized eigenvalue problems that generally require a matrix-free approach and large computer clusters. Starting from this discretization, we construct a TT representation of the PDE fields and discrete operators, followed by a QTT representation of the TT cores. We then solve the tensorized generalized eigenvalue problem using a fixed-point scheme with tensor network optimization techniques. We validate our approach by applying the method to two examples of 3D neutron transport problems, currently solved by the Los Alamos National Laboratory PARallel TIme-dependent SN (PARTISN) solver.1 We demonstrate that our TT/QTT method, executed on a standard desktop computer, leads to large compression. This allows for the storage of terrabyte-sized neutron angular flux eigenvectors in megabytes. Additionally, we create megabyte-sized full access TT representations of yottabyte-sized transport matrix operators. By leveraging the TT operators and solution methods, we obtain a 7500 times speedup when compared to the PARTISN solution time with an error of less than 10−5.

Stability of a photosurfactant-laden viscous liquid thread under illumination

This paper investigates the effects of a light-actuated photosurfactant on the canonical problem of the linear stability of a viscous thread surrounded by a dynamically passive fluid. A model consisting of the Navier–Stokes equations and a set of molar concentration equations is presented that capture light-induced switching between two stable surfactant isomer states, trans and cis. These two states display significantly different interfacial properties, allowing for some external control of the stability behaviour of the thread via incident light. Normal modes are used to generate a generalized eigenvalue problem for the growth rate which is solved with a hybrid analytical and numerical method. The results are validated with appropriate analytical solutions of increasing complexity, beginning with a solution to a clean interface, then analytical solutions for one insoluble surfactant, one soluble surfactant and a special case of two photosurfactants with a spatially uniform undisturbed state. Presenting each of these cases allows for a holistic discussion of the effect of surfactants in general on the stability of a liquid thread. Finally, the numerical solutions in the presence of two photosurfactants that display radially non-uniform undisturbed states are presented, and details of the impact of the illumination on the linear stability of the thread are discussed.

Non-Bloch band theory of generalized eigenvalue problems

Non-Bloch band theory of generalized eigenvalue problems

An application of stability charts to prediction of buckling instability in tapered columns via Galerkin’s method

This work tackles the mathematical modeling of buckling problem to obtain their critical loads in tapered columns subjected to concentrated and axial distributed loads. The governing model is a general eigenvalue problem that has no exact solution due to some new terms included. A semi-analytical technique satisfying the boundary conditions is proposed for the solution procedure. The minimum residual Galerkin’s method is suggested due to its effectiveness as a semi-analytical tool for the buckling problem to obtain the buckling shape modes by using admissible periodic functions. The study investigates the buckling instability and the responses of tapered columns with different periodic trial shape functions as approximations to the exact solutions. Based on the eigenvalue problem, Galerkin’s method is employed to obtain the transition curves to represent the critical loads. The stability charts (Ince–Strutt diagrams) among the parameters of the problem are proposed to explain the elastic stability of different tapered columns subjected to different shapes of cross sections and distributed weights. Consequently, the influences of the included parameters on the critical buckling loads are discussed. Among the different tapered columns presented, some parameters in the proposed distributions have a big influence on the critical buckling load and the creation of the instability regions in the chart for the clamped-clamped boundary conditions. The results are verified using the analytical solutions for some specific known problems.

Design sensitivity analysis of modal frequencies of elastic structures submerged in an infinite fluid domain

SummaryThis paper presents a numerical approach for the design sensitivity analysis of modal frequencies of elastic structures submerged in an unbounded heavy fluid domain. Because the feedback of sound pressure onto the fluid‐loaded structures has to be taken into account in this case, a fully coupled scheme which combines the structural finite element method (FEM) and the acoustic boundary element method (BEM) is employed in the numerical simulation. The resulting nonlinear eigenvalue problem created by the coupled finite element and boundary element (FE‐BE) method is converted into a generalized eigenvalue problem (GEP) of reduced dimension by using a contour integral method. The design sensitivity formulation for non‐repeated eigenvalues is derived based on an adjoint method which uses both the left and right eigenvectors, and a small GEP is formed to calculate the gradients of repeated eigenvalues. The Burton‐Miller formulation is employed to shift the fictitious eigenfrequencies and their derivatives, and the parameter setting which is beneficial to the filtering of fictitious eigenfrequencies is investigated. Numerical examples are given to verify the accuracy and effectiveness of the developed method.

Effects of Rotation and Magnetic Field on Rayliegh Benard Convection

In this paper, a numerical method based on the Chebyshev tau method is applied to analyze the effects of rotation and magnetic fields on Rayleigh-Bénard convection. The rotation and magnetic fields are assumed to be parallel to the vertical direction. The perturbation equations and boundary conditions are analyzed using normal mode analysis. The equations are then converted into a non-dimensional form and transformed into a generalized eigenvalue problem of the form AX=RBX, where R represents the eigenvalue corresponding to the Rayleigh number. The MATLAB software package is utilized to determine the relationship between the Rayleigh number and the Taylor number (rate of rotation), as well as the relationship between the Rayleigh number and the magnetic parameter (strength of the magnetic field) for different boundary conditions (free-free, rigid-rigid, or one free and the other rigid). The numerical and graphical results are presented and found to be in full agreement with the results obtained from previous analytical and numerical studies of the problem.

