Linear Algebra Techniques Research Articles

On Saturation of Zero-Dimensional Ideals

In this paper, we explore the effective computation as well as arithmetic complexity of saturating a zero-dimensional polynomial ideal with respect to a given polynomial when a Gröbner basis of the ideal is provided. For this purpose, we first focus on studying in more detail a lesser-known algorithm due to Traverso for updating a zero-dimensional Gröbner basis by adding a new polynomial. Based on this algorithm and by applying linear algebra techniques, we propose two new algorithms for calculating the saturation of a zero-dimensional ideal with respect to a polynomial. These algorithms have been implemented in Maple and their efficiency compared to the classical method (using the Rabinowitsch trick) for computing ideal saturation is evaluated through a set of benchmark polynomial ideals.

Efficient Implementation of Interior-Point Methods for Quantum Relative Entropy

Quantum relative entropy (QRE) programming is a recently popular and challenging class of convex optimization problems with significant applications in quantum computing and quantum information theory. We are interested in modern interior-point (IP) methods based on optimal self-concordant barriers for the QRE cone. A range of theoretical and numerical challenges associated with such barrier functions and the QRE cones have hindered the scalability of IP methods. To address these challenges, we propose a series of numerical and linear algebraic techniques and heuristics aimed at enhancing the efficiency of gradient and Hessian computations for the self-concordant barrier function, solving linear systems, and performing matrix-vector products. We also introduce and deliberate about some interesting concepts related to QRE such as symmetric quantum relative entropy. We design a two-phase method for performing facial reduction that can significantly improve the performance of QRE programming. Our new techniques have been implemented in the latest version (DDS 2.2) of the software package Domain-Driven Solver (DDS). In addition to handling QRE constraints, DDS accepts any combination of several other conic and nonconic convex constraints. Our comprehensive numerical experiments encompass several parts, including (1) a comparison of DDS 2.2 with Hypatia for the nearest correlation matrix problem, (2) using DDS 2.2 for combining QRE constraints with various other constraint types, and (3) calculating the key rate for quantum key distribution (QKD) channels and presenting results for several QKD protocols. History: Accepted by Giacomo Nannicini, Area Editor for Quantum Computing and Operations Research. Accepted for Special Issue. Funding: This work was supported by the National Science Foundation [Grant CMMI-2347120] and Discovery Grants from the Natural Sciences and Engineering Research Council of Canada. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2024.0570 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2024.0570 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Effective resistance, random walk and algorithms

Efficiently determining the effective resistance of large-scale electrical networks is crucial for optimizing energy distribution and assessing network robustness. In this paper, we propose a novel approach that combines random walk simulations and advanced linear algebra techniques to compute the effective resistance of complex electrical networks. Our method leverages the concept of random walks to simulate the flow of current through the network, capturing its behavior under various conditions. Subsequently, we employ graph Laplacian-based linear algebra algorithms to analyze the resulting data, enabling accurate computation of the effective resistance. In our model, we aim to deploy these combined algorithm to develop a novel method to deal with such methods.

Singular value decomposition: A useful technique for image denoising

A key function of image processing is picture denoising, which improves the quality of images by eliminating extraneous noise while keeping crucial information in tact. Singular Value Decomposition (SVD) is a linear algebraic technique that reduces the original datas complexity and scale by breaking down the matrices and extracting the important information. With the power of decomposition which utilizes the non-local self-similarity property of an image to achieve satisfactory denoising performance, SVD denoising has become a potent tool in image processing. In this paper, SVD is outlined and its working, applications, and challenges as a denoising technique in image denoising are discussed. The author discovered that Singular Value Decomposition can be a significant factor in image denoising by applying it to the image. As a result, Singular Value Decomposition could be thought as a helpful image denoising approach in the image processing sequence that will raise the images Peak Signal-to-Noise Ration (PSNR) and improve the quality of the image.

Model Predictive Control (MPC) of a Countercurrent Flow Plate Heat Exchanger in a Virtual Environment.

