Matrix Eigenvalue Problem Research Articles

Simplification of the Gram Matrix Eigenvalue Problem for Quadrature Amplitude Modulation Signals.

In quantum information science, it is very important to solve the eigenvalue problem of the Gram matrix for quantum signals. This allows various quantities to be calculated, such as the error probability, mutual information, channel capacity, and the upper and lower bounds of the reliability function. Solving the eigenvalue problem also provides a matrix representation of quantum signals, which is useful for simulating quantum systems. In the case of symmetric signals, analytic solutions to the eigenvalue problem of the Gram matrix have been obtained, and efficient computations are possible. However, for asymmetric signals, there is no analytic solution and universal numerical algorithms that must be used, rendering the computations inefficient. Recently, we have shown that, for asymmetric signals such as amplitude-shift keying coherent-state signals, the Gram matrix eigenvalue problem can be simplified by exploiting its partial symmetry. In this paper, we clarify a method for simplifying the eigenvalue problem of the Gram matrix for quadrature amplitude modulation (QAM) signals, which are extremely important for applications in quantum communication and quantum ciphers. The results presented in this paper are applicable to ordinary QAM signals as well as modified QAM signals, which enhance the security of quantum cryptography.

Partial-wave approach to the Stark resonance problem of the water molecule

A partial-wave method is developed to deal with small molecules dominated by a central atom as an extension of earlier single-center methods. In particular, a model potential for the water molecule is expanded over a basis of spherical harmonics. A finite-element method is employed to generate local polynomial functions in subintervals to represent the radial part of the wave function. The angular parts of the wave function are represented by spherical harmonics. The problem of Stark resonances is treated with the exterior complex scaling method which incorporates a wave-function discontinuity at the scaling radius. The resultant non-Hermitian matrix eigenvalue problem yields resonance positions and widths (decay rates). We present these dc Stark shifts and exponential decay rates for the valence orbitals $1{b}_{1}, 3{a}_{1}$, and the bonding orbital $1{b}_{2}$. Furthermore, comparison is made with total molecular decay rates and dc shifts obtained recently within the Hartree-Fock and coupled-cluster approaches.

Dark-dark solitons, soliton molecules and elastic collisions in the mixed three-level coupled Maxwell–Bloch equations

We study the integrable three-level coupled Maxwell–Bloch equations with the mixed focusing-defocusing case. The n-fold Darboux transformation is constructed on basis of a linear 3×3 matrix eigenvalue problem. As an application, the n-dark-dark soliton solution in a simple determinant form is presented and the asymptotic behavior of the n-dark-dark soliton solution is rigorously given. In particular, under the resonant mechanism, the unusual dark-dark soliton molecules which represent the soliton bound states are analytically demonstrated. The double- and triple-dark-dark soliton molecules that are composed of two and three solitons propagating with identical velocities are graphically shown, respectively. The elastic collisions between a double- or triple-dark-dark soliton molecule and a usual dark-dark soliton, and the elastic collision of two double-dark-dark soliton molecules are discussed via the standard asymptotic analysis method.

The Inverse Eigenvalue Problem for a Class of Sub-periodic Generalized Jacobi Matrices

The inverse eigenvalue problem of matrix has practical significance in engineering and scientific calculation. In this paper, the inverse eigenvalue problem of sub-periodic generalized Jacobi matrices is studied. Two eigenvalues and corresponding eigenvectors are used to construct generalized Jacobi matrices. The existence and uniqueness conditions of the solution are discussed, the solution algorithm for the inverse problem is given, and the effectiveness of the algorithm is verified by a numerical example.

An efficient interpolating wavelet collocation scheme for quasi‐exactly solvable Sturm–Liouville problems in ℝ+

This investigation is an attempt to obtain a highly accurate approximation of the spectrum of Sturm–Liouville problems in by representing the unknown solution of the model in the interpolating wavelet basis of . To accomplish the goal, the domain has been stretched to to avoid the additional care of the elements in the basis containing boundary point 0. In addition, such transformation may judiciously be utilized to eliminate (up to quadratic) the singularity of the equation. The equation in the new variable has been subsequently transformed into a generalized matrix eigenvalue problem by approximating the new (unknown) function in an appropriate (truncated) basis comprising interpolating scale functions generated by scale functions in Daubechies family. The (interpolating wavelet‐collocation) scheme developed here has been applied to some solvable and quasi‐exactly solvable Sturm–Liouville problems in appearing in quantum mechanical modeling in flat and curved spaces. It is observed that the approximation of eigenfunctions in the (compact support) interpolating wavelet basis obtained by using the collocation method can be reliably used to reveal a hidden spectrum of quasi‐exactly solvable Sturm–Liouville problems in with high accuracy.