Effect of Oldroyd-B fluid on thermal convection stability in a bidispersive porous media with relatively large macropores

This study investigates the effect of Oldroyd-B fluids on the thermal convection instability in a bi-disperse porous medium with relatively large macropores. The modified momentum equation of Oldroyd-B fluid in the macro- and micropores is first derived, assuming local thermal equilibrium. The Orr–Sommerfeld (O–S) equation is derived using the new modified incompressible Navier–Stokes equations of Oldroyd-B fluid. The Chebyshev collocation method is employed to solve the generalized eigenvalue problem, and the neutral stability curves and temporal growth rates are obtained. The results demonstrate that the relaxation time enhances the convective instability, while the retardation time suppresses it. Furthermore, the effect of relaxation time is generally small for smaller wave number, but the effect of relaxation time is significant for large wave number, and this effect becomes accentuated as the wave number increases. The retardation time does not have any influence when the wavenumber is small, and the influence of retardation time becomes small with the large relaxation time. Moreover, the convection instability is enhanced with the higher micropermeability and a higher ratio of momentum transfer coefficient between the macro- and microphases will suppress the instability.

A family of Nonoverlapping Spectral Additive Schwarz methods (NOSAS) and their economic versions

NOSAS (Nonoverlapping Spectral Additive Schwarz) methods are nonoverlapping iterative domain decomposition methods for efficiently solving large sparse linear systems arising from elliptic problems with heterogeneous coefficients. NOSAS methods are of additive Schwarz types with local Dirichlet solver on each nonoverlapping subdomain Ωi, plus a coarse problem on the interface of the subdomains. The coarse problem has global and local interaction components which are associated respectively to low-frequency and high-frequency modes obtained locally from a generalized eigenvalue problem defined on each subdomain’s interface. In this paper, we propose several variants of the NOSAS methods by selecting different generalized eigenvalue problems and different coarse space extensions. Additionally, we propose an economic variant to reduce the complexity of these generalized eigenvalue problems. The analysis of all the variants is established, and numerical tests are provided.

Extraction of deflating subspaces using disk function of a matrix pencil via matrix sign function with application in generalized eigenvalue problem

The objective of this study is to extract two pairs of left and right deflating subspaces of a regular matrix pencil A−λB corresponding to the eigenvalues lying inside and outside the unit disk via computation of left and right disk functions diskL(A−λB) and diskR(A−λB) which are obtained from matrix sign evaluations of two suitable matrices. We propose a novel fifth order iterative method for evaluating matrix sign function which can be implemented to compute disk functions. Convergence analysis of the proposed method has been investigated globally along with the illustration of attraction basins. Asymptotic stability of the method is also presented. Then an application of deflating subspaces in solving generalized eigenvalue problem has been explored. Numerical test problems have been performed by considering matrices of various sizes to justify the efficiency and superiority of the proposed method.

A universal variational quantum eigensolver for non-Hermitian systems

Many quantum algorithms are developed to evaluate eigenvalues for Hermitian matrices. However, few practical approach exists for the eigenanalysis of non-Hermintian ones, such as arising from modern power systems. The main difficulty lies in the fact that, as the eigenvector matrix of a general matrix can be non-unitary, solving a general eigenvalue problem is inherently incompatible with existing unitary-gate-based quantum methods. To fill this gap, this paper introduces a Variational Quantum Universal Eigensolver (VQUE), which is deployable on noisy intermediate scale quantum computers. Our new contributions include: (1) The first universal variational quantum algorithm capable of evaluating the eigenvalues of non-Hermitian matrices—Inspired by Schur’s triangularization theory, VQUE unitarizes the eigenvalue problem to a procedure of searching unitary transformation matrices via quantum devices; (2) A Quantum Process Snapshot technique is devised to make VQUE maintain the potential quantum advantage inherited from the original variational quantum eigensolver—With additional O(log2N)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(log_{2}{N})$$\\end{document} quantum gates, this method efficiently identifies whether a unitary operator is triangular with respect to a given basis; (3) Successful deployment and validation of VQUE on a real noisy quantum computer, which demonstrates the algorithm’s feasibility. We also undertake a comprehensive parametric study to validate VQUE’s scalability, generality, and performance in realistic applications.