This research proposes advanced model-based control strategies for a countercurrent flow plate heat exchanger in a virtual environment. A virtual environment with visual and auditory effects is designed, which requires a mathematical model describing the real dynamics of the process; this allows parallel fluid movement in different directions with hot and cold temperatures at the outlet, incorporating control monitoring interfaces as communication links between the virtual heat exchanger and control applications. A multivariable and non-linear process like the plate and countercurrent flow heat exchanger requires analysis in the controller design; therefore, this work proposes and compares two control strategies to identify the best-performing one. The first controller is based on the inverse model of the plant, with linear algebra techniques and numerical methods; the second controller is a model predictive control (MPC), which presents optimal control actions that minimize the steady-state errors and aggressive variations in the actuators, respecting the temperature constraints and the operating limits, incorporating a predictive model of the plant. The controllers are tested for different setpoint changes and disturbances, determining that they are not overshot and that the MPC controller has the shortest settling time and lowest steady-state error.

Quantum Computing Techniques for Numerical Linear Algebra in Computational Mathematics

Quantum computing is a new and exciting area of computational mathematics that has the ability to solve very hard problems that traditional computing methods have not been able to solve for a long time. This abstract goes into detail about how quantum computing can be used in numerical linear algebra, which is an important part of computational mathematics that is used in many fields, such as science, engineering, and data analysis. The idea behind quantum computing is new. It uses quantum bits (qubits) and quantum gates to do calculations in a very different way, based on the rules of quantum physics. Because of this change in thinking, quantum computers might be able to solve some numerical linear algebra problems a lot faster than traditional computers. The outline then talks about some of the most important quantum computing methods and algorithms that are made for numerical linear algebra problems. There are quantum versions of standard algorithms like matrix multiplication, Gaussian elimination, and singular value decomposition. There are also new algorithms that are intended to use quantum parallelism and interference to speed up computations. For situations where full-scale quantum gear is not yet available, quantum-inspired methods are also talked about. These blend conventional and quantum techniques. The abstract talks about the difficulties and chances of using quantum algorithms for numerical linear algebra on quantum hardware systems that are available now and in the near future. It talks about things like error correction, qubit coherence times, and making efficient quantum circuit implementations. Lastly, the abstract talks about how quantum computing might change computational mathematics. It imagines that advances in quantum computing could help solve numerical linear algebra problems that were previously impossible to solve. This could have huge effects on many fields, from machine learning and optimization to quantum chemistry and cryptography. In the end, it stresses how important it is for mathematicians, computer scientists, physicists, and engineers to work together across disciplines in order to fully utilize the transformative power of quantum computing in numerical linear algebra and other areas.

Robust elastic full-waveform inversion using an alternating direction method of multipliers with reconstructed wavefields

Elastic full-waveform inversion (EFWI) is a process used to estimate subsurface properties by fitting seismic data while satisfying wave propagation physics. The problem is formulated as a least-squares data fitting minimization problem with two sets of constraints: partial-differential equation (PDE) constraints governing elastic wave propagation and physical model constraints implementing prior information. The alternating direction method of multipliers is used to solve the problem, resulting in an iterative algorithm with well-conditioned subproblems. Although wavefield reconstruction is the most challenging part of the iteration, sparse linear algebra techniques can be used for moderate-sized problems and frequency domain formulations. The Hessian matrix is blocky with diagonal blocks, making model updates fast. Gradient ascent is used to update Lagrange multipliers by summing PDE violations. Various numerical examples are used to investigate algorithmic components, including model parameterizations, physical model constraints, the role of the Hessian matrix in suppressing interparameter cross-talk, computational efficiency with the source sketching method, and the effect of noise and near-surface effects.

Subspace Acceleration for a Sequence of Linear Systems and Application to Plasma Simulation

We present an acceleration method for sequences of large-scale linear systems, such as the ones arising from the numerical solution of time-dependent partial differential equations coupled with algebraic constraints. We discuss different approaches to leverage the subspace containing the history of solutions computed at previous time steps in order to generate a good initial guess for the iterative solver. In particular, we propose a novel combination of reduced-order projection with randomized linear algebra techniques, which drastically reduces the number of iterations needed for convergence. We analyze the accuracy of the initial guess produced by the reduced-order projection when the coefficients of the linear system depend analytically on time. Extending extrapolation results by Demanet and Townsend to a vector-valued setting, we show that the accuracy improves rapidly as the size of the history increases, a theoretical result confirmed by our numerical observations. In particular, we apply the developed method to the simulation of plasma turbulence in the boundary of a fusion device, showing that the time needed for solving the linear systems is significantly reduced.

