Matrix Operations Research Articles

Three-phase power flow using binary tree search algorithm for unbalanced distribution networks

The main objective of this study is to examine a binary tree (BT) three-phase (3Φ) power flow analysis method for unbalanced distribution networks. A simple and efficient 3Φ power flow solution method for unbalanced radial distribution networks was proposed. This method is based on the BT search structure. Large-scale distribution network problems can be evaluated using current-injection techniques and two-port network models. The power flow solution process was simplified into a set of 3 × 3 matrix operations using the BT-based method. The solution complexity remained unchanged, regardless of the size of the power system. In other words, the performance and robustness of the proposed algorithm were maintained at relatively high values with increasing system size. Furthermore, programming complexity was reduced by standardising the solution process. The effectiveness and accuracy of the proposed model and algorithm were validated using four IEEE standard benchmark feeders and a series of random cases.

Alpha evolution: An efficient evolutionary algorithm with evolution path adaptation and matrix generation

Metaheuristics involve information extraction and utilization processes to generate more promising solutions. However, the background of excessive metaphor has led to ambiguity in the computational process. To solve this problem, this paper proposes a novel evolutionary algorithm called alpha evolution (AE). It updates the solution using the alpha operator with the adaptive base vector and the random and adaptive step sizes. First, sample candidate solutions to construct the evolution matrix. Estimate the population state through diagonal or weighted operations of the evolution matrix. To enhance the correlation of estimates for each generation, two evolution paths accumulate the estimate results and achieve the base vector adaptation. Second, the composite differential operation constructs the adaptive step size to estimate the problem gradient, which is used to accelerate the convergence of AE. Finally, the attenuation factor alpha adaptively adjusts the random step size generated based on search space to balance exploration and exploitation. AE was comprehensively verified regarding its search bias, invariance, scalability, parameter sensitivity, search behavior, qualitative indicators, exploration and exploitation, convergence, statistics, and complexity. In numerical simulation, AE was compared with 106 algorithms on the CEC’17 benchmark announced at the 2017 congress on evolutionary computation (CEC). Furthermore, AE was applied to solve multiple sequence alignment and engineering design problems. The evidence shows that AE is competitive in exploration and exploitation, convergence speed and accuracy, avoiding local optima, applicability, and reliability. The source code of AE is publicly available at https://github.com/tsingke/AlphaEvolution.

Energy-saving mechanism of series–parallel hybrid transmissions — A dynamic programming method based on matrix operations

The economic evaluation of hybrid transmission is a complicated numerical calculation problem in the finite time domain. Currently, employing traditional dynamic programming methods to evaluate a single design scheme consumes a lot of time, making it arduous to effectively compare performance disparities across various design schemes. This paper introduces a dynamic programming method based on matrix operations, enabling the determination of maximum energy-saving potential in hybrid transmission while reducing the evaluation time for a single design scheme from hours to less than 10 s. It unifies the design variables, constraints, and optimization objectives for multi-gear and multi-mode series–parallel hybrid transmissions (MGMM-SPHTs) during economic evaluation. By solving the performance of tens of thousands of design schemes, it is found that increasing the number of engine direct drive gears from 1 to 2 enhances the economy and power performance of series–parallel hybrid transmissions. For the optimal design scheme of Type-B vehicles, this enhancement results in a 1.712% increase in energy-saving efficiency and a 30.5% increment in power performance. Furthermore, this paper elucidates the adaptive relationship between MGMM-SPHTs and different vehicle types, revealing the energy-saving mechanisms under varied design schemes. This paper holds the promise of offering theoretical underpinnings and methodological instruments for selecting technical routes and optimizing design parameters for MGMM-SPHTs.

Development of an in-situ real-time laser targeting algorithm for a 3D LiDAR-integrated galvanometric laser scanner system

A Galvanometric laser scanner utilizes mirrors attached to dual galvanometers to direct laser beams. GLS systems are widely employed across various industrial sectors due to their compact size, high resolution, and high positional repeatability. Extensive research has been conducted to calibrate these systems to steer lasers to desired positions accurately. However, much of this research has traditionally focused on fixed working distances, which does not align well with the portable capabilities that GLS systems offer in non-destructive evaluation fields. This study explores an algorithm that enables real-time laser targeting using a 3D LiDAR camera integrated into a GLS system. The research involved developing and calculating a projection matrix that projects 3D camera coordinates into 2D bit coordinates input to each galvanometer, based on linear assumptions. To address the non-linear errors that emerge from the linear assumption process, compensation was applied through 3rd-order regression and Artificial neural networks. As a result, the algorithm facilitated highly accurate laser targeting within a range of 0.5 m to 2.5 m based on coordinates acquired by the LiDAR camera.

Sensitivity analysis of frequency response functions with imaginary parts decoupling based on multicomplex-step perturbation

Imaginary perturbation is used in the complex step differentiation method to compute first-order derivatives, widely known as an effective approach for sensitivity analysis in structural dynamics. However, coupling of imaginary parts occurs in the damped frequency response functions when employing this method. To mitigate this coupling, a novel approach for sensitivity analysis based on multicomplex-step perturbation is proposed in this paper, for sensitivity analysis of Frequency Response Functions in structural dynamics. The structural parameters are perturbed in multicomplex domain, the dimensions of structural matrices are expanded using the Cauchy Riemann matrix representation, the equation of motion for sensitivity analysis in frequency domain is transformed to matrix operation in field of real numbers, imaginary term will not exist in the equation of motion for sensitivity analysis, the imaginary part of the frequency response function and the imaginary part of the perturbation are decoupled, the structural frequency response functions and the corresponding sensitivities are obtained from the dimension-expanded equation of motion. A truss structure and a solar wing are adopted to verify the accuracy of the proposed method. Results show that the sensitivity of FRFs can be effectively calculated using the proposed method. Compare to the finite difference method, the proposed method is not depended on the step-size selection procedure. The multi-order and mixed-order sensitivity matrices, especially Hessian matrix can also be obtained using the proposed method.

Existence, uniqueness and Ulam–Hyers stability result for variable order fractional predator-prey system and it's numerical solution

This study presents an approximate numerical technique for solving time fractional advection-diffusion-reaction predator-prey equations with variable order (VO), where the analyzed fractional derivatives of VO are in the Caputo sense. Results for Ulam–Hyers stability are shown, as well as the existence and uniqueness of solutions. It is suggested to use a numerical approximation based on the shifted second kind of airfoil polynomials to solve the equations under consideration. A fractional derivative operational matrix with VO is derived for shifted airfoil polynomials, which will be used to compute the unknown function. The main equations are transformed into a set of algebraic equations by substituting the aforementioned operational matrix into the equations under consideration and utilizing the properties of the shifted airfoil polynomial along with the collocation points. A numerical solution is obtained by solving the acquired set of algebraic equations. To verify the accuracy and efficiency of the discussed scheme, several illustrative examples have been considered. The results obtained by the proposed method demonstrate the efficiency and superiority of the method compared to other existing methods.

A Three-Dimensional Reconstruction Method Based on Telecentric Epipolar Constraints

When calibrating a microscopic fringe projection profile system with a telecentric camera, the orthogonality of the camera causes an ambiguity in the positive and negative signs of its external parameters. A common solution is to introduce additional constraints, which often increase the level of complexity and the calibration cost. Another solution is to abandon the internal/external parameter models derived from the physical imaging process and obtain a numerically optimal projection matrix through the least squares solution. This paper proposes a novel calibration method, which derives a telecentric epipolar constraint model from the conventional epipolar constraint relationship and uses this constraint relationship to complete the stereo calibration of the system. On the one hand, since only the camera’s intrinsic parameters are needed, there is no need to introduce additional constraints. On the other hand, the solution is optimized based on the full consideration of the imaging model to make the parameters confirm to the physical model. Our experiments proved the feasibility and accuracy of the method.

Model-Based Inference of Flame Transfer Matrices from Acoustic Measurements in an Aero-Engine Test Rig

Abstract Flame dynamics, represented as a flame transfer matrix (FTM), is not directly measurable in test rigs and must be deduced from transfer matrix measurements of the combustion system. The burner-flame transfer matrix (BFTM) approach for FTM estimation is based on local pressure signals from microphones located upstream and downstream of the combustor. It combines acoustic measurements in non-reacting and reacting conditions, with the latter implicitly including flame dynamics. A simple matrix operation yields the FTM. However, this approach assumes loss-free wave propagation at constant speed of sound with no change in cross-sectional area between the microphones and the burner/flame. The present work demonstrates the limitations of these assumptions when applied to a test rig with effusion cooling, bypass annulus, and end contraction. This work proposes a method to infer the FTM for complex combustors by combining reactive transfer matrix measurements of the entire combustor with an accurate low-order model (LOM) of the test rig. This generalized method reduces to the BFTM approach as a special case. The Rolls-Royce SCARLET test rig, operating under realistic engine conditions, is used to analyze the capabilities of the proposed model-based inference method and the limitations of the BFTM approach. First, a LOM based on SCARLET's geometry and operating point is formulated using a generic FTM. This model visualizes the limitations of the BFTM approach concerning various physical and geometrical parameters. Finally, experimental data is used to infer the FTM of SCARLET using the proposed approach.

Solving the periodic coupled operator matrix equations via BCR iterative method and its application in periodic state feedback pole assignment

PurposeThe paper aims to study the constraint solutions of the periodic coupled operator matrix equations by the biconjugate residual algorithm. The new algorithm can solve a lot of constraint solutions including Hamiltonian solutions and symmetric solutions, as special cases. At the end of this paper, the new algorithm is applied to the pole assignment problem.Design/methodology/approachWhen the studied periodic coupled operator matrix equations are consistent, it is proved that constraint solutions can converge to exact solutions. It is demonstrated that the solutions of the equations can be obtained by the new algorithm with any arbitrary initial matrices without rounding error in a finite number of iterative steps. In addition, the least norm-constrained solutions can also be calculated by selecting any initial matrices when the equations of the periodic coupled operator matrix are inconsistent.FindingsNumerical examples show that compared with some existing algorithms, the proposed method has higher convergence efficiency because less data are used in each iteration and the data is sufficient to complete an update. It not only has the best convergence accuracy but also requires the least running time for iteration, which greatly saves memory space.Originality/valueCompared with previous algorithms, the main feature of this algorithm is that it can synthesize these equations together to get a coupled operator matrix equation. Although the equation of this paper contains multiple submatrix equations, the algorithm in this paper only needs to use the information of one submatrix equation in the equation of this paper in each iteration so that different constraint solutions of different (coupled) matrix equations can be studied for this class of equations. However, previous articles need to iterate on a specific constraint solution of a matrix equation separately.

A method for calibrating multi-camera systems based on sparse reconstruction of a 3D object

This paper presents a novel calibration method for multi-camera systems that effectively addresses the complexities and inaccuracies associated with traditional calibration techniques. The method relies on the sparse point cloud of a 3D object. By establishing correspondences between 2D points in the images and 3D points in the sparse point cloud, we compute the projection matrix of the camera to be calibrated. We sequentially place the calibration object at multiple positions within the coverage area of the multi-camera system’s field of view, capturing images simultaneously with all cameras. By constructing an undirected graph that reflects the positional relationships between the cameras and the calibration object, and solving for the shortest paths between the camera vertices, we complete the calibration of the multi-camera system. This method’s advantage is its ability to simultaneously obtain each camera intrinsic parameters and the relative extrinsic parameters between cameras in a single capture process.

Fibonacci Wavelet Collocation Method for Solving Dengue Fever SIR Model

The main focus in this manuscript is to find a numerical solution of a dengue fever disease model by using the Fibonacci wavelet method. The operational matrix of integration has been obtained using Fibonacci wavelets. The proposed method is called Fibonacci wavelet collocation method (FWCM). This biological model has been transformed into a system of nonlinear algebraic equations by using the Fibonacci wavelet collocation scheme. Afterward, this system has been solved by using the Newton–Raphson method. Finally, we provide evidence that our results are better than those obtained by various current approaches, both numerically and graphically, demonstrating the method’s accuracy and efficiency.

Bricks: A high-performance portability layer for computations on block-structured grids

From partial differential equations to the convolutional neural networks in deep learning, to matrix operations in dense linear algebra, computations on structured grids dominate high-performance computing and machine learning. The performance of such computations is key to effective utilization of the billions of US dollar’s worth of GPU-accelerated systems such computations are run on. Concurrently, the end of Moore’s law and Dennard scaling are driving the specialization of compute and memory architectures. This specialization often makes performance brittle (small changes in function can have severe ramifications on performance), non-portable (vendors are increasingly motivated to develop their programming models tailored for their specialized architectures), and not performance portable (even a given computation may perform very differently from one architecture to the next). The mismatch between computations that reference data that is logically neighboring in N-dimensional space but physically distant in memory motivated the creation of Bricks — a novel data-structure transformation for multi-dimensional structured grids that reorders data into small, fixed-sized bricks of contiguously-packed data. Whereas a cache-line naturally captures spatial locality in only one dimension of a structured grid, Bricks can capture spatial locality in three or more dimensions. When coupled with a Python interface, a code-generator, and autotuning, the resultant BrickLib software provides not only raw performance, but also performance portability across multiple CPUs and GPUs, scalability in distributed memory, user productivity, and generality across computational domains. In this paper, we provide an overview of BrickLib and provide a series of vignettes on how it delivers on the aforementioned metrics.

Unsupervised feature selection based on bipartite graph and low-redundant regularization

Unsupervised feature selection (UFS) has attracted increasing attention because of the difficulty and high cost of obtaining data labels. Since the ignorance of redundancy between features and the use of a fixed similarity matrix, current UFS algorithms do not perform well. Moreover, because of the high computational complexity, existing UFS algorithms usually cannot handle large-scale data. A novel UFS method is proposed to solve these problems through bipartite graph learning with low-redundant regularization (BGLR). Firstly, BGLR uses a variance-based de-correlation anchor selection method to ensure that selected anchors can be evenly distributed across classes. Secondly, BGLR builds an adaptive bipartite graph by using selected anchors and original data in the projection space. L2,0-norm constraint is applied to the projection matrix. Thus, the feature subset can be obtained directly. Finally, a regularization term is applied to control the redundancy of selected features. Moreover, BGLR is solved by an iterative method, which can converge quickly. Experiments are performed on public datasets in comparison with some relevant methods. The experimental results demonstrate the BGLR’s effectiveness.

An Effective Matrix Technique for the Numerical Solution of Second Order Differential Equations

In this paper, an effective technique for solving differential equations with initial conditions is presented. The method is based on the use of the Legendre matrix of derivatives defined on the close interval [-1,1]. Properties of the polynomial are outlined and further used to obtain the matrix of derivative which was used in transforming the differential equation into systems of linear and nonlinear algebraic equations. The systems of these algebraic equations were then solved using Gaussian elimination method to determine the unknown parameters required for approximating the solution of the differential equation. The advantage of this technique over other methods is that, it has less computational manipulations and complexities and also its availability for application on both linear and nonlinear second-order initial value problems is impressive. Other advantage of the algorithm is that high accurate approximate solutions are achieved by using a greater number of terms of the Legendre polynomial and once the operational matrix is obtained, it can be used to solve differential equations of higher order by introducing just a little manipulation on the operational matrix. Some existing sample problems from literature were solved and the results were compared to show the validity, simplicity and applicability of the proposed method. The results obtained validate the simplicity and applicability of the method and it also reveals that the method perform better than most existing methods.

Euclidean operator radius inequalities of d-tuple operators and operator matrices

Abstract We study Euclidean operator radius inequalities of d-tuple operators as well as the sum and the product of d-tuple operators. A power inequality for the Euclidean operator radius of d-tuple operators is also studied. Further, we study the Euclidean operator radius inequalities of 2 × 2 operator matrices whose entries are d-tuple operators.

F-Norm-Based Soft LDA Algorithm for Fault Detection in Chemical Production Processes.

In chemical production processes, outliers are inevitable. Many existing feature extraction algorithms are overly sensitive to outliers and excessively focus on secondary features while ignoring the key features in the data. To address this problem, the Frobenius norm based soft linear discriminant analysis algorithm (FBSLA) is proposed in this paper. Specifically, FBSLA uses the Frobenius norm instead of its square as a metric to enhance the robustness of the algorithm. Furthermore, a nonreduced dimensionality projection matrix is introduced to make the training data features more obvious. Additionally, soft constraint is adopted instead of the traditional hard constraint to reduce the sensitivity caused by outliers. To validate the effectiveness of FBSLA, in this paper, experiments are conducted on the Tennessee Eastman Process and the Penicillin Fermentation Process data sets. According to experimental results, FBSLA significantly outperforms other state-of-the-art algorithms in terms of fault detection accuracy.

Computationally-efficient locking-free isogeometric discretizations of geometrically nonlinear Kirchhoff–Love shells

Discretizations based on the Bubnov-Galerkin method and the isoparametric concept suffer from membrane locking when applied to Kirchhoff–Love shell formulations. Membrane locking causes not only smaller displacements than expected, but also large-amplitude spurious oscillations of the membrane forces. Continuous-assumed-strain (CAS) elements were originally introduced to remove membrane locking in quadratic NURBS-based discretizations of linear plane curved Kirchhoff rods (Casquero et al., CMAME, 2022). In this work, we propose CASs and CASns elements to overcome membrane locking in quadratic NURBS-based discretizations of geometrically nonlinear Kirchhoff–Love shells. CASs and CASns elements are interpolation-based assumed-strain locking treatments. The assumed strains have C0 continuity across element boundaries and different components of the membrane strains are interpolated at different interpolation points. CASs elements use the assumed strains to obtain both the physical strains and the virtual strains, which results in a global tangent matrix which is a symmetric matrix. CASns elements use the assumed strains to obtain only the physical strains, which results in a global tangent matrix which is a non-symmetric matrix. To the best of the authors’ knowledge, CASs and CASns elements are the first assumed-strain treatments to effectively overcome membrane locking in quadratic NURBS-based discretizations of geometrically nonlinear Kirchhoff–Love shells while satisfying the following important characteristics for computational efficiency: (1) No additional unknowns are added, (2) No additional systems of algebraic equations need to be solved, (3) The same elements are used to approximate the displacements and the assumed strains, (4) No additional matrix operations such as matrix inversions or matrix multiplications are needed to obtain the stiffness matrix, and (5) The nonzero pattern of the stiffness matrix is preserved. Analogously to the interpolation-based assumed-strain locking treatments for Lagrange polynomials that are widely used in commercial FEA software, the implementation of CASs and CASns elements only requires to modify the subroutine that computes the element residual vector and the element tangent matrix. The benchmark problems show that CASs and CASns elements, using either 2 × 2 or 3 × 3 Gauss–Legendre quadrature points per element, are effective locking treatments since these element types result in more accurate displacements for coarse meshes and excise the spurious oscillations of the membrane forces.

Sparse feature selection using hypergraph Laplacian-based semi-supervised discriminant analysis

Feature selection, as a dimension reduction technique in data mining and pattern recognition, aims to select the most discriminative features and improve the learning performance. With an abundance of unlabeled data readily available across various applications, semi-supervised feature selection has emerged as a promising approach. While most semi-supervised feature selection methods rely on simple graphs to preserve the geometrical structure of data, this approach often fails in capturing the high-order relationships present in many real-world applications. In contrast, hypergraphs offer the ability to encode more complex structures of data beyond what a simple graph can achieve. In this paper, we propose a feature selection method formulated in the trace ratio form, integrating hypergraph Laplacian-based semi-supervised discriminant analysis (SDA) and the mixed convex and non-convex ℓ2,p-norm (0<p≤1) regularization. The proposed trace ratio-based method, called HSDAFS, leverages the discriminative information from labeled data to maximize class separability while also utilizing the hypergraph Laplacian to capture the geometrical structure and high-order relationships within both labeled and unlabeled data. The ℓ2,p-norm regularization in the proposed HSDAFS provides improved sparsity over the ℓ2,1-norm. It ensures that the projection matrix is row-sparse, enabling the effective joint selection of discriminative features across all data. To solve the trace ratio-based HSDAFS method, we convert it into a trace difference method and propose an iterative algorithm. Experiments on several datasets demonstrate that HSDAFS is more effective in selecting the most discriminative features compared to other methods.

Enhancing geotechnical damage detection with deep learning: a convolutional neural network approach.

Most natural disasters result from geodynamic events such as landslides and slope collapse. These failures cause catastrophes that directly impact the environment and cause financial and human losses. Visual inspection is the primary method for detecting failures in geotechnical structures, but on-site visits can be risky due to unstable soil. In addition, the body design and hostile and remote installation conditions make monitoring these structures inviable. When a fast and secure evaluation is required, analysis by computational methods becomes feasible. In this study, a convolutional neural network (CNN) approach to computer vision is applied to identify defects in the surface of geotechnical structures aided by unmanned aerial vehicle (UAV) and mobile devices, aiming to reduce the reliance on human-led on-site inspections. However, studies in computer vision algorithms still need to be explored in this field due to particularities of geotechnical engineering, such as limited public datasets and redundant images. Thus, this study obtained images of surface failure indicators from slopes near a Brazilian national road, assisted by UAV and mobile devices. We then proposed a custom CNN and low complexity model architecture to build a binary classifier image-aided to detect faults in geotechnical surfaces. The model achieved a satisfactory average accuracy rate of 94.26%. An AUC metric score of 0.99 from the receiver operator characteristic (ROC) curve and matrix confusion with a testing dataset show satisfactory results. The results suggest that the capability of the model to distinguish between the classes 'damage' and 'intact' is excellent. It enables the identification of failure indicators. Early failure indicator detection on the surface of slopes can facilitate proper maintenance and alarms and prevent disasters, as the integrity of the soil directly affects the structures built around and above it.

Wavelet Collocation Method for HIV-1/HTLV-I Co-Infection Model Using Hermite Polynomial.

In this study, the dynamic behavior of fractional order co-infection model with human immunodeficiency virus type 1 (HIV-1) and human T-lymphotropic virus type I (HTLV-I) is analyzed using operational matrix of Hermite wavelet collocation method. Also, the uniqueness and existence of solutions are calculated based on the fixed point hypothesis. For the fractional order co-infection model, its positivity and boundedness are demonstrated. Furthermore, different types of Ulam-Hyres stability are also discussed. The numerical solution of the model are obtained by using the operational matrix of the Hermite wavelet approach. This scheme is used to solve the system of nonlinear equationsthat are very fruitful and easy to implement. Additionally, the stability analysis of the numerical scheme is explained. The mathematical model taken in this work incorporates the biological characteristics of both HIV-1 and HTLV-I. After that all the equilibrium points of the fractional order co-infection model are found and their existence conditions are explored with the help of the Caputo derivative. The global stability of all equilibrium points of this model are determined with the help of Lyapunov functions and the LaSalle invariance principle. Convergence analysis is also discussed. Hermite wavelet operational matrix methods are more accurate and convergent than other numerical methods. Lastly, variations in model dynamics are found when examining different fractional order values. These findings will be valuable to biologists in the treatment of HIV-1/HTLV-I.