Eigenvalue Decomposition Research Articles

Control-Oriented Galerkin-Spectral Model for 3-D Thermal Diffusion of Pouch-Type Batteries

Control-oriented thermal models are essential for onboard temperature monitoring of lithium-ion batteries in automobile applications. This work develops a Galerkin-spectral model for the three-dimensional (3-D) thermal diffusion in pouch cells. First, a full-order model that depicts the battery thermal phenomenon is introduced from a physical point of view. Considering different physical properties in each area, we apply a space decomposition approach to decouple the interactive thermal effects between the cell core and tabs. Then, mild approximations are made to generate a more succinct model governed by the partial and ordinary differential equations. Finally, a low-order representation is extracted from the original infinite-dimensional system by employing spectral expansion on the spatiotemporal variable. Experimental and simulation studies indicate satisfactory reduced-order performance and practical validity of the proposed model.

Two Rapid Power Iterative DOA Estimators for UAV Emitter Using Massive/Ultra-Massive Receive Array

To provide rapid direction finding (DF) for unmanned aerial vehicle (UAV) emitters in future wireless networks, a low-complexity direction of arrival (DOA) estimation architecture for massive multiple-input multiple-output (MIMO) receiver arrays is constructed. In this paper, we propose two strategies to address the extremely high complexity caused by eigenvalue decomposition of the received signal covariance matrix. Firstly, a rapid power iterative rotational invariance (RPI-RI) method is proposed, which adopts the signal subspace generated by power iteration to obtain the final direction estimation through rotational invariance between subarrays. RPI-RI causes a significant complexity reduction at the cost of a substantial performance loss. In order to further reduce the complexity and provide good directional measurement results, the rapid power iterative polynomial rooting (RPI-PR) method is proposed, which utilizes the noise subspace combined with the polynomial solution method to obtain the optimal direction estimation. In addition, the influence of initial vector selection on convergence in the power iteration is analyzed, especially when the initial vector is orthogonal to the incident wave. Simulation results show that the two proposed methods outperform the conventional DOA estimation methods in terms of computational complexity. In particular, the RPI-PR method achieves more than two orders of magnitude lower complexity than conventional methods and achieves performance close to the Cramér–Rao Lower Bound (CRLB). Moreover, it is verified that the initial vector and the relative error have a significant impact on the performance with respect to the computational complexity.

A Novel Low-Complexity Method for Near-Field Sources Based on an S-IMISC Array Model

Array optimization has recently received significant attention owing to its several advantages, such as larger array aperture and greater degrees of freedom (DOFs). However, current works focus on far-field sources, while array optimization for near-field sources has not been adequately investigated. Therefore, this work develops a new symmetry sparse array model for near-field sources based on the improved maximum inter-element spacing constraint (IMISC). The proposed symmetry IMISC (S-IMISC) array model has all the advantages of traditional sparse array models. Compared with traditional sparse array models, the S-IMISC array model affords more uniform DOFs and is less affected by mutual coupling. Additionally, in order to improve the real-time performance of near-field sources localization, the characteristic equation-based method (CEM) is used to obtain the azimuth information of near-field sources which can avoid eigenvalue decomposition (EVD), and a spectrum peak search and compression scheme is used to obtain the distance information by searching the partial area instead of the whole Fresnel area, thereby significantly reducing computation complexity. Extensive simulations verify the advantages of the proposed algorithm and the S-IMISC array model.

On the spectral theorem of Langlands

We show that the Hilbert subspace of L2(G(F)﹨G(A)) generated by wave packets of Eisenstein series built from discrete series is the whole space. Together with the work of Lapid [17], it achieves a proof of the spectral theorem of R.P. Langlands ([16], [19]) based on the work of J. Bernstein and E. Lapid [6] on the meromorphic continuation of Eisenstein series built from discrete data. I use truncation on compact sets as J. Arthur did for the local trace formula in [2].

An efficient null-broadening adaptive beamformer based on magnitude response constraints

To reduce computational complexity of the minimum power distortionless response (MPDR) beamformer, a novel efficient null-broadening method based on magnitude response constraints is proposed in this paper. We convert the original quadratic power constraint to multiple first-order magnitude response constraints using Gauss-Legendre quadrature (GLQ) rather than eigenvalue decomposition (EVD). In addition, the number of first-order constraints only depends on the order of Legendre polynomials. Furthermore, based on the conjugate symmetry of the beamforming weight vector on centro-symmetric array, the nonlinear modular operation in magnitude response constraints is transformed into equivalent linear real operation. Finally, we relax the inequality constraints in the model into equality constraints, which can be solved by Lagrange multiplier method. Simulation shows that the proposed magnitude response constrained MPDR (MRC-MPDR) beamformer can achieve the same dynamic interference suppression capability as other beamformers with shorter calculation time.

Kirchhoff Index and Degree Kirchhoff Index of Tetrahedrane-Derived Compounds

Tetrahedrane-derived compounds consist of n crossed quadrilaterals and possess complex three-dimensional structures with high symmetry and dense spatial arrangements. As a result, these compounds hold great potential for applications in materials science, catalytic chemistry, and other related fields. The Kirchhoff index of a graph G is defined as the sum of resistive distances between any two vertices in G. This article focuses on studying a type of tetrafunctional compound with a linear crossed square chain shape. The Kirchhoff index and degree Kirchhoff index of this compound are calculated, and a detailed analysis and discussion is conducted. The calculation formula for the Kirchhoff index is obtained based on the relationship between the Kirchhoff index and Laplace eigenvalue, and the number of spanning trees is derived for linear crossed quadrangular chains. The obtained formula is validated using Ohm’s law and Cayley’s theorem. Asymptotically, the ratio of Kirchhoff index to Wiener index approaches one-fourth. Additionally, the expression for the degree Kirchhoff index of the linear crossed quadrangular chain is obtained through the relationship between the degree Kirchhoff index and the regular Laplace eigenvalue and matrix decomposition theorem.

Equivalence theorem for the spectral density of light waves scattered from different types ofanisotropic media.

The spectral density equivalence theorem of light waves on scattering from different types of anisotropic media is discussed. The possibility of producing identical spectral density distribution as light waves are scattered by different anisotropic media is also discussed. Moreover, the conditions under which a determinate anisotropic medium and random anisotropic medium may produce identical spectral densities are obtained. It is shown that the complete equivalence theorem in whole space or the partial equivalence theorem in some special planes can be realized under particular conditions.

Anchor-based multi-view subspace clustering with graph learning

Multi-view subspace clustering (MVSC) has been drawn wide attentions in the area of pattern recognition and data mining. However, for a multi-view dataset with n samples and V views from k clusters, MVSC commonly requires O(Vn2) memory for storing the view-specific graph matrices and O(n3) time for the eigenvalue decomposition of a shared graph matrix. Hence, most of MVSC methods are difficult to handle the large-scale multi-view data problem. To address this issue, this paper proposes an Anchor-based Multi-View Subspace Clustering with Graph Learning (AMVSCGL) method. Instead of constructing a n×n graph matrix, our method generates a shared coefficient matrix with the size of n×k based on few learned view-specific anchors. Moreover, through further merging a graph learning term, this shared coefficient matrix can simultaneously capture the global and local information among multiple views and few learned view-specific anchors for clustering. Experimental results on seven large-scale multi-view data verify our AMVSCGL’s effectiveness and superiority in comparison with some state-of-the-art methods.

Stability and asymptotics for fractional delay diffusion‐wave equations

The asymptotic stability and long‐time decay rates of solutions to linear Caputo time‐fractional ordinary differential equations (FODEs) with order both and are known to be completely determined by the eigenvalues of the coefficient matrices. Very different from the exponential decay of solutions to classical ordinary differential equations, solutions of FODEs decay only polynomially, leading to the so‐called Mittag–Leffler stability, which was already extended to linear fractional delay differential equations (FDDEs) with order . However, the study on the long‐time decay rates of the solutions for FDDEs with order is scarce. Hence, this paper is devoted to the asymptotical stability and long‐time decay rates for a class of FDDEs with , that is, fractional delay diffusion‐wave equations. We reformulate the fractional delay diffusion‐wave equations to infinite‐dimensional FDDEs by the eigenvalue decomposition method. Through a detailed analysis for the characteristic equations, a novel necessary and sufficient stability condition is established in a coefficient criterion. Moreover, we are able to deal with the awful singularities caused by delay and fractional exponent by introducing a novel integral path, and hence to give an accurate estimation for the fractional resolvent operators. We show that the long‐time decay rates of the solutions for both linear FDDEs and fractional delay diffusion‐wave equations are . Finally, an example of an application model with feedback control is provided to show the effectiveness of our theoretical results.

Deep learning-based reconstruction of missing long-term girder-end displacement data for suspension bridge health monitoring

The flexibility of a suspension bridge usually results in significant longitudinal displacement at the end of the main girder under the joint action of environmental factors and traffic loads. Fluctuation amplitude and accumulation of girder-end displacement (GED) are primary data for the performance evaluation of suspension bridge elements, including expansion joints, supports, and dampers. However, the performance of the bridge structural health monitoring (SHM) system is frequently deteriorated by the long-term continuous GED data loss or anomaly. In this study, this issue of the suspension bridge SHM system was resolved by proposing a deep learning-based framework for reconstructing the missing GED data. Under this framework, a long short-term memory (LSTM) network-based regression model was first built between the ambient temperature data and the thermal-induced low-frequency components of GED data. Next, a U-net-based spectral bandwidth expansion model was applied for the step-by-step generation of vehicle/wind-induced high-frequency GED data terms based on the available low-frequency ones. Finally, a statistical correction strategy was employed to improve the prediction accuracy for the high-frequency GED data fluctuation amplitudes. The trained model for the reconstruction of the missing GED data uses only the air temperature input values in the data loss period. This method is especially lucrative when GED data in all sensor channels are missing. The reliability and efficiency of the proposed model were demonstrated by a case study. Specifically, the problem of reconstructing the missing GED data in the SHM system of the Jiangyin Yangtze River Bridge in China was considered. Using the relative error of cumulative GED as the performance indicator, a reconstruction precision within 10% was obtained.

On the Pinsky Phenomenon for B-Elliptic Operators

Necessary conditions for the summability of spectral expansions in eigenfunctions of an elliptic operator with a Bessel operator in one of the variables in an arbitrary-dimensional domain adjacent to the singularity hypersurface are obtained. It is proved that if the spectral expansion of an arbitrary function at some point of this hypersurface is summable by Riesz means, then its average value over the half-ball centered at the specified point has generalized smoothness.

Data-driven estimation, tracking, and system identification of deterministic and stochastic optical spot dynamics.

Stabilization, disturbance rejection, and control of optical beams and optical spots are ubiquitous problems that are crucial for the development of optical systems for ground and space telescopes, free-space optical communication terminals, precise beam steering systems, and other types of optical systems. High-performance disturbance rejection and control of optical spots require the development of disturbance estimation and data-driven Kalman filter methods. Motivated by this, we propose a unified and experimentally verified data-driven framework for optical-spot disturbance modeling and tuning of covariance matrices of Kalman filters. Our approach is based on covariance estimation, nonlinear optimization, and subspace identification methods. Also, we use spectral factorization methods to emulate optical-spot disturbances with a desired power spectral density in an optical laboratory environment. We test the effectiveness of the proposed approaches on an experimental setup consisting of a piezo tip-tilt mirror, piezo linear actuator, and a CMOS camera.

1-D and 2-D Direction of Arrival Estimation in a Conical Conformal Array: Design and Implementation.

Direction of arrival (DOA) estimation for conformal arrays is challenging due to non-omnidirectional element patterns and shadow effects. Conical conformal array (CCA) can avoid the shadow effect at small elevation angles. So CCA is suitable for DOA estimation on both azimuth and elevation angles at small elevation angles. However, the element pattern in CCA cannot be obtained by conventional directional element coordinate transformation. Its local element pattern also has connection with the cone angle. The paper establishes the CCA radiation pattern in local coordinate system using 2-D coordinate transformation. In addition, in the case of large elevation angle, only half elements of the CCA can receive signal due to the shadow effect. The array degrees of freedom (DOF) are reduced by halves. We introduce the difference coarray method, which increases the DOF. Moreover, we propose a more accurate propagator method for 2-D cases. This method constructs a new propagation matrix and reduces the estimation error. In addition, this method reduces computational complexity by using linear computations instead of eigenvalue decomposition (EVD) and avoids spectral search. Simulation and experiment verify the estimation performance of the CCA. Both demonstrate the CCA model established in this paper is corresponding to the designed CCA antenna, and the proposed algorithms meet the needs of CCA angle detection. When the number of array elements is 12, the estimation accuracy is about 5 degrees.

High-Performance of Eigenvalue Decomposition on FPGA for the DOA Estimation

For the direction of arrival (DOA) in array signal processing, eigenvalue decomposition (EVD) is one key issue in hardware implementation of the multiple signal classification (MUSIC) algorithm. Therefore, we introduce the look-ahead sim- plified one-sided Jacobi's method to efficiently decompose those symmetric matrices in this article and prove that the new method has the best orthogonality of eigenvector and locates eigenvectors closest to the true solution in theory. Both the numerical perform- ance and real-time are important in engineering, so we present the novel flexible hardware architecture in single floating point arithmetic for EVD on field-programmable gate arrays (FPGAs). Finally, the simulated and raw data are used to investigate the performance of some different approaches in the context of both the EVD and MUSIC algorithm. The experimental results show that our proposed method has the best performance.

INTERDISCIPLINARY APPROACH OF LINEAR ALGEBRA IN QUANTUM COMPUTING

In this paper we discuss the interdisciplinary approach of Linear Algebra in quantum computing. Everything in quantum computing, from the representation of Qubits and gates to circuit's functionality can be described using various forms of Linear Algebra. The most advanced topics Inner product spaces, Vectors, Spectral theorem, Hilbert Spaces, exponential matrices, tensor product etc., of linear algebra are required to predict what a quantum computer does in response to a sequence of instructions

Local nonlinear dimensionality reduction via preserving the geometric structure of data

Dimensionality reduction has many applications in data visualization and machine learning. Existing methods can be classified into global ones and local ones. The global methods usually learn the linear relationship in data, while the local ones learn the manifold intrinsic geometry structure, which has a significant impact on pattern recognition. However, most of existing local methods obtain an embedding with eigenvalue or singular value decomposition, where the computational complexities are very high in a large amount of high-dimensional data. In this paper, we propose a local nonlinear dimensionality reduction method named Vec2vec, which employs a neural network with only one hidden layer to reduce the computational complexity. We first build a neighborhood similarity graph from the input matrix, and then define the context of data points with the random walk properties in the graph. Finally, we train the neural network with the context of data points to learn the embedding of the matrix. We conduct extensive experiments of data classification and clustering on nine image and text datasets to evaluate the performance of our method. Experimental results show that Vec2vec is better than several state-of-the-art dimensionality reduction methods, except that it is equivalent to UMAP on data clustering tasks in the statistical hypothesis tests, but Vec2vec needs less computational time than UMAP in high-dimensional data. Furthermore, we propose a more lightweight method named Approximate Vec2vec (AVec2vec) with little performance degradation, which employs an approximate method to build the neighborhood similarity graph. AVec2vec is still better than some state-of-the-art local dimensionality reduction methods and competitive with UMAP on data classification and clustering tasks in the statistical hypothesis tests.

RIS-Assisted Space-Air-Ground Integrated Networks: New Horizons for Flexible Access and Connectivity

Space-air-ground integrated network (SAGIN) has emerged as a paradigm shifting architecture that offers largescale, flexible wireless coverage and seamless, high-rate connectivity to complement terrestrial communications. Nonetheless, unlocking the potentials of SAGIN is subject to addressing nontrivial challenges associated with their inherent time-variability, self-organization, and heterogeneity. Meanwhile, the concept of reconfigurable intelligent surfaces (RISs) is recognized as a disruptive technology that supports an unprecedented control of electromagnetic waves propagation and potentially offers significant enhancements in spectral efficiency, coverage expansion, and security, all achieved in a low-cost and energy-efficient manner. We anticipate that the integration of RISs into SAGIN will not only play a fundamental role in improving the quality of interand intra- layer communications, but will also provide complex interactions among the three network segments, and hence, opens the horizons for a new breed of applications across all industries. In this article, we explore the underlying opportunities and challenges of integrating RIS-enabled communications into SAGIN, and present a forward-looking overview of the crosslayer operational elements supported by RISs. Finally, we outline major enabling technologies and present a look ahead toward addressing open research issues.

K-space dispersion error compensators for the fractional spatial derivatives based constant-Q viscoelastic wave equation modeling

Unlike the fractional temporal derivatives based constant-Q viscoelastic wave equation, the fractional spatial derivatives based one is preferred for the wavefield simulation due to its tractable memory consumption. This viscoelastic wave equation is often solved by the staggered-grid pseudospectral method, which has the spatial spectral accuracy but temporal low order accuracy. The k-space method is one of the potential remedies for the temporal discretization errors. However, it is difficult to fully remove the temporal dispersion errors by directly applying the elastic k-space error compensators to the viscoelastic wave equation, which contains additional temporal derivatives associated with the amplitude decay terms of the P- and S- waves. To tackle the issue, we have derived a new wavenumber-time domain viscoelastic k-space equation by the eigenvalue decomposition method and by the solution to the matrix differential equations. The error compensation terms inside our k-space equation can simultaneously correct the discretization errors of different types of temporal derivatives and are wave mode adaptive. The accuracy of the k-space wave equation is verified by the viscoelastic Green's function's solution. To further boost the simulation efficiency of the k-space method in heterogeneous media, we have proposed an improved k-space method, which decreases the number of the costly mixed-domain operators. The simulation results from the simple and complex models demonstrate that our two k-space methods are superior to the conventional staggered-grid pseudospectral method for the viscoelastic wave modeling.

Bidiagonal factorization of tetradiagonal matrices and Darboux transformations

Recently a spectral Favard theorem for bounded banded lower Hessenberg matrices that admit a positive bidiagonal factorization was presented. These type of matrices are oscillatory. In this paper the Lima–Loureiro hypergeometric multiple orthogonal polynomials and the Jacobi–Piñeiro multiple orthogonal polynomials are discussed at the light of this bidiagonal factorization for tetradiagonal matrices. The Darboux transformations of tetradiagonal Hessenberg matrices is studied and Christoffel formulas for the elements of the bidiagonal factorization are given, i.e., the bidiagonal factorization is given in terms of the recursion polynomials evaluated at the origin.

Estimates for the Weyl coefficient of a two-dimensional canonical system

For a two-dimensional canonical system $y'(t)=zJH(t)y(t)$ on some interval $(a,b)$ whose Hamiltonian $H$ is a.e. positive semi-definite and which is regular at $a$ and in the limit point case at $b$, denote by $q_H$ its Weyl coefficient. De~Branges' inverse spectral theorem states that the assignment $H\mapsto q_H$ is a bijection between Hamiltonians (suitably normalised) and Nevanlinna functions. We give upper and lower bounds for $|q_H(z)|$ and $\Im q_H(z)$ when $z$ tends to $i\infty$ non-tangentially. These bounds depend on the Hamiltonian $H$ near the left endpoint $a$ and determine $|q_H(z)|$ up to universal multiplicative constants. We obtain that the growth of $|q_H(z)|$ is independent of the off-diagonal entries of $H$ and depends monotonically on the diagonal entries in a natural way. The imaginary part is, in general, not fully determined by our bounds (in forthcoming work we shall prove that for most Hamiltonians also $\Im q_H(z)$ is fully determined). We translate the asymptotic behaviour of $q_H$ to the behaviour of the spectral measure $\mu_H$ of $H$ by means of Abelian--Tauberian results and obtain conditions for membership of growth classes defined by weighted integrability condition (Kac classes) or by boundedness of tails at $\pm\infty$ w.r.t. a weight function. Moreover, we apply our results to Krein strings and Sturm--Liouville equations.