Genuinely nonabelian partial difference sets

AbstractStrongly regular graphs (SRGs) provide a fertile area of exploration in algebraic combinatorics, integrating techniques in graph theory, linear algebra, group theory, finite fields, finite geometry, and number theory. Of particular interest are those SRGs with a large automorphism group. If an automorphism group acts regularly (sharply transitively) on the vertices of the graph, then we may identify the graph with a subset of the group, a partial difference set (PDS), which allows us to apply techniques from group theory to examine the graph. Much of the work over the past four decades has concentrated on abelian PDSs using the powerful techniques of character theory. However, little work has been done on nonabelian PDSs. In this paper we point out the existence of genuinely nonabelian PDSs, that is, PDSs for parameter sets where a nonabelian group is the only possible regular automorphism group. We include methods for demonstrating that abelian PDSs are not possible for a particular set of parameters or for a particular SRG. Four infinite families of genuinely nonabelian PDSs are described, two of which—one arising from triangular graphs and one arising from Krein covers of complete graphs constructed by Godsil—are new. We also include a new nonabelian PDS found by computer search and present some possible future directions of research.

Applying Laplace Transformation on Epidemiological Models as Caputo Derivatives

This paper delves into the application of fractional calculus, with a focus on Caputo derivatives, in epidemiological models using ordinary differential equations. It highlights the critical role Caputo derivatives play in modeling intricate systems with memory effects and assesses various epidemiological models, including SIR variants, demonstrating how Caputo derivatives capture fractional-order dynamics and memory phenomena found in real epidemics. The study showcases the utility of Laplace transformations for analyzing systems described by ordinary differential equations with Caputo derivatives. This approach facilitates both analytical and numerical methods for system analysis and parameter estimation. Additionally, the paper introduces a tabular representation for epidemiological models, enabling a visual and analytical exploration of variable relationships and dynamics. This matrix-based framework permits the application of linear algebra techniques to assess stability and equilibrium points, yielding valuable insights into long-term behavior and control strategies. In summary, this research underscores the significance of Caputo derivatives, Laplace transformations, and matrix representation in epidemiological modeling. We assume that by using this type of methodology we can get analytic solutions by hand when considering a function as constant in certain cases and it will not be necessary to search for numerical methods.

Complexity of Some Types of Cyclic Snake Graphs

The number of spanning trees in graphs (networks) is a crucial invariant, and it is also an important measure of the reliability of a network. Spanning trees are special subgraphs of a graph that have several important properties. First, <I>T</I> must span <I>G</I>, which means it must contain every vertex in graph <I>G</I>, if <I>T</I> is a spanning tree of graph <I>G</I>. <I>T</I> needs to be a subgraph of <I>G</I>, second. Stated differently, any edge present in <I>T</I> needs to be present in <I>G </I>as well. Third, <I>G</I> is the same as <I>T</I> if each edge in <I>T</I> is likewise present in <I>G</I>. In path-finding algorithms like Dijkstra's shortest path algorithm and A* search algorithm, spanning trees play an essential part. In those approaches, spanning trees are computed as component components. Protocols for network routing also take advantage of it. In numerous techniques and applications, minimum spanning trees are highly beneficial. Computer networks, electrical grids, and water networks all frequently use them. They are also utilized in significant algorithms like the min-cut max-flow algorithm and in graph issues like the travelling salesperson problem. In this paper, we use matrix analysis and linear algebra techniques to obtain simple formulas for the number of spanning trees of certain kinds of cyclic snake graphs.