Solving Block Low-Rank Matrix Eigenvalue Problems

To solve large-scale matrix eigenvalue problems (EVPs), a two-step tridiagonalization method using the block Householder transformation (HT) is often employed. Although the method based on dense matrix arithmetic requires a memory storage of O(N2) and an arithmetic operations of O(N3), in this study, these were reduced by approximating the method using block low-rank (BLR-) matrices. A special block HT for BLR-matrices and a two-step tridiagonalization method using it are proposed to solve an EVP with a real symmetric BLR-matrix. In the proposed block HT, block Householder vectors are also formed using BLR-matrices. It is demonstrated how the block size m in the BLR-matrix should be determined and confirmed that the memory and arithmetic complexities of the proposed method were O(N5/3) and O(N7/3), respectively, for typical cases when using an appropriate block size m ∝ N1/3. In numerical experiments of a string free vibration problem with known analytical solutions, for large eigenvalues, the calculated eigenvalues using the proposed method converge toward the analytical ones in accordance with the theoretical convergence curves. Owing to the reduced complexity, an EVP of a matrix was solved with about N =300,000, which is significantly larger than the limit of the conventional method with dense matrices, within a reasonable amount of time on a CPU core. For the calculation time, the proposed method was faster than the conventional method when the matrix size N was larger than a few tens of thousands.

Quorum-sensing induced transitions between bistable steady-states for a cell-bulk ODE-PDE model with lux intracellular kinetics.

Intercellular signaling and communication are used by bacteria to regulate a variety of behaviors. In a type of cell-cell communication known as quorum sensing (QS), which is mediated by a diffusible signaling molecule called an autoinducer, bacteria can undergo sudden changes in their behavior at a colony-wide level when the density of cells exceeds a critical threshold. In mathematical models of QS behavior, these changes can include the switch-like emergence of intracellular oscillations through a Hopf bifurcation, or sudden transitions between bistable steady-states as a result of a saddle-node bifurcation of equilibria. As an example of this latter type of QS transition, we formulate and analyze a cell-bulk ODE-PDE model in a 2-D spatial domain that incorporates the prototypical LuxI/LuxR QS system for a collection of stationary bacterial cells, as modeled by small circular disks of a common radius with a cell membrane that is permeable only to the autoinducer. By using the method of matched asymptotic expansions, it is shown that the steady-state solutions for the cell-bulk model exhibit a saddle-node bifurcation structure. The linear stability of these branches of equilibria are determined from the analysis of a nonlinear matrix eigenvalue problem, called the globally coupled eigenvalue problem. The key role on QS behavior of a bulk degradation of the autoinducer field, which arises from either a Robin boundary condition on the domain boundary or from a constant bulk decay, is highlighted. With bulk degradation, it is shown analytically that the effect of coupling identical bacterial cells to the bulk autoinducer diffusion field is to create an effective bifurcation parameter that depends on the population of the colony, the bulk diffusivity, the membrane permeabilities, and the cell radius. QS transitions occur when this effective parameter passes through a saddle-node bifurcation point of the Lux ODE kinetics for an isolated cell. In the limit of a large but finite bulk diffusivity, it is shown that the cell-bulk system is well-approximated by a simpler ODE-DAE system. This reduced system, which is used to study the effect of cell location on QS behavior, is easily implemented for a large number of cells. Predictions from the asymptotic theory for QS transitions between bistable states are favorably compared with full numerical solutions of the cell-bulk ODE-PDE system.

The Weierstrass iterative method as a Petrov–Galerkin method for solving eigenvalue problem

In this study, we obtain another representation of a relatively new method for solving the eigenvalue problem of the Frobenius companion matrix. We show that the iterative scheme under consideration is equivalent to the Weierstrass method. Based on this dependence, we derive new proofs of some known results and get new theoretical properties for the Weierstrass method.

A method for long-time integration of Lyapunov exponent and vectors along fluid particle trajectories

Finite-time Lyapunov exponents (FTLEs) and Lyapunov vectors (LVs) are powerful tools to illustrate Lagrangian coherent structures (LCSs) in experiments and numerical simulations of fluid flows. To obtain the FTLEs and LVs with the flow simulation simultaneously, we computed the eigenvectors and eigenvalues of the left Cauchy–Green tensor along the trajectories of fluid particles separately instead of computing deformation gradient tensor directly. The method proposed in the present study not only avoids solving the eigenvalue problem of the singular matrix at each time step but also guarantees a stable simulation for a long time. The method is applied in the computation of FTLEs and LVs in two-/three-dimensional (2D/3D) compressible/incompressible cases. In 2D cases, we found that LCSs are folded as fine filaments induced by vortices, while LCSs are sheet-like structures among the vortices for 3D cases. Meanwhile, the directions of stretching and compression of LVs are tangent and normal to the FTLE ridges (2D)/iso-surfaces (3D), respectively.

Effect of surfactants on the long-wave stability of two-layer oscillatory film flow

The stability of the two-layer film flow driven by an oscillatory plate under long-wave disturbances is studied. The influence of key factors, such as thickness ratio ( $n$ ), viscosity ratio ( $m$ ), density ratio ( $r$ ), oscillatory frequency ( $\beta$ ) and insoluble surfactants on the stability behaviours is studied systematically. Four special Floquet patterns are identified, and the corresponding growth rates are obtained by solving the eigenvalue problem of the fourth-order matrix. A small viscosity ratio ( $m\le 1$ ) may stabilize the flow but it depends on the thickness ratio. If the viscosity ratio is not very small ( $m>0.1$ ), in the $(\beta ,n)$ -plane, stable and unstable curved stripes appear alternately. In other words, under the circumstances, if the two-layer film flow is unstable, slightly adjusting the thickness of the upper film may make it stable. In particular, if the upper film is thin enough, even under high-frequency oscillation, the flow is always stable. The influence of density ratio is similar, i.e. there are curved stable and unstable stripes in the $(\beta ,r)$ -planes. Surface surfactants generally stabilize the flow of the two-layer oscillatory membrane, while interfacial surfactants may stabilize or destabilize the flow but the effect is mild. It is also found that gravity can generally stabilize the flow because it narrows the bandwidth of unstable frequencies.

Data-driven sensitivity analysis in surface structure determination using total-reflection high-energy positron diffraction (TRHEPD)

The present article proposes a data analysis method for experimentally-derived measurements, which consists of an auto-optimization procedure and a sensitivity analysis. The method was applied to the results of a total-reflection high-energy positron diffraction (TRHEPD) experiment, a novel technique of determining surface structures or the position of the atoms near the material surface. This method solves numerically the partial differential equation in the fully-dynamical quantum diffraction theory with many trial surface structures. In the sensitivity analysis, we focused on the experimental uncertainties and the variation over individual fitting parameters, which was analyzed by solving the eigenvalue problem of the variance-covariance matrix. A modern massively parallel supercomputer was used to complete the analysis within a moderate computational time. The sensitivity analysis provides a basis for the choice of variables in the data analysis for practical reliability. The effectiveness of the present analysis method was demonstrated in the structure determination of a Si4O5N3 / 6H-SiC(0001)-(3×3) R30∘ surface. Furthermore, this analysis method is applicable to many experiments other than TRHEPD.

Dressed energy of the XXZ chain in the complex plane

We consider the dressed energy varepsilon of the XXZ chain in the massless antiferromagnetic parameter regime at 0< Delta < 1 and at finite magnetic field. This function is defined as a solution of a Fredholm integral equation of the second kind. Conceived as a real function over the real numbers, it describes the energy of particle–hole excitations over the ground state at fixed magnetic field. The extension of the dressed energy to the complex plane determines the solutions to the Bethe Ansatz equations for the eigenvalue problem of the quantum transfer matrix of the model in the low-temperature limit. At low temperatures, the Bethe roots that parametrize the dominant eigenvalue of the quantum transfer matrix come close to the curve mathrm{Re}, varepsilon (lambda ) = 0. We describe this curve and give lower bounds to the function mathrm{Re}, varepsilon in regions of the complex plane, where it is positive.

Random matrix eigenvalue problems in structural dynamics: An iterative approach

Uncertainties need to be taken into account in the dynamic analysis of complex structures. This is because in some cases uncertainties can have a significant impact on the dynamic response and ignoring it can lead to unsafe design. For complex systems with uncertainties, the dynamic response is characterised by the eigenvalues and eigenvectors of the underlying generalised matrix eigenvalue problem. This paper aims at developing computationally efficient methods for random eigenvalue problems arising in the dynamics of multi-degree-of-freedom systems. There are efficient methods available in the literature for obtaining eigenvalues of random dynamical systems. However, the computation of eigenvectors remains challenging due to the presence of a large number of random variables within a single eigenvector. To address this problem, we project the random eigenvectors on the basis spanned by the underlying deterministic eigenvectors and apply a Galerkin formulation to obtain the unknown coefficients. The overall approach is simplified using an iterative technique. Two numerical examples are provided to illustrate the proposed method. Full-scale Monte Carlo simulations are used to validate the new results.

Newton's method for the eigenvalue problem of a symmetric matrix

Newton's method for calculating the eigenvalue and the corresponding eigenvector of a symmetric real matrix is considered. The nonlinear system of equations solved by Newton's method consists of an equation that determines the eigenvalue and eigenvector of the matrix and the normalization condition for the eigenvector. The method allows someone to simultaneously calculate the eigenvalue and the corresponding eigenvector. Initial approximations for the eigenvalue and the corresponding eigenvector can be found by the power method or by the reverse iteration with shift. A simple proof of the convergence of Newton's method in a neighborhood of a simple eigenvalue is proposed. It is shown that the method has a quadratic convergence rate. In terms of computational costs per iteration, Newton's method is comparable to the reverse iteration method with the Rayleigh ratio. Unlike reverse iteration, Newton's method allows to compute the eigenpair with better accuracy.

Two Chebyshev Spectral Methods for Solving Normal Modes in Atmospheric Acoustics.

The normal mode model is important in computational atmospheric acoustics. It is often used to compute the atmospheric acoustic field under a time-independent single-frequency sound source. Its solution consists of a set of discrete modes radiating into the upper atmosphere, usually related to the continuous spectrum. In this article, we present two spectral methods, the Chebyshev-Tau and Chebyshev-Collocation methods, to solve for the atmospheric acoustic normal modes, and corresponding programs are developed. The two spectral methods successfully transform the problem of searching for the modal wavenumbers in the complex plane into a simple dense matrix eigenvalue problem by projecting the governing equation onto a set of orthogonal bases, which can be easily solved through linear algebra methods. After the eigenvalues and eigenvectors are obtained, the horizontal wavenumbers and their corresponding modes can be obtained with simple processing. Numerical experiments were examined for both downwind and upwind conditions to verify the effectiveness of the methods. The running time data indicated that both spectral methods proposed in this article are faster than the Legendre-Galerkin spectral method proposed previously.

Adaptive threshold split Bregman algorithm based on magnetic induction tomography for brain injury monitoring imaging

Objective. Traditional magnetic induction tomography (MIT) algorithms have problems in reconstruction, such as large area error (AE), blurred boundaries of reconstructed targets, and considerable image noise (IN). As the size and boundary of a lesion greatly affect the treatment plan, more accurate algorithms are necessary to meet clinical needs. Approach. In this study, adaptive threshold split Bregman (ATSB) is proposed for brain injury monitoring imaging in MIT. We established a 3D brain MIT simulation model with the actual anatomical structure and a phantom model and obtained the reconstructed images of single targets in different positions and multiple targets, using the Tikhonov, eigenvalue threshold regularisation (ETR), split Bregman (SB), and ATSB algorithms. Main results. Compared with the Tikhonov and ETR algorithms, the ATSB algorithm reduced the AE by 95% and the IN by 17% in a simulation and reduced the AE by 87% and IN by 6% in phantom experiments. Compared with the SB algorithm, the ATSB algorithm can reduce the difficulty of adjusting parameters and is easier to use in clinical practice. The simulation and phantom experiments results showed that the ATSB algorithm could reconstruct the target size more accurately and could distinguish multiple targets more effectively than the other three algorithms. Significance. The ATSB algorithm could improve the image quality of MIT and better meet the needs of clinical applications and is expected to promote brain injury monitoring imaging via MIT.

