Computation Of Eigenvectors Research Articles

Extraordinarily Time- and Memory-Efficient Large-Scale Canonical Correlation Analysis in Fourier Domain: From Shallow to Deep.

Canonical correlation analysis (CCA) is a correlation analysis technique that is widely used in statistics and the machine-learning community. However, the high complexity involved in the training process lays a heavy burden on the processing units and memory system, making CCA nearly impractical in large-scale data. To overcome this issue, a novel CCA method that tries to carry out analysis on the dataset in the Fourier domain is developed in this article. Appling Fourier transform on the data, we can convert the traditional eigenvector computation of CCA into finding some predefined discriminative Fourier bases that can be learned with only element-wise dot product and sum operations, without complex time-consuming calculations. As the eigenvalues come from the sum of individual sample products, they can be estimated in parallel. Besides, thanks to the data characteristic of pattern repeatability, the eigenvalues can be well estimated with partial samples. Accordingly, a progressive estimate scheme is proposed, in which the eigenvalues are estimated through feeding data batch by batch until the eigenvalues sequence is stable in order. As a result, the proposed method shows its characteristics of extraordinarily fast and memory efficiencies. Furthermore, we extend this idea to the nonlinear kernel and deep models and obtained satisfactory accuracy and extremely fast training time consumption as expected. An extensive discussion on the fast Fourier transform (FFT)-CCA is made in terms of time and memory efficiencies. Experimental results on several large-scale correlation datasets, such as MNIST8M, X-RAY MICROBEAM SPEECH, and Twitter Users Data, demonstrate the superiority of the proposed algorithm over state-of-the-art (SOTA) large-scale CCA methods, as our proposed method achieves almost same accuracy with the training time of our proposed method being 1000 times faster. This makes our proposed models best practice models for dealing with large-scale correlation datasets. The source code is available at https://github.com/Mrxuzhao/FFTCCA.

A novel approach to crack modeling using extended finite element and substructures methods

In the present work, a combination of Extended Finite Element Method (XFEM) and substructures methods has been utilized for crack modeling. Structures are divided into several substructures, and XFEM is employed to model the presence of cracks within each substructure. In another method, crack modeling has been carried out without using the Extended Finite Element Method. Instead, cracks are modeled by eliminating boundary constraints from two substructures. The Krylov-Schur algorithm is utilized to accelerate the computation of structural modes by preparing stiffness and mass matrices for eigenvalue and eigenvector calculations related to the superstructure. For evaluating this method, four examples have been considered: In the first example, an experimental L-shaped steel structure comprising a beam and a column was examined. The second example involved an experimental steel beam with a box-shaped cross-section. In the third example, a steel beam with simple supports at both ends and a rectangular cross-section was proposed. In the fourth example, a real two-span bridge (Turtle Mountain Bridge) is also presented to evaluate the proposed method. The method was applied for modeling, and the obtained results were validated using experimental and analytical results. In this work, the Structural Similarity Index Method (SSIM) for mode shape similarity assessment has been employed for crack identification. Based on the results obtained from the modeling and crack identification evaluations, it can be concluded that the current method can model multiple cracks in structures with variable cross-sections, real concrete bridges, frame-shaped structures, and diverse structures by assembling substructures. Despite simplifying the problems, it maintains an acceptable level of accuracy in validation. This advantage enables the identification of cracks through a straightforward comparison of mode shapes.

Special measures to promote supplementary professional education services: formation and evaluation of effectiveness

Introduction. The problem of promotion of supplementary professional education services is one of the important issues for attracting consumers of professional retraining and advanced development programmes, contributing to marketing attainments and a company’s efficiency in general. The solution largely depends on the developmental level of the market of supplementary professional education services at a regional level, the specifics and quality of training courses, as well as the relevance of promotion tools used by the organisation. Purpose of the research: development and approbation of the methodology for forming and evaluating the effectiveness of a special tool for the promotion of supplementary professional education services. Materials and methods. To determine the format of a special survey, an online questioning of 1,035 potential consumers was carried out; ten marketing and advertising experts were interviewed and selected by the Delphi method modified by J. Fowles; a questioning of 319 participants and 20 organisers of the special survey was undertaken. The expert judgement was arranged using the Hierarchy Analysis Method by T. Saati, including the processing of the data obtained in the course of the experts interviewing and the calculation of eigenvectors of the pairwise comparison matrix. Results. A methodology for forming and evaluating the effectiveness of a special survey was developed. As a result of the approbation, eleven qualitative indicators were evaluated, scoring 7.92 out of nine, totalling 88%. The quantitative indicators, particularly, the engagement index was 56.8%; the key performance indicator was 8.5% and the return on investment was 336.8%. As a result of the special survey, 27 participants acquired the Internet marketing course. Conclusion. The proposed methodology can extend the opportunities of organisations providing supplementary special education services in terms of selecting, planning and realisation of efficient promotion tools. The methodology can be applied to the formation and evaluation of different-format special surveys with regard to their specifics.

Rational Kriging

This article proposes a new kriging that has a rational form. It is shown that the generalized least squares estimator of the mean from rational kriging is much more well behaved than that of ordinary kriging. Parameter estimation and uncertainty quantification for rational kriging are proposed using a Gaussian process framework. A generalized version of rational kriging is also proposed, which includes ordinary and rational kriging as special cases. Extensive simulations carried out over a wide class of functions show that the generalized rational kriging performs on par or better than both ordinary and rational kriging in terms of prediction and uncertainty quantification. The only extra step needed for generalized rational kriging over ordinary kriging is the computation of Perron eigenvector of an augmented correlation matrix which can be computed in near linear time and therefore, its overall computational complexity is no more than that of ordinary kriging. The potential applications of the new kriging methods in the emulation of computationally expensive models and model calibration problems are illustrated with real and simulated examples. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

Privacy-preserving eigenvector computation with applications in spectral clustering

Privacy-preserving eigenvector computation with applications in spectral clustering

Quantum Value Valuation Continuum

Price, cost, and income (PCI) methods are traditionally used to approximate the value state of an economic commodity such as a property. Based on the estimates of these methods, we explore how quantum theory represents the fundamental process of value valuation in practice. We propose that the mathematical formalism of quantum theory is a promising view and measure of economic value. To ground our exploration, we first map traditional PCI estimates onto three-dimensional spherical coordinates, which were then transformed into two-dimensional quantum states using the Bloch sphere. This step enabled the computation of eigenvalues and eigenvectors of the Hamiltonian matrix, from which the value state measures were derived. The results exhibit practical applications as well as fundamental insights into potential connections between economic and quantum value states.

Revisiting Kassman’s path deletion procedure for the eigenvector coefficients of molecular graphs

Kassman’s path deletion procedure for the determination of eigenvector polynomials (EPs) and hence the eigenvector or molecular orbital (MO) coefficients of a molecular graph is revisited with required proofs and illustrations. As EPs vanish for n-fold degenerate eigenvalues, the calculation of eigenvector (MO) coefficients for such an eigenvalue requires (n-1)-th derivative of each EP. The eigenvector coefficients for other degenerate eigenvalues are then obtained by exploiting the inherent symmetry of the molecular graph. The method of symmetry-factorisation followed by the path deletion procedure makes such calculations straightforward and simple as symmetry-factorisation distributes the degenerate eigenvalues (if there are any) in the fragmented graphs and the problem of degeneracy is thus avoided in calculating the MO coefficients for the individual fragments. The procedure is illustrated with the molecular graphs having non-degenerate and degenerate eigenvalues.

Advanced Dielectric Anomaly Detection for Non-Destructive Fruit Inspection Using Electromagnetic Microwave Technique

This paper presents a comprehensive investigation into the application of an electromagnetic microwave technique combined with the dielectric anomaly approach algorithm for the non-destructive inspection of fruits. The proposed antenna configuration comprises eight elements arranged in a circular array, enabling the collection of signals scattered by objects, specifically fruits, placed in their path. The collected data undergoes a series of processing steps, including Fast Fourier Transform, covariance matrix estimation, eigenvalue and eigenvector computation, and spatial spectrum construction. The dielectric anomaly algorithm is then applied to detect defects in the fruits. The study covers five different types of fruits, both healthy and defective, and gathers essential dielectric properties. Furthermore, a novel hybrid IQR method is introduced for outlier detection in the dielectric data. The results demonstrate the effectiveness of the proposed methodology in providing valuable insights into the internal structures of fruits and detecting anomalies, contributing to the enhancement of quality assessment in fruit inspection processes.