Multivariate Stochastic Volatility Modeling via Integrated Nested Laplace Approximations: A Multifactor Extension

This study introduces a multivariate extension to the class of stochastic volatility models, employing integrated nested Laplace approximations (INLA) for estimation. Bayesian methods for estimating stochastic volatility models through Markov Chain Monte Carlo (MCMC) can become computationally burdensome or inefficient as the dataset size and problem complexity increase. Furthermore, issues related to chain convergence can also arise. In light of these challenges, this research aims to establish a computationally efficient approach for estimating multivariate stochastic volatility models. We propose a multifactor formulation estimated using the INLA methodology, enabling an approach that leverages sparse linear algebra and parallelization techniques. To evaluate the effectiveness of our proposed model, we conduct in-sample and out-of-sample empirical analyses of stock market index return series. Furthermore, we provide a comparative analysis with models estimated using MCMC, demonstrating the computational efficiency and goodness of fit improvements achieved with our approach.

Surface science insight note: A linear algebraic approach to elucidate native films on Fe3O4 surface

Standard materials are often used to obtain spectra that can be compared to those from unknown samples. Spectra measured from these known substances are also used as a means of computing sensitivity factors to allow quantification by X‐ray photoelectron spectroscopy (XPS) of less well‐defined materials. Spectra from known materials also provide line shapes suitable for inclusion in spectral models which, when fitted to spectra, permit the chemical state for a sample to be assessed. Both types of information depend on isolating photoemission signals from the inelastically scattered signal. In this Insight note, technical issues associated with the use of XPS of as received Fe3O4 powder sample surface are discussed. The Insight note is designed to show how linear algebraic techniques applied to data collected from a sample marketed as pure Fe3O4 powder are used to verify that XPS has been performed on chemistry representative of the sample. The methods described in this Insight note can further be utilized in elucidating complex XPS data obtained from thin films formed or evolved during cyclic/non‐steady use of complex (electro)catalyst surfaces, especially in the presence of contaminants.

Characteristics of restricted selection indices and geometrical interpretation of restricted breeding values.

Restricted selection is used to control genetic changes in one or more characters. Three main selection indices are adopted for this purpose. First, Kempthorne's index is used to maximize aggregate breeding value (BV) with changes in some traits restricted to zero; second, Harville's index is used to maximize aggregate BV with proportional changes for some traits; and third, Yamada's index is mathematically used to achieve the relative desired changes for all traits. Kempthorne's index is equivalent to Harville's index. However, the relationship between Kempthorne's and Yamada's indices has not been clarified. In addition, the characteristics of restricted selection indices and the relationship between BV and restricted BV (RBV) are also unknown. The aim of this study was to clarify the characteristics of restricted selection indices and describe the relationship between BV and RBV by using linear algebra and geometric techniques, respectively. First, I proved that Yamada's index is part of Kempthorne's index. Second, I investigated the relationship between BVs that were estimated using an ordinary selection index (EBVs) and RBVs estimated using a restricted selection index (ERBVs) and proved that the ERBVs of the restricted traits are proportional to the relative desired changes. Third, I proved that RBV is represented by a linear function of BV and geometrically represented the relationship between BV and RBV. In this study, new findings on restricted selection indices and RBV were obtained. This useful clarification of the relationship between BV and RBV will make it possible to evaluate the response to selection using not only a restricted selection index, but also a restricted BLUP in computer simulation studies.

Surface Analysis Insight Note: Observations relating to photoemission peak shapes, oxidation state, and chemistry of titanium oxide films