Highly accurate calculation of higher energy eigenvalues for the radial Schrödinger eigenproblems

In this paper, we derive an efficient numerical scheme for approximating the energy eigenvalues and corresponding wavefunctions of the radial Schrödinger eigenproblems defined on a semi-infinite domain for arbitrary values of the angular momentum number. The numerical scheme is based on the Chebyshev spectral collocation differentiation matrix method. In this scheme, at first, we redefine the radial Schrödinger eigenproblem on a finite interval by adopting an appropriate change of the independent variable. Then, by expanding the wavefunctions of the problem as a series of the Chebyshev polynomials of the first kind, as well as employing the differentiation matrices in order to determine the derivatives of Chebyshev polynomials at the collocation points, considered problem becomes a generalized eigenvalue problem. The convergence behavior and excellent performance of the proposed technique are illustrated through some numerical experiments of the most significant potentials in quantum mechanics. Compared with exact solutions (when available), and those reported previously in the literature, the current scheme achieves higher accuracy and efficiency even when high-index energy eigenvalues are computed.

Joint Optimization for Secure and Reliable Communications in Finite Blocklength Regime

To realize ultra-reliable low latency communications with high spectral efficiency and security, we investigate a joint optimization problem for downlink communications with multiple users and eavesdroppers in the finite blocklength (FBL) regime. We formulate a multi-objective optimization problem to maximize a sum secrecy rate by developing a secure precoder and to minimize a maximum error probability and information leakage rate. The main challenges arise from the complicated multi-objective problem, non-tractable back-off factors from the FBL assumption, non-convexity and non-smoothness of the secrecy rate, and the intertwined optimization variables. To address these challenges, we adopt an alternating optimization approach by decomposing the problem into two phases: secure precoding design, and maximum error probability and information leakage rate minimization. In the first phase, we obtain a lower bound of the secrecy rate and derive a first-order Karush–Kuhn–Tucker (KKT) condition to identify local optimal solutions with respect to the precoders. Interpreting the condition as a generalized eigenvalue problem, we solve the problem by using a power iteration-based method. In the second phase, we adopt a weighted-sum approach and derive KKT conditions in terms of the error probabilities and leakage rates for given precoders. Simulations validate the proposed algorithm.

Porosity effects on the dynamic response of arbitrary restrained FG nanobeam based on the MCST

Abstract In this study, two different general eigenvalue problems for nanobeams made of functionally graded material with pores in their sections according to Rayleigh beam theory using modified couple stress theory are established. Fourier sine series and Stokes transformation are used for the solution. First, the partial differential equation of motion of the problem is discretized into an ordinary differential equation. Then, the Fourier sine series of infinite series is substituted into this ordinary differential equation to determine the Fourier coefficient. Using the force boundary conditions of the system, Stokes’ transformation is performed at both ends to include elastic spring parameters. The unknown displacement terms are discretized to form two eigenvalue problems. By solving these eigenvalue problems, vibration frequencies for different boundary conditions can be found analytically. The variations of some parameters are discussed in a series of graphs.

Stochastic free vibration analysis of FG-CNTRC plates based on a new stochastic computational scheme

Stochastic analysis can provide more accurate results compared to deterministic analysis. In this study, the spatial variability of material parameters is introduced into the free vibration analysis of functionally graded carbon nanotube reinforced composite (FG-CNTRC) plates to achieve a higher degree of accuracy, and a new stochastic computational scheme is proposed to handle the low-level uncertainties. First-order shear deformation theory (FSDT) is employed to establish the displacement field of the plates. The Young's moduli of carbon nanotube (CNT) and matrix are regarded as one-dimensional (1D) and two-dimensional (2D) random fields, respectively, which is discretized by Karhunen-Loève (K-L) expansion. The obtained random variables are substituted into modified perturbation stochastic method (MPSM) and radial point interpolation method (RPIM) to calculate the first two order estimates of the stochastic non-dimensional natural frequencies ω¯. The sensitivity of the first to fourth ω¯ to the random fields is analyzed, and the corresponding stochastic bands are plotted. The results indicate that the presented approach can accurately solve the generalized eigenvalue problem in stochastic free vibration; ω¯ is sensitive to random fields E11CNT and Em, and insensitive to random field E22CNT; the CNT distribution mode also affects the sensitivity.

A robust numerical strategy for finding surface waves in flows of non-Newtonian liquids

Gravity-driven flows of liquid films are frequent in nature and industry, such as in landslides, lava flow, cooling of nuclear reactors, and coating processes. In many of these cases, the liquid is non-Newtonian and has particular characteristics. In this paper, we analyze numerically the temporal stability of films of non-Newtonian liquids falling by gravity, on the onset of instability. The liquid flows over an incline, where surface waves appear under certain conditions, and we do not fix a priori its rheological behavior. For that, we made used of the Carreau–Yasuda model without assigning specific values to its constants, and we compute general stability solutions. The numerical strategy is based on expansions of Chebyshev polynomials for discretizing the Orr–Sommerfeld equation and boundary conditions, and a Galerkin method for solving the generalized eigenvalue problem. In addition, an Inverse Iteration method was implemented to increase accuracy and improve computational time. The result is a robust and light numerical tool capable of finding the critical conditions for different types of fluids, which we use to analyze some key fluids. We show that the outputs of the general code match previous solutions obtained for specific computations. Besides increasing our knowledge on surface-wave instabilities in non-Newtonian liquids, our findings provide a new tool for obtaining comprehensive solutions on the onset of instability.