A boundary penalization technique to remove outliers from isogeometric analysis on tensor-product meshes

We introduce a boundary penalization technique to improve the spectral approximation of isogeometric analysis (IGA). The method removes the outliers appearing in the high-frequency region of the approximate spectrum when using the Cp−1,p-th (p≥3) order isogeometric elements. We focus on the classical Laplacian (Dirichlet) eigenvalue problem in 1D to illustrate the idea and then use the tensor-product structure to generate the stiffness and mass matrices for multidimensional problems. To remove the outliers, we penalize the higher-order derivatives from both the solution and test spaces at the domain boundary. Intuitively, we construct a better approximation by weakly imposing features of the exact solution. Effectively, we add terms to the variational formulation at the boundaries with minimal extra-computational cost. We then generalize the idea to remove the outliers for the isogeometric analysis of the Neumann eigenvalue problem (for p≥2). The boundary penalization does not change the test and solution spaces. In the limiting case, when the penalty goes to infinity, we perform the dispersion analysis of C2 cubic elements for the Dirichlet eigenvalue problem and C1 quadratic elements for the Neumann eigenvalue problem. We obtain the analytical eigenpairs for the resulting matrix eigenvalue problems. Numerical experiments show optimal convergence rates for the eigenvalues and eigenfunctions of the discrete operator.

Shooting method for linear inviscid bi-global stability analysis of non-axisymmetric jets

The shooting method is commonly used to solve the linear parallel-flow stability problem for axisymmetric jets, i.e., a flow having one inhomogeneous direction. The present extension to two inhomogeneous directions – i.e., a bi-global stability problem – is motivated by inviscid non-axisymmetric jets. The azimuthal direction is Fourier transformed to obtain a set of coupled one-dimensional shooting problems that are solved by two-way integration from both radial boundaries – centreline and far field. The overall problem is formulated as one of iterative root-finding to match the solutions from the two integrations. The approach is validated against results from the well-established matrix method that discretizes the domain to obtain a matrix eigenvalue problem. We demonstrate very good agreement in two jet problems – an offset dual-stream jet, and a jet exiting from a nozzle with chevrons. A disadvantage of the shooting method is its sensitivity to the initial guess of the solution; however, this becomes an advantage when the need arises to track an eigensolution in a sweep over a problem parameter – say with increasing offset in the dual-stream jet, or with downstream distance from the nozzle exit. We demonstrate the performance of the shooting method in such tracking tasks.

Copy-move forgery detection of medical images using golden ball optimization

The medical images can be tampered with by attackers with a malevolent goal of hiding or creating multiple copies of lesions, resulting in wrong treatment, false insurance claim, defame public figures, and so on. Such forgery demands authentication of digital images before starting the diagnosis and treatment of the patients. The existing key-point (KP) based forgery detection methods may not uniformly distribute the KPs on the images, thereby making the detection process to fail for images with forgeries in smooth regions. This paper attempts to employ a minimum Eigen-value algorithm that distributes the KPs in the entire image region and apply speeded up robust features and singular value decomposition for obtaining reduced descriptors at identified KPs. Besides, the method applies golden ball-based optimization, inspired by the behaviour of players in team-based sports games, for optimal clustering of the evaluated features. This paper studies the performances of the suggested method on 300 medical images and compares the results with the existing methods. It exhibits that the precision, specificity, sensitivity, and accuracy of the developed method are superior to the existing methods. Though the suggested method outperforms, it can further be improved by combining it with block-based methods.

An extended shift-invert residual Arnoldi method

Matrix eigenvalue problems are frequently encountered in engineering and science. In this paper, we propose an extended shift-invert residual Arnoldi method for solving some smaller eigenvalues and the corresponding eigenvectors of a large and symmetric matrix, based on the idea of perfect Krylov subspace method proposed by Bai and Miao. We prove that the projected subspace of the proposed method is the same as the perfect Krylov subspace in exact computation. Based on this result, we also prove the convergence of the proposed method. Numerical experiments show that this method is superior to the perfect Krylov subspace method and the shift-invert residual Arnoldi method.