It is common practice to describe the coordination of metal atoms in a binding configuration with their nearest neighbors in terms of oxidation state, a measure by which the number of electrons redistributed between atoms forming chemical bonds. In XPS terms, change to an oxidation state is commonly inferred by correlating photoemission signal with binding energy. The assumption, when classifying photoemission signals into distinct spectral shapes, is that a distribution of intensities shifted to lower binding energy is evidence of a reduction in oxidation state. In this Insight note, we raise the prospect that changes in photoemission peak shape may occur without obvious changes, determined by XPS in stoichiometry for a material. It is well known that TiO2 measured by XPS yields reproducible Ti 2p photoemission peaks. However, on exposing TiO2 to ion beams, Ti 2p photoemission evolves to complex distributions in intensity, which are particularly difficult to analyze by traditional fitting of bell‐shaped curves to these data. For these reasons, in this Insight note, a thin film of TiO2 deposited on a silicon substrate is chosen for analysis by XPS and linear algebraic techniques. Alterations in spectral shapes created from modified TiO2, which might be interpreted as the change in oxidation state, are assessed in terms of relative proportions of titanium to oxygen. It is found through detailed analysis of spectra that quantification by XPS, using procedures routinely used in practice, is not in accord with the typical interpretations of photoemission shapes. The data processing methods used and results presented in this work are of particular relevance to elucidating fundamental phenomena governing the surface evolution of materials‐enabled energy processes where cyclic/non‐steady usage changes the nature of bonding, especially in the presence of contaminants.

Constructing a straight line intersecting four lines

In this paper, we determine the set S of straight lines L0 that have intersections with four given distinct lines L1,…,L4 in R3. If any two of the four given lines are skew, i.e., not co-planar, Bielinski and Lapinska used techniques in projective geometry to show that there are either zero, one, or two elements in the set S. Using linear algebra techniques, we determine S and show that there are no, one, two or infinitely many elements L0 in S, where the last case was overlooked in the earlier paper. For the sake of completeness, we provide a comprehensive determination of all the elements L0 in S if at least two of the four given lines are co-planar. In this scenario, there may also be zero, one, two, or infinitely many solutions.

A Comparison of Latent Semantic Analysis and Latent Dirichlet Allocation in Educational Measurement

Topic models are mathematical and statistical models used to analyze textual data. The objective of topic models is to gain information about the latent semantic space of a set of related textual data. The semantic space of a set of textual data contains the relationship between documents and words and how they are used. Topic models are becoming more common in educational measurement research as a method for analyzing students’ responses to constructed-response items. Two popular topic models are latent semantic analysis (LSA) and latent Dirichlet allocation (LDA). LSA uses linear algebra techniques, whereas LDA uses an assumed statistical model and generative process. In educational measurement, LSA is often used in algorithmic scoring of essays due to its high reliability and agreement with human raters. LDA is often used as a supplemental analysis to gain additional information about students, such as their thinking and reasoning. This article reviews and compares the LSA and LDA topic models. This article also introduces a methodology for comparing the semantic spaces obtained by the two models and uses a simulation study to investigate their similarities.

Ideal uniform multipartite secret sharing schemes

Multipartite secret sharing schemes are those having a multipartite access structure, in which the set of participants is divided into several parts and all participants in the same part play an equivalent role. Secret sharing schemes for these access structures have gained attention, as they generalize threshold secret sharing. This work focuses on the construction of linear secret sharing schemes for ideal uniform multipartite access structures, while avoiding the inefficiencies and randomness of known constructions. To achieve this, we have developed two approaches. The first approach combines polymatroid-based techniques with Gabidulin codes, which efficiently realize a particular family of ideal uniform multipartite access structures. The efficiency of this method is derived from the unique properties of Gabidulin codes, which allow for polynomial share and secret sizes relative to the number of participants. Our second approach applies linear algebraic techniques and polymatroid-based techniques to explicitly design linear secret sharing schemes for arbitrary ideal uniform multipartite access structures. The efficiency of this method stems from the properties of polynomials over finite fields. The schemes from this method are efficient for smaller numbers of parts relative to the number of participants. In summary, we present explicit constructions of linear secret sharing schemes for ideal uniform multipartite access structures, which have important applications in cryptography.

Decompositions of optimal averaged Gauss quadrature rules

Optimal averaged Gauss quadrature rules provide estimates for the quadrature error in Gauss rules, as well as estimates for the error incurred when approximating matrix functionals of the form uTf(A)v with a large matrix A∈RN×N by low-rank approximations that are obtained by applying a few steps of the symmetric or nonsymmetric Lanczos processes to A; here u,v∈RN are vectors. The latter process is used when the measure associated with the Gauss quadrature rule has support in the complex plane. The symmetric Lanczos process yields a real tridiagonal matrix, whose entries determine the recursