Matrix Operations Research Articles

The Galerkin Bell method to solve the fractional optimal control problems with inequality constraints

In this manuscript, we adopt the Caputo fractional derivative approach and employ the Galerkin-Bell method to tackle fractional optimal control problems (FOCPs) with equality and inequality constraints in multi-dimensional settings. We derive the Riemann–Liouville (RL) operational matrix for Bell polynomials to facilitate our analysis. By leveraging these matrices and utilizing the Galerkin method, we transform the FOCP into a system of algebraic equations that can be readily solved. We delve into the Bell polynomials’ convergence analysis and error estimation and introduce a residual correction procedure for error estimation. To assess the effectiveness and applicability of our proposed method, we conduct experiments on four different examples and compare our results with those previously reported in the literature. Our findings reveal that the obtained results are highly satisfactory, and in some instances, we achieve the exact solution.

Nonlinear model updating of the rotor-bearing system by multi-harmonic balance method and analytical sensitivity derivation

Responses of the bearing system are usually predicted or tested, to analyze its nonlinear dynamic characteristics. However, responses predicted from the bearing model constructed by Hertz contact theory usually show differences with their tested counterparts, denoting the improper setting of the nonlinear contact parameters. It is compulsory to reduce such differences and obtain a precise bearing model using nonlinear model updating techniques. In this paper, a novel nonlinear model updating framework is proposed following the Multi-Harmonic Balance Method (MHBM) and the analytical sensitivity analysis, for updating of the rotor-bearing system using the nonlinear responses in the frequency domain. With the derivation of bearing response sensitivities to the nonlinear contact parameters, increments of the nonlinear parameters can be ascertained by the sensitivity inverse and updating of the nonlinear bearing model can be implemented in an iterative way, until a precise bearing model with the response differences reduced to minima is obtained. Firstly, a novel MHBM model is derived for the bearing system, which is fully described by matrix operations and used to further deduce the analytical sensitivity formulations of the bearing responses to the design parameters. Afterwards, the bearing responses and sensitivities in the frequency domain are predicted by the continuation strategy. Finally, model updating is conducted for the bearing system using the traditional inverse-sensitivity framework. Two cases are used to demonstrate the feasibility of the method. The first case is to update bearing parameters of contact stiffness, clearance and eccentricity of a 2-DOF bearing system in order to match the predicted bearing responses with the measured ones in strong nonlinearity with the multivalued phenomenon. The next is to update the Hertz contact parameters of the left and right bearings, together with the disc eccentricity, of a Finite Element (FE) rotor-bearing system. These results show the validity and superiority of the proposed method.

Towards a unified framework for graph-based multi-view clustering

Recently, clustering data collected from various sources has become a hot topic in real-world applications. The most common methods for multi-view clustering can be divided into several categories: Spectral clustering algorithms, subspace multi-view clustering algorithms, matrix factorization approaches, and kernel methods. Despite the high performance of these methods, they directly fuse all similarity matrices of all views and separate the affinity learning process from the multiview clustering process. The performance of these algorithms can be affected by noisy affinity matrices. To overcome this drawback, this paper presents a novel method called One Step Multi-view Clustering via Consensus Graph Learning and Nonnegative Embedding (OSMGNE). Instead of directly merging the similarity matrices of different views, which may contain noise, a step of learning a consensus similarity matrix is performed. This step forces the similarity matrices of different views to be too similar, which eliminates the problem of noisy data. Moreover, the use of the nonnegative embedding matrix (soft cluster assignment matrix makes it possible to directly obtain the final clustering result without any extra step. The proposed method can solve five subtasks simultaneously. It jointly estimates the similarity matrix of all views, the similarity matrix of each view, the corresponding spectral projection matrix, the unified clustering indicator matrix, and automatically gives the weight of each view without the use of hyper-parameters. In addition, another version of our method is also studied in this paper. This method differs from the first one by using a consensus spectral projection matrix and a consensus Laplacian matrix over all views. An iterative algorithm is proposed to solve the optimization problem of these two methods. The two proposed methods are tested on several real datasets, which prove their superiority.

Operational matrix based numerical scheme for the solution of time fractional diffusion equations

Operational matrix based numerical scheme for the solution of time fractional diffusion equations

Massively parallel implementation of iterative eigensolvers in large-scale plane-wave density functional theory

The Kohn-sham density functional theory (DFT) is a powerful method to describe the electronic structures of molecules and solids in condensed matter physics, computational chemistry and materials science. However, large and accurate DFT calculations within plane waves process a cubic-scaling computational complexity, which is usually limited by expensive computation and communication costs. The rapid development of high performance computing (HPC) on leadership supercomputers brings new opportunities for developing plane-wave DFT calculations for large-scale systems. Here, we implement parallel iterative eigensolvers in large-scale plane-wave DFT calculations, including Davidson, locally optimal block preconditioned conjugate gradient (LOBPCG), projected preconditioned conjugate gradient (PPCG) and the Chebyshev subspace iteration (CheFSI) algorithms, and analyze the performance of these algorithms in massively parallel plane-wave computing tasks. We adopt a two-level parallelization strategy that combines the message passing interface (MPI) with open multi-processing (OpenMP) parallel programming to handle data exchange and matrix operations in the construction and diagonalization of large-scale Hamiltonian matrix within plane waves. Numerical results illustrate that these iterative eigensolvers can scale up to 42,592 processing cores with high peak performance of 30% on leadship supercomputers to study the electronic structures of bulk silicon systems containing 10,648 atoms. Program summaryProgram Title: Plane wave density functional theory (PWDFT)CPC Library link to program files:https://doi.org/10.17632/c8v2mx5vn4.1Developer's repository link:https://bitbucket.org/berkeleylab/scalesLicensing provisions: BSD 3-clauseProgramming language: C++Nature of problem: PWDFT is used for electronic structure calculations based on Kohn-Sham density functional theory. The key challenge to address is a constrained energy minimization problem, which can also be formulated as a nonlinear eigenvalue problem. MPI/OpenMP-based approaches are employed to provide multi-core acceleration for the study of the chemical and material properties of larger-scale molecules and solids.Solution method: PWDFT implements self-consistent field (SCF) iterations and direct constrained minimization algorithms with various acceleration strategies. It is written in C++ and offers parallel acceleration based on MPI/OpenMP.

Symplectic solutions of the plane annular sectors in micropolar elasticity

The symplectic approach is utilized to derive the solutions of the plane annular sector in micropolar elasticity. According to the Hellinger-Reissner variational principle, the Hamiltonian canonical equations are obtained for the plane problem in an annular sectorial region. Through the method of separation of variables, we obtain an eigenvalue equation of the Hamiltonian operator matrix which is a system of linear differential equations with variable coefficients. By the method of series solutions, its general solutions can be expressed as the linear combination of Bessel functions and power functions. The corresponding eigensolutions for three types of homogeneous boundary conditions are derived. According to the adjoint symplectic orthogonality of the eigensolutions, the solution of the micropolar plane annular sector is expressed as a series expansion of eigensolutions. Numerical analyses are presented to demonstrate the validity of the proposed method and confirm its ability to accurately capture the size effect.

Higher‐order inverse mass matrices for the explicit transient analysis of heterogeneous solids

AbstractNew methods are presented for the direct computation of higher‐order inverse mass matrices (also called reciprocal mass matrices) that are used for explicit transient finite element analysis. The motivation of this work lies in the need of having appropriate sparse inverse mass matrices, which present the same structure as the consistent mass matrix, preserve the total mass, predict suitable frequency spectrum and dictate sufficiently large critical time step sizes. For an efficient evaluation of the reciprocal mass matrix, the projection matrix should be diagonal. This condition can be satisfied by adopting dual shape functions for the momentum field, generated from the same shape functions used for the displacement field. A theoretically consistent derivation of the inverse mass matrix is based on the three‐field Hamilton principle and requires the projection matrix to be evaluated from the integral of these shape functions. Unfortunately, for higher‐order FE shape functions and serendipity FE elements, the projection matrix is not positive definitive and can not be employed. Therefore, we study several lumping procedures for higher order reciprocal mass matrices considering their effect on total‐mass preserving, frequency spectra and accuracy in explicit transient simulations. The article closes with several numerical examples showing suitability of the direct inverse mass matrix in dynamics.

Robotis Can Create Numerical Inverse Kinematic Problems Because Continuously Replace Humans

Robotic process automation is an smart way to manage work across those machines which AI can easily be categorize. Robot Automation is a sequence of steps that gives result in a meaningful action without interference of human being. robotics, design, construction, and use of machines (robots) to perform tasks done traditionally by human beings. Robots are widely used in such industries as automobile manufacture to perform simple repetitive tasks, and in industries where work must be performed in environments hazardous to humans. Medical robotics is an interdisciplinary field that focuses on developing electromechanical devices for clinical applications. The goal of this field is to enable new medical techniques by providing new capabilities to the physician or by providing assistance during surgical procedures. robotics is revolutionizing the future of work in a variety of industries. From manufacturing and logistics to healthcare and education, robots are improving efficiency, productivity, and safety. The application of the inverse kinematics method will herald a new approach in the robot nondestructive testing industry. Meanwhile, complicated matrix operations greatly limit traditional inverse kinematics methods.

Verification and identification of the speech signal using algebraic analysis and DWT

We presented a flowchart, of a new approach that relies on the mathematical operations of matrices and to verify and identifies the speaker, this is a new method difficult to detect by attackers, and this is due to the difficulty to determine its base matrix used in the speaker verification and identification process, Although there are many artificial intelligence programs that can imitate voices and generating false conversations, which poses security and hacking problems.

Music curriculum integration and reconstruction model based on advanced music waveform iterative reconstruction algorithm

To improve the effect of intelligent teaching in music classrooms, this paper combines the advanced music waveform iterative reconstruction algorithm to analyze the integration and reconstruction of the music curriculum. Aiming at the problem that the projection matrix occupies a large space and takes a long time to calculate in iterative reconstruction, a fast and real-time incremental method for generating a music wave matrix is proposed. The improved method avoids the judgment and comparison calculations performed by the incremental method when calculating the length and number of each voxel that the ray passes through. The research results show that the music curriculum integration and reconstruction model based on the advanced music waveform iterative reconstruction algorithm can effectively improve the teaching effect of modern music classrooms.

Comparison of predictors under constrained general linear model and its future observations

. This study deals with some basic inference problems about future observations in a general linear model (GLM) with linear parameter constraints, known as a constrained general linear model (CGLM). Combining the CGLM and its future observations, the author turns the model into a reparameterized form. Using some quadratic matrix optimization methods, the author derives analytical formulas for calculating the best linear unbiased predictors (BLUPs) of all unknown parameter matrices under a CGLM and its future observations. In particular, the author next gives a comprehensive search on the comparison of dispersion matrices of BLUPs of unknown vectors by establishing various equalities and inequalities for dispersion matrices of BLUPs under the model by using elementary block matrix operations and some formulas of rank and inertia of block matrices.

On normal complements

Although the theory of completion has a long history, there are not many results related to completing operator matrices to a normal operator. In this paper, we consider the problem of completion of the upper-triangular operator matrix [A?0B⁎] to normality, where A∈B(H) and B∈B(K) are called normal complements. We give different characterizations of normal complements and explore their joint spectral properties. We also investigate the similarity between A, B, and MC with respect to some essential properties such as regularity, Fredholmness, and others. Finally, we consider the self-duality property of Aluthge and Duggal transforms of hyponormal and semi-hyponormal operators.

Mittag-Leffler wavelets and their applications for solving fractional optimal control problems

Herein, we design a new scheme for finding approximate solutions to fractional optimal control problems (OCPs) with and without delay. In this strategy, we introduce Mittag-Leffler wavelet functions and develop a new Riemann–Liouville fractional integral operator for these functions utilizing the hypergeometric function. The properties of the operational matrix have reflected well in the process of the numerical method and affect the accuracy of the proposed method directly. Employing the Riemann–Liouville fractional integral operator, delay operational matrix, and Galerkin method, the considered problems lead to systems of algebraic equations. An error analysis is proposed. Finally, some illustrative numerical tests are given to show the precision and validity of the suggested technique. The proposed method is very efficient for solving the OCPs with delay and without delay, and gives very accurate results.

Culturo-Behavioral Contributions to a Sustainable Market: The Interplay of Producers’ and Consumers’ Practices

Sustainability is a relevant and pressing theme in which concepts and tools of cultural systems analysis should be applied to a greater extent. The misalignment of the incentive system for producers and consumers raises the issue of sustainable production and consumption of goods and services. A change of paradigm may be needed for shifting defective practices into better informed choices that are consistent with environmental targets and avoid the overexploitation of our planet’s resources. Previous efforts include the 2010 special issue of The Behavior Analyst and the Matrix Project, which advocated a greater involvement of behavior analysts and scientists in sustainability issues. In this conceptual analysis, we continue this effort by analyzing and discussing some of the contingencies underlying: i. manufacturers’ drive for offering environmentally friendly goods; ii. consumers’ bounded choices; and iii. their interaction in the market for achieving sustainable goals. We provide examples of virtuous and defective practices and relate to the Green Shift in Norway. The potential impact of this work consists of informing and influencing policymakers and advocacy organizations as agents of change. Their work is key to shape producers’ and consumers’ preferences for implementing sustainable practices and achieving common environmental goals.

A study on Chlamydia transmission in United States through the Haar wavelet technique

In this study, a novel method known as the Haar wavelet collocation method (HWCM) is used to analyze the mathematical model of Chlamydia transmission in the United States. We use dependent variables with various parameters, such as birth rate, mortality rate, and recovery rate, etc., to explore the nature of the Chlamydia disease in susceptible unvaccinated individuals, susceptible vaccinated individuals, infected, treated, and recovered individuals. The ordinary differential equations (ODEs) in this model are coupled nonlinear. In this method, the Chlamydia model is converted into a system of non-linear algebraic equations using properties of the operational matrix of Haar wavelets. Later, the Newton–Raphson approach is used to extract the unknown coefficients. Mathematica software has been used for all calculations. Tables and graphs contain tabulated results of calculations. As a result, it can be seen that the current approach is more precise than those used in the literature. Also, we discuss the effect of different parameters on susceptible unvaccinated individuals, susceptible vaccinated individuals, infected, treated, and recovered individuals through graphs.

Research on collaborative development mechanism of multiple stakeholders in medical institutions based on rough set

Reducing doctor-patient conflict is an important part of coordinating doctor-patient disputes and easing doctor-patient relationship, which is conducive to building a harmonious medical environment and promoting the healthy development of medical undertakings. This paper constructs a multi-decision-maker mixed conflict model based on rough set theory, puts forward the matrix operation expression of the conflict degree theory in the Pawlak model, and gives a more objective and scientific evaluation function. Combined with hot issues of doctor-patient conflict, the proposed multi-decision-maker mixed conflict model is applied to doctor-patient conflict, examines the doctor-patient relationship in the medical institution system from multiple internal perspectives, and calculates feasible solutions in the conflict system. The results show that high medical quality, high standardize medication, high institutional efficiency, high staff efficiency, high hospital benefits, high hospital revenue, medium employee development, medium equipment development, or high medical quality, high standardize medication, high institutional efficiency, medium staff efficiency, medium hospital benefits, high hospital revenue, high employee development, and high equipment development are important conditions for building a harmonious medical environment and reducing doctor-patient conflicts.

Application of matrix in signal processing

Signal processing, a foundational discipline in modern technology, encompasses a diverse array of applications, ranging from audio and image processing to communication systems and medical imaging. This review investigates how matrix-based techniques are widely used to advance signal processing methodologies. In order to discretize continuous-time signals for digital processing, which occurs in the first section of the paper, matrices play a crucial role in signal sampling. A key principle, the Nyquist-Shannon Sampling Theorem, directs appropriate sampling rates to prevent aliasing, with matrices permitting effective signal representation. The effectiveness of matrix-based filtering methods for frequency modulation and noise reduction, such as convolution and correlation, is then investigated. By utilising matrix operations, these methods enable real-time signal processing. The Fourier Transform and Wavelet Transform are also featured in matrix-driven signal transformation, providing insights into frequency analysis and non-stationary signal characterization. By reducing noise components, matrix-based approaches, particularly Singular Value Decomposition (SVD) denoising, are essential for improving signal quality. Additionally, image compression employs SVD. Matrix-based compressive sensing revolutionises signal recovery from sparse data and results in data-efficient reconstruction. Signal processing has been transformed by matrix-based approaches, which have enabled previously unheard-of levels of efficiency, accuracy, and adaptability. The review highlights their significant influence on several signal processing fields.

MxPL: A Programming Language for Matrix-Related Operations

It is important to establish solid mathematical knowledge throughout the education period. Matrices are important mathematical objects commonly used in many diverse disciplines, including mathematics, engineering, and science. Most problems encountered in such disciplines are represented by mathematical models with various types of matrices, and solved through some applications of matrix algebra. Although simple or advanced operations of matrices can be performed by using modern programming languages, it usually results in a large fragment of code with a low level of readability due to a complicated sequence of control statements. On the other hand, special-purpose languages handle these operations via library functions, presenting poor integration with other programming environments, and less programming flexibility and practice. This paper addresses the design and development of a programming language, called MxPL, which supports matrix-related mathematics with the provision of some basic structures and functions. Firstly, a grammar that is compatible with the usual notations of matrices is constructed and the parser is produced. Then, the code verifier and interpreter for MxPL programs are implemented. Some code examples are presented to illustrate the performance of several sophisticated matrix operations. The comparative analysis of MxPL is conducted with modern programming languages based on language features.

MLFLHMDA: predicting human microbe-disease association based on multi-view latent feature learning.

A growing body of research indicates that microorganisms play a crucial role in human health. Imbalances in microbial communities are closely linked to human diseases, and identifying potential relationships between microbes and diseases can help elucidate the pathogenesis of diseases. However, traditional methods based on biological or clinical experiments are costly, so the use of computational models to predict potential microbe-disease associations is of great importance. In this paper, we present a novel computational model called MLFLHMDA, which is based on a Multi-View Latent Feature Learning approach to predict Human potential Microbe-Disease Associations. Specifically, we compute Gaussian interaction profile kernel similarity between diseases and microbes based on the known microbe-disease associations from the Human Microbe-Disease Association Database and perform a preprocessing step on the resulting microbe-disease association matrix, namely, weighting K nearest known neighbors (WKNKN) to reduce the sparsity of the microbe-disease association matrix. To obtain unobserved associations in the microbe and disease views, we extract different latent features based on the geometrical structure of microbes and diseases, and project multi-modal latent features into a common subspace. Next, we introduce graph regularization to preserve the local manifold structure of Gaussian interaction profile kernel similarity and add -norms to the projection matrix to ensure the interpretability and sparsity of the model. The AUC values for global leave-one-out cross-validation and 5-fold cross validation implemented by MLFLHMDA are 0.9165 and 0.8942+/-0.0041, respectively, which perform better than other existing methods. In addition, case studies of different diseases have demonstrated the superiority of the predictive power of MLFLHMDA. The source code of our model and the data are available on https://github.com/LiangzheZhang/MLFLHMDA_master.

State‐dependent indirect hp‐pseudospectral method for rigid spacecraft attitude optimal control problem

AbstractThe state‐dependent indirect ‐pseudospectral (SDIHP) method for rigid spacecraft attitude control is presented in this paper. The optimal controller is obtained by the SDIHP method through classifying the attitude optimal control problem as a nonlinear quadratic optimal control problem. In the proposed control scheme, successive state‐dependent coefficient (SDC) parameterization is initially applied to the spacecraft dynamics to derive an available linear system to decrease the calculation cost and speed. Then, the indirect ‐adaptive spectral discretization is developed to convert the resultant optimal control problem into a sequence of linear equations, which can improve the global computational efficiency and guarantee the solving precision. Hence, the optimal control can be efficiently solved using basic matrix operations under the necessary collocation points based on the ‐adaptive features. Considering the inertia uncertainties and external disturbances, a state feedback controller is generated by the receding horizon control (RHC) framework to improve the robustness. Simulation results are included to show the effectiveness and performance of the suggested optimal method.