Matrix Operations Research Articles

Krylov complexity of density matrix operators

Quantifying complexity in quantum systems has witnessed a surge of interest in recent years, with Krylov-based measures such as Krylov complexity (CK) and Spread complexity (CS) gaining prominence. In this study, we investigate their interplay by considering the complexity of states represented by density matrix operators. After setting up the problem, we analyze a handful of analytical and numerical examples spanning generic two-dimensional Hilbert spaces, qubit states, quantum harmonic oscillators, and random matrix theories, uncovering insightful relationships. For generic pure states, our analysis reveals two key findings: (I) a correspondence between moment-generating functions (of Lanczos coefficients) and survival amplitudes, and (II) an early-time equivalence between CK and 2CS. Furthermore, for maximally entangled pure states, we find that the moment-generating function of CK becomes the Spectral Form Factor and, at late-times, CK is simply related to NCS for N ≥ 2 within the N-dimensional Hilbert space. Notably, we confirm that CK = 2CS holds across all times when N = 2. Through the lens of random matrix theories, we also discuss deviations between complexities at intermediate times and highlight subtleties in the averaging approach at the level of the survival amplitude.

A novel collocation method with a coronavirus optimization algorithm for the optimal control of COVID-19: A case study of Wuhan, China

In this study, we develop a numerical optimization approach to address the challenge of optimal control in the spread of COVID-19. We evaluate the impact of various control strategies aimed at reducing the number of exposed and infectious individuals. Our novel approach employs Legendre wavelets, their derivative operational matrix, and a collocation method to transform the COVID-19 transmission optimal control model into a nonlinear programming (NLP) problem. To solve this problem, we employ a coronavirus optimization algorithm (COVIDOA) to determine the optimal control, state variables, and objective value. We investigate three control plans for this highly contagious disease, focusing on individual protection, rapid detection and treatment, detection with delay in treatment, and environmental viral dispersion as time-based control functions. These strategies are applied within an SEIR-type control model specific to COVID-19 in China, designed to mitigate disease spread. Lastly, we analyze the effects of various parameters within the COVID-19 spread model. Our numerical results highlight the significant impact of strategies that minimize the number of exposed and infectious individuals, particularly those related to rapid detection, detection delay, and environmental viral dispersion, in controlling and preventing the transmission of the COVID-19 virus.

Stock Trend Prediction with Machine Learning: Incorporating Inter-Stock Correlation Information through Laplacian Matrix

Predicting stock trends in financial markets is of significant importance to investors and portfolio managers. In addition to a stock’s historical price information, the correlation between that stock and others can also provide valuable information for forecasting future returns. Existing methods often fall short of straightforward and effective capture of the intricate interdependencies between stocks. In this research, we introduce the concept of a Laplacian correlation graph (LOG), designed to explicitly model the correlations in stock price changes as the edges of a graph. After constructing the LOG, we will build a machine learning model, such as a graph attention network (GAT), and incorporate the LOG into the loss term. This innovative loss term is designed to empower the neural network to learn and leverage price correlations among different stocks in a straightforward but effective manner. The advantage of a Laplacian matrix is that matrix operation form is more suitable for current machine learning frameworks, thus achieving high computational efficiency and simpler model representation. Experimental results demonstrate improvements across multiple evaluation metrics using our LOG. Incorporating our LOG into five base machine learning models consistently enhances their predictive performance. Furthermore, backtesting results reveal superior returns and information ratios, underscoring the practical implications of our approach for real-world investment decisions. Our study addresses the limitations of existing methods that miss the correlation between stocks or fail to model correlation in a simple and effective way, and the proposed LOG emerges as a promising tool for stock returns prediction, offering enhanced predictive accuracy and improved investment outcomes.