Approximated decompositions for computational continuum mechanics

Solid and fluid mechanics problems often involve the computation of eigenvalues and eigenvectors. One has short algebraic expressions for the Euler equations and the perfect gas equation of state. However, algebraic expressions for solid mechanics models can become substantial. Computing the eigenvalues and eigenvectors algebraically for different models can become cumbersome. Also, larger formulae may result in increased overall computational time. Internal energy plays a significant role in modeling the medium, and thermodynamic consistency is indispensable. The models should correctly capture heat transfer, mechanical response, and fluid flow, e.g., in mechanocaloric and bone remodeling (BR) problems. In mechanocaloric problems, temperature variation occurs because of load and unload stresses. In BR problems, several mathematical models were developed for modeling its behavior. Flexibility for testing different models is an essential requirement. Hence, our goal is to propose and test approximated eigensystem and singular value (SVD) decompositions for continuum mechanics problems. We tested the resolution and accuracy of our propositions in benchmarking CFD problems. For the solid mechanics, we solved a few classical problems, verified the accuracy, and solved mechanocaloric tests and bar under tension. Although more expensive, the approximated eigensystem decomposition is more flexible. It is high-order and high-resolution, can capture heat transfer, fluid flow, discontinuities, and mechanical response, and can model real-world applications. The SVD is a viable possibility and can be used to adjust the computational time of severe situations, but it needs to be improved.

Acceleration of Approximate Matrix Multiplications on GPUs.

Matrix multiplication is important in various information-processing applications, including the computation of eigenvalues and eigenvectors, and in combinatorial optimization algorithms. Therefore, reducing the computation time of matrix products is essential to speed up scientific and practical calculations. Several approaches have been proposed to speed up this process, including GPUs, fast matrix multiplication libraries, custom hardware, and efficient approximate matrix multiplication (AMM) algorithms. However, research to date has yet to focus on accelerating AMMs for general matrices on GPUs, despite the potential of GPUs to perform fast and accurate matrix product calculations. In this paper, we propose a method for improving Monte Carlo AMMs. We also give an analytical solution for the optimal values of the hyperparameters in the proposed method. The proposed method improves the approximation of the matrix product without increasing the computation time compared to the conventional AMMs. It is also designed to work well with parallel operations on GPUs and can be incorporated into various algorithms. Finally, the proposed method is applied to a power method used for eigenvalue computation. We demonstrate that, on an NVIDIA A100 GPU, the computation time can be halved compared to the conventional power method using cuBLAS.

Technique for reducing time in the calculation of eigenvalues and eigenvectors applied in neutron transport fixed-source problems

In this work, a mathematical technique to accelerate the eigenvalue calculations, originally proposed for the Analytical Discrete Ordinates method (ADO), is applied to the coarse-mesh method Modified Spectral Deterministic (MSD), used in neutron shielding problems (fixed-source) in one-dimensional geometry. This technique consists in an arrangement of the neutron transport equation which leads to a reduced eigenvalue calculation on the intranodal analytical solution, where only the positive half of the eigenvalues and its respective eigenvectors are calculated, hence reducing the methods execution time. Precision and performance tests were performed in 2 model-problems, reducing around 80% of the execution time. All the codes were developed using the programming language C++.

Simultaneous diagonalization of nearly commuting Hermitian matrices: do-one-then-do-the-other

Abstract Commuting Hermitian matrices may be simultaneously diagonalized by a common unitary matrix. However, the numerical aspects are delicate. We revisit a previously rejected numerical approach in a new algorithm called ‘do-one-then-do-the-other’. One of two input matrices is diagonalized by a unitary similarity, and then the computed eigenvectors are applied to the other input matrix. Additional passes are applied as necessary to resolve invariant subspaces associated with repeated eigenvalues and eigenvalue clusters. The algorithm is derived by first developing a spectral divide-and-conquer method and then allowing the method to break the spectrum into, not just two invariant subspaces, but as many as safely possible. Most computational work is delegated to a black-box eigenvalue solver, which can be tailored to specific computer architectures. The overall running time is a small multiple of a single eigenvalue-eigenvector computation, even on difficult problems with tightly clustered eigenvalues. The article concludes with applications to a structured eigenvalue problem and a highly sensitive eigenvector computation.

Spectral Ranking in Complex Networks Using Memristor Crossbars

Various centrality measures have been proposed to identify the influence of each node in a complex network. Among the most popular ranking metrics, spectral measures stand out from the crowd. They rely on the computation of the dominant eigenvector of suitable matrices related to the graph: EigenCentrality, PageRank, Hyperlink Induced Topic Search (HITS) and Stochastic Approach for Link-Structure Analysis (SALSA). The simplest algorithm used to solve this linear algebra computation is the Power Method. It consists of multiple Matrix-Vector Multiplications (MVMs) and a normalization step to avoid divergent behaviours. In this work, we present an analog circuit used to accelerate the Power Iteration algorithm including current-mode termination for the memristor crossbars and a normalization circuit. The normalization step together with the feedback loop of the complete circuit ensure stability and convergence of the dominant eigenvector. We implement a transistor level peripheral circuitry around the memristor crossbar and take non-idealities such as wire parasitics, source driver resistance and finite memristor precision into account. We compute the different spectral centralities to demonstrate the performance of the system. We compare our results to the ones coming from the conventional digital computers and observe significant energy savings while maintaining a competitive accuracy.

Efficient parallel reduction of bandwidth for symmetric matrices

Bandwidth reduction can be a first step in the computation of eigenvalues and eigenvectors for a wide-banded complex Hermitian (or real symmetric) matrix. We present algorithms for this reduction and the corresponding back-transformation of the eigenvectors. These algorithms rely on blocked Householder transformations, thus enabling level 3 BLAS performance, and they feature two levels of parallelism. The efficiency of our approach is demonstrated with numerical experiments.

An Efficient FPGA Implementation of MUSIC Processor Using Cyclic Jacobi Method: LiDAR Applications

LiDAR is a technology that uses lasers to measure the position of elements. Measuring the laser travel time and calculating the distance between the LiDAR and the surface requires the calculation of eigenvalues and eigenvectors of the convergence matrix. SVD algorithms have been proposed to solve an eigenvalue problem, which is computationally expensive. As embedded systems are resource-constrained hardware, optimized algorithms are needed. This is the subject of our paper. The first part of this paper presents the methodology and the internal architectures of the MUSIC processor using the Cyclic Jacobi method. The second part presents the results obtained at each step of the FPGA processing, such as the complex covariance matrix, the unitary and inverse transformation, and the value and vector decomposition. We compare them to their equivalents in the literature. Finally, simulations are performed to select the way that guarantees the best performance in terms of speed, accuracy and power consumption.

Robust level-3 BLAS Inverse Iteration from the Hessenberg Matrix

Inverse iteration is known to be an effective method for computing eigenvectors corresponding to simple and well-separated eigenvalues. In the non-symmetric case, the solution of shifted Hessenberg systems is a central step. Existing inverse iteration solvers approach the solution of the shifted Hessenberg systems with either RQ or LU factorizations and, once factored, solve the corresponding systems. This approach has limited level-3 BLAS potential since distinct shifts have distinct factorizations. This paper rearranges the RQ approach such that data shared between distinct shifts can be exploited. Thereby the backward substitution with the triangular R factor can be expressed mostly with matrix–matrix multiplications (level-3 BLAS). The resulting algorithm computes eigenvectors in a tiled, overflow-free, and task-parallel fashion. The numerical experiments show that the new algorithm outperforms existing inverse iteration solvers for the computation of both real and complex eigenvectors.

МЕТОДИКА ОЦЕНКИ МОДАЛЬНЫХ ПАРАМЕТРОВ ПЛАНЕТАРНЫХ РЕДУКТОРОВ МОТОР-КОЛЕС КАРЬЕРНЫХ САМОСВАЛОВ ПРИ ИХ ЭКСПЛУАТАЦИОННОЙ ВИБРОДИАГНОСТИКЕ

Technical condition diagnostics of the planetary reducers of motor wheels in mining dump trucks electromechanical power units requires the study of dynamic processes caused by shock pulses in gearings when tooth remating. The levels of the occurring vibrations are determined by the natural frequencies and natural modes of reducer. The article describes calculation method based on the study of the developed dynamic model of a two-row planetary reducer of a motor wheel, describing the torsional vibrations of its elements and differing from the known ones by taking into account the compound motion of the planetary row satellites, the axes of which rotate together with the carrier. The calculation of natural frequencies and modes of vibrations is reduced to the computation of eigenvalues and eigenvectors of a matrix of special form based on the application of the iterative numerical method of Jacobi rotations. As an example, the calculation is given for gears of the motor wheel reducer of BELAZ mining dump trucks of the 7530 and 7531 series with a load capacity of 220 and 240 tons, respectively. The calculation of the natural frequencies of reducer performed makes it possible to determine those harmonic components of vibration-inducing disturbances in the form of shock pulses which cause resonance phenomena in the mechanism contributing to a significant increase of dynamic loads levels and accelerating the processes of lifetime expense of gear trains. In addition, the analysis of natural modes parameters makes it possible to determine the reducer elements with the highest vibration activity and monitor during vibration monitoring the technical condition of gear trains that limit the reliability of the mechanism.

Connected and automated vehicle platoon formation control via differential games

In this study, the connected and automated vehicles (CAVs) platooning problem is resolved under a differential game framework. Three information topologies are considered here. Firstly, Predecessor-following (PF) topology is utilized where the vehicles control the distance with respect to the merely nearest predecessor via a sensor link-based information flow. Secondly, Two-predecessor-following topology (TPF) is exploited where each vehicle controls the distance with respect to the two nearest predecessors. In this topology, the second predecessor is communicated via a Vehicle-to-vehicle (V2V) link. The individual trajectories of CAVs under the Nash equilibrium are derived in closed-form for these two information topologies. Finally, general information topology is examined and the differential game is formulated in this context. In all these options, Pontryagin's principle is employed to investigate the existence and uniqueness of the Nash equilibrium and obtain its corresponding trajectories. In the general topology, we suppose numerical computation of eigenvalues and eigenvectors. All these approaches represent promising and powerful analytical representations of the CAV platoons under the differential games. Simulation experiments have verified the efficiency of the proposed models and their solutions.

Invited Tutorial: Analog Matrix Computing With Crosspoint Resistive Memory Arrays

Matrix computation is ubiquitous in modern scientific and engineering fields. Due to the high computational complexity in conventional digital computers, matrix computation represents a heavy workload in many data-intensive applications, e.g., machine learning, scientific computing, and wireless communications. For fast, efficient matrix computations, analog computing with resistive memory arrays has been proven to be a promising solution. In this Tutorial, we present analog matrix computing (AMC) circuits based on crosspoint resistive memory arrays. AMC circuits are able to carry out basic matrix computations, including matrix multiplication, matrix inversion, pseudoinverse and eigenvector computation, all with one single operation. We describe the main design principles of the AMC circuits, such as local/global or negative/positive feedback configurations, with/without external inputs. Mapping strategies for matrices containing negative values will be presented. The underlying requirements for circuit stability will be described via the transfer function analysis, which also defines time complexity of the circuits towards steady-state results. Lastly, typical applications, challenges, and future trends of AMC circuits will be discussed.

Program Synthesis of Sparse Algorithms for Wave Function and Energy Prediction in Grid-Based Quantum Simulations.

We have recently shown how program synthesis (PS), or the concept of “self-writing code”, can generate novel algorithms that solve the vibrational Schrödinger equation, providing approximations to the allowed wave functions for bound, one-dimensional (1-D) potential energy surfaces (PESs). The resulting algorithms use a grid-based representation of the underlying wave function ψ(x) and PES V(x), providing codes which represent approximations to standard discrete variable representation (DVR) methods. In this Article, we show how this inductive PS strategy can be improved and modified to enable prediction of both vibrational wave functions and energy eigenvalues of representative model PESs (both 1-D and multidimensional). We show that PS can generate algorithms that offer some improvements in energy eigenvalue accuracy over standard DVR schemes; however, we also demonstrate that PS can identify accurate numerical methods that exhibit desirable computational features, such as employing very sparse (tridiagonal) matrices. The resulting PS-generated algorithms are initially developed and tested for 1-D vibrational eigenproblems, before solution of multidimensional problems is demonstrated; we find that our new PS-generated algorithms can reduce calculation times for grid-based eigenvector computation by an order of magnitude or more. More generally, with further development and optimization, we anticipate that PS-generated algorithms based on effective Hamiltonian approximations, such as those proposed here, could be useful in direct simulations of quantum dynamics via wave function propagation and evaluation of molecular electronic structure.