On Efficient Large Sparse Matrix Chain Multiplication

Sparse matrices are often used to model the interactions among different objects and they are prevalent in many areas including e-commerce, social network, and biology. As one of the fundamental matrix operations, the sparse matrix chain multiplication (SMCM) aims to efficiently multiply a chain of sparse matrices, which has found various real-world applications in areas like network analysis, data mining, and machine learning. The efficiency of SMCM largely hinges on the order of multiplying the matrices, which further relies on the accurate estimation of the sparsity values of intermediate matrices. Existing matrix sparsity estimators often struggle with large sparse matrices, because they suffer from the accuracy issue in both theory and practice. To enable efficient SMCM, in this paper we introduce a novel row-wise sparsity estimator (RS-estimator), a straightforward yet effective estimator that leverages matrix structural properties to achieve efficient, accurate, and theoretically guaranteed sparsity estimation. Based on the RS-estimator, we propose a novel ordering algorithm for determining a good order of efficient SMCM. We further develop an efficient parallel SMCM algorithm by effectively utilizing multiple CPU threads. We have conducted experiments by multiplying various chains of large sparse matrices extracted from five real-world large graph datasets, and the results demonstrate the effectiveness and efficiency of our proposed methods. In particular, our SMCM algorithm is up to three orders of magnitude faster than the state-of-the-art algorithms.

Robust autonomous vehicle control by leveraging multi-stage MPC and quantized CNN in HIL Framework

This study presents a novel path planning and path-following system designed explicitly for real-world driving conditions in autonomous vehicles. The pipeline considers uncertainties in the dynamic parameters of the vehicle, which are common in real-world scenarios. To address these uncertainties, a robust multi-stage model predictive control approach is developed to generate optimal control outputs. A customized convolutional neural network, Nano-FastSCNN, quantized to half-precision, is employed for accurately segmenting left and right lane boundaries. To ensure that our visual planner is robust against adverse weather conditions, we have rigorously evaluated its generalization capabilities under challenging weather types. The centerline of these lanes is then selected as the desired path and transformed into the camera reference frame by leveraging the camera projection matrix. The controller utilizes this reference trajectory to guide the vehicle along the desired trajectory. To demonstrate real-time processing capabilities, the entire pipeline, including the vision system module and controller, is implemented in a hardware-in-the-loop setup utilizing three NVIDIA Jetson devices to demonstrate the scalability of our architecture. Comprehensive evaluations demonstrate that the pipeline achieves an inference speed of 15.6 FPS, even on the less powerful NVIDIA Jetson Nano. Moreover, the system successfully alleviates the detrimental effects of uncertainties and reliably follows the desired path.

Quantum No-signalling Correlations and Non-local Games

We introduce and examine three subclasses of the family of quantum no-signalling (QNS) correlations introduced by Duan and Winter: quantum commuting, quantum and local. We formalise the notion of a universal TRO of a block operator isometry, define an operator system, universal for stochastic operator matrices, and realise it as a quotient of a matrix algebra. We describe the classes of QNS correlations in terms of states on the tensor products of two copies of the universal operator system and specialise the correlation classes and their representations to classical-to-quantum correlations. We study various quantum versions of synchronous no-signalling correlations and show that they possess invariance properties for suitable sets of states. We introduce quantum non-local games as a generalisation of non-local games. We define the operation of quantum game composition and show that the perfect strategies belonging to a certain class are closed under channel composition. We specialise to the case of graph colourings, where we exhibit quantum versions of the orthogonal rank of a graph as the optimal output dimension for which perfect classical-to-quantum strategies of the graph colouring game exist, as well as to non-commutative graph homomorphisms, where we identify quantum versions of non-commutative graph homomorphisms introduced by Stahlke.

A new approach of B-spline wavelets to solve fractional differential equations

This paper presents a groundbreaking method for solving the multi-order fractional differential (M-OFD), both linear and nonlinear, as well as fractional partial differential equations (FPDE)s. This approach involves constructing an operational matrix of fractional derivatives using linear B-spline (LB-S) wavelet functions with perfect subtlety. The new method has two crucial features. Firstly, it simplifies the problem by converting it into a set of algebraic equations, which is a significant advantage. This makes the method highly accurate and reliable. Secondly, it uses thresholding to dramatically reduce the computational workload in linear problems. This leads to lightning-fast and highly efficient problem-solving. The newly developed scheme underwent a thorough examination of its error estimates and convergence, revealing remarkable results in terms of accuracy and efficiency. The analysis provides a comprehensive understanding of the scheme’s performance, highlighting its potential as a dependable and effective method. Based on the findings, it is evident that the proposed method not only delivers exceptional precision but also operates with remarkable efficiency.

The use of Hermite wavelet collocation method for fractional cancer dynamical system

Tumour is a very serious and dangerous health risk, which is defined as a mass or lump of tissue formalized by the aggregation of abnormal cells. In this paper, we have presented a comparative and chaotic investigation of tumour and effector cells using non-integer order tumour immune dynamical model. Also, we have looked at the interactions between different tumour cell populations and immunological composition using a model of a real-world medical research problem. Additionally, stability analysis of the proposed model is established with the help of fixed-point iteration. Further bifurcation diagrams, as well as phase portraits, have been used to study the proposed system numerically and to analyse its behaviour. For numerical simulations, we have used the Hermite wavelet operational matrix and Taufik-Atangana methods using Caputo fractional derivatives. models. The main objective of the paper is to convert nonlinear fractional systems into algebraic equations. Our findings will be valuable to biologists in cancer treatment.

Boubaker operational matrix method for solving fractional weakly singular two-dimensional partial Volterra integral equation

Boubaker operational matrix method for solving fractional weakly singular two-dimensional partial Volterra integral equation

Numerical study of variable order model arising in chemical processes using operational matrix and collocation method

This article introduces the fractional variable order (VO) Gray–Scott model using the notion of VO fractional derivative in the Caputo sense. An efficient numerical method has been designed based on the Vieta–Lucas polynomial and the spectral collocation method for solving this model. The designed technique converts the concerned model into a nonlinear algebraic system of equations, which can be solved by Newton’s iterative method. In this article, we have illustrated the convergence analysis of the approximation and shown that a high order of convergence can be achieved despite a smaller number of approximations. A few numerical results are presented in order to verify the reliability and accuracy of the demonstrated scheme. The results of absolute errors for the considered Gray–Scott model with its exact solution show that the technique is very suitable for finding the solutions to the said kind of complex physical problem.

Inequalities for the Euclidean Operator Radius of n-Tuple Operators and Operator Matrices in Hilbert C∗-Modules

This study takes a detailed look at various inequalities related to the Euclidean operator radius. It examines groups of n-tuple operators, studying how they add up and multiply together. It also uncovers a unique power inequality specific to the Euclidean operator radius. The research broadens its scope to analyze how n-tuple operators, when used as parts of 2×2 operator matrices, illustrate inequalities connected to the Euclidean operator radius. By using the Euclidean numerical radius and Euclidean operator norm for n-tuple operators, the study introduces a range of new inequalities. These inequalities not only set limits for the addition, multiplication, and Euclidean numerical radius of n-tuple operators but also help in establishing inequalities for the Euclidean operator radius. This process involves carefully examining the Euclidean numerical radius inequalities of 2×2 operator matrices with n-tuple operators. Additionally, a new inequality is derived, focusing specifically on the Euclidean operator norm of 2×2 operator matrices. Throughout, the research keeps circling back to the idea of finding and understanding symmetries in linear operators and matrices. The paper highlights the significance of symmetry in mathematics and its impact on various mathematical areas.

Operational Matrix Framework for Energy Balance Analysis for Early Stage Design of Complex Vessels

Considering vital ship systems or distributed ship service systems at the early stage of complex vessels is a challenging task. The recent UCL Network Block Approach aimed to enable ship designer to address ship systems design synthesis simultaneously as a logical network using MATLAB with a CPLEX Toolbox in MATLAB and representative piping, cabling, and trunking routings on the physical description of the ship using a proven CASD tool PARAMARINESURFCON. This was possible due to adopting a set of frameworks, as part of this comprehensive approach. The paper presents one of the frameworks: the Operational Matrix, to formulate distributed ship service systems network in the early stage design of complex vessels. The application of the framework could take on many forms and can be manipulated to suit a specific distributed ship service system’s design. In this paper, a tutorial is given, leading from the simplest application of the Operational Matrix Framework to an example of a complex Operational Matrix application for the 3D multiplex submarine systems problem. The use of the proposed Operational Matrix Framework is shown to reveal the relationship between objective functions, constraints, bounds, and solutions of that linear programming formulation. The Operational Matrix Framework can enable the solvers (CPLEX Toolbox in MATLAB) to be very efficient in advancing early stage ship design applications. The Framework could be developed further for investigating the analysis of energy balances for new systems to achieve net zero energy demands for future naval vessels.

An operational matrix strategy for time fractional Fokker-Planck equation in an unbounded space domain

In this study, the Caputo fractional derivative is used to define time fractional Fokker-Planck equation in an unbounded domain. To solve this equation, the Jacobi polynomials together with the tanh-Jacobi functions are employed. The operational matrices of the classical and fractional derivatives of these basis functions are obtained to use them in constructing a numerical method for the expressed equation. In the proposed method, the introduced basis functions are used simultaneously to approximate the equation’s unknown solution. More precisely, the shifted Jacobi polynomials are applied to approximate the solution in the temporal direction and the tanh-Jacobi functions are utilized to approximate the solution in the spatial direction. By substitute the expressed approximation into the equation and employing the introduced operational matrix, solving the problem under consideration transforms into solving an algebraic system of equations, which can be solved easily. The accuracy and efficiency of the presented method are investigated numerically by solving some numerical examples. The reported results confirms the high accuracy of the established method.

A holistic exploration of the optimal control strategies on an enhanced mathematical model for the co-infection of HIV/AIDS and varicella-zoster

Because of its high contagiousness and correlation with HIV/AIDS complaints, the virus that causes varicella-zoster virus and its interactions have major consequences for a considerable portion of people worldwide. The primary aim of this work is to suggest and examine optimal control methods for managing the transmission dynamics of HIV/AIDS and Varicella-Zoster co-infection, using an integer model approach. The mathematical analyses of the proposed integer order model places particular emphases on the boundedness and non-negativity of the model solutions, scrutinizing equilibrium points, determining the models basic reproduction ratios (the models basic reproduction numbers) through the next-generation matrix operator method, and assessing the model equilibrium points existences and stabilities in local approach by considering the local stability conditions of Routh and Hurwitz. Additionally, it incorporates an optimal control framework to enhance our understanding of the dynamics involved in the spreading of HIV/AIDS and Varicella-Zoster co-infection within a considered population. This entails determining preventative measures that can be deliberately put into place to lessen the effects of these co-infections. The solutions of the HIV/AIDS and Varicella-Zoster co-infection model converges to the co-infection endemic equilibrium point whenever the associated basic reproduction number is greater than unity, as verified by numerical simulation results. Including optimal management gives the research an innovative viewpoint and helps identify tactical ways to mitigate the negative effects of this co-infection on the public health. The results highlight how crucial it is to address these complex structures in order to protect and improve public health outcomes. Implementing the proposed protection measures and treatment measures simultaneously has most effective result to minimize and eliminate the HIV/AIDS and Varicella-Zoster co-infection disease throughout the population.

Accelerating Genome- and Phenome-Wide Association Studies using GPUs - A case study using data from the Million Veteran Program.

The expansion of biobanks has significantly propelled genomic discoveries yet the sheer scale of data within these repositories poses formidable computational hurdles, particularly in handling extensive matrix operations required by prevailing statistical frameworks. In this work, we introduce computational optimizations to the SAIGE (Scalable and Accurate Implementation of Generalized Mixed Model) algorithm, notably employing a GPU-based distributed computing approach to tackle these challenges. We applied these optimizations to conduct a large-scale genome-wide association study (GWAS) across 2,068 phenotypes derived from electronic health records of 635,969 diverse participants from the Veterans Affairs (VA) Million Veteran Program (MVP). Our strategies enabled scaling up the analysis to over 6,000 nodes on the Department of Energy (DOE) Oak Ridge Leadership Computing Facility (OLCF) Summit High-Performance Computer (HPC), resulting in a 20-fold acceleration compared to the baseline model. We also provide a Docker container with our optimizations that was successfully used on multiple cloud infrastructures on UK Biobank and All of Us datasets where we showed significant time and cost benefits over the baseline SAIGE model.

CNUCTRAN: A program for computing final nuclide concentrations using a direct simulation approach

It is essential to precisely determine the evolving concentrations of radioactive nuclides within transmutation problems. It is also a crucial aspect of nuclear physics with widespread applications in nuclear waste management and energy production. This paper introduces CNUCTRAN, a novel computer program that employs a probabilistic approach to estimate nuclide concentrations in transmutation problems. CNUCTRAN directly simulates nuclei transformations arising from various nuclear reactions, diverging from the traditional deterministic methods that solve the Bateman equation using matrix exponential approximation. This approach effectively addresses numerical challenges associated with solving the Bateman equations, therefore, circumventing the need for matrix exponential approximations that risk producing nonphysical concentrations. Our sample calculations using CNUCTRAN shows that the concentration predictions of CNUCTRAN have a relative error of less than 0.001% compared to the state-of-the-art method, CRAM, in different test cases. This makes CNUCTRAN a valuable alternative tool for transmutation analysis. Program summaryProgram Title:CNUCTRANCPC Library link to program files:https://doi.org/10.17632/b484w2vx52.1Developer's repository link:https://github.com/rabieomar92/cnuctran/releasesLicensing provisions: MITProgramming language: C++Nature of problem:CNUCTRAN simulates the transmutation of various nuclides such as decays, fissions, and neutron induced reactions using a direct simulation approach. It has the capability of predicting the final concentration of a large system of nuclides altogether after a specified time step, tf.Solution method:CNUCTRAN works based on the novel probabilistic method such that it does not compute the final nuclide concentrations by solving Bateman equations. Instead, it statistically tracks nuclide transformations into one another in a transmutation problem. The technique encapsulates various possible nuclide transformations into a sparse transfer matrix, T, whose elements are made up of various nuclear reaction probabilities. Next, T serves as a matrix operator acting on the initial nuclide concentrations, y(0), producing the final nuclide concentrations, y.

Time-Varying Function Matrix Projection Synchronization of Caputo Fractional-Order Uncertain Memristive Neural Networks with Multiple Delays via Mixed Open Loop Feedback Control and Impulsive Control

This paper shows solicitude for the generalized projective synchronization of Caputo fractional-order uncertain memristive neural networks (FOUMNNs) with multiple delays. By extending the constant scale factor to the time-varying function matrix, we establish an extraordinary synchronization mode called time-varying function matrix projection synchronization (TFMPS), which is a generalized version of traditional matrix projection synchronization, modified projection synchronization, complete synchronization, and anti-synchronization. To achieve the goal of TFMPS, we design a novel mixed controller including the open loop feedback control and impulsive control, which employs the state information in the time-delayed interval and the sampling information at the impulse instants. It has a prominent advantage that impulse intervals are not restricted by time delays. To establish the connection between the error system and the auxiliary system, a generalized fractional-order comparison theorem with time-varying coefficients and impulses is established. Applying the stability theory, the comparison theorem, and the Laplace transform, new synchronization criteria of FOUMNNs are acquired under the mixed impulsive control schemes, and the derived synchronization theorem and corollary can effectively expand the correlative synchronization achievements of fractional-order systems.

Numerical computation of fractional Bloch equation by using Jacobi operational matrix

In this work, we present a numerical scheme based on the operational matrix of fractional Caputo-Fabrizio (CF) integration for handling fractional Bloch equation (FBE) in nuclear magnetic resonance (NMR). The understanding of Bloch equation provides us a fundamental framework for describing magnetic resonance phenomena, facilitating breakthrough in diverse fields such as medical diagnostics, quantum computing and materials characterization. The non-integer order derivative and integration are presented in the Caputo-Fabrizio sense. To construct the operational matrix, Jacobi polynomial is used as a basis. The fractional Bloch equation is transformed into a set of algebraic equations by using the operational matrix. In order to examine the fractional order problem, we obtain an approximate solution for FBE and present the numerical results in graphical and tabular forms.

Improving Measurement Accuracy of Deep Hole Measurement Instruments through Perspective Transformation.

Deep hole measurement is a crucial step in both deep hole machining and deep hole maintenance. Single-camera vision presents promising prospects in deep hole measurement due to its simple structure and low-cost advantages. However, the measurement error caused by the heating of the imaging sensor makes it difficult to achieve the ideal measurement accuracy. To compensate for measurement errors induced by imaging sensor heating, this study proposes an error compensation method for laser and vision-based deep hole measurement instruments. This method predicts the pixel displacement of the entire field of view using the pixel displacement of fixed targets within the camera's field of view and compensates for measurement errors through a perspective transformation. Theoretical analysis indicates that the perspective projection matrix changes due to the heating of the imaging sensor, which causes the thermally induced measurement error of the camera. By analyzing the displacement of the fixed target point, it is possible to monitor changes in the perspective projection matrix and thus compensate for camera measurement errors. In compensation experiments, using target displacement effectively predicts pixel drift in the pixel coordinate system. After compensation, the pixel error was suppressed from 1.99 pixels to 0.393 pixels. Repetitive measurement tests of the deep hole measurement instrument validate the practicality and reliability of compensating for thermal-induced errors using perspective transformation.

Correction: Solution of linear and nonlinear singular value problems using operational matrix of integration of Fibonacci wavelets

Correction: Solution of linear and nonlinear singular value problems using operational matrix of integration of Fibonacci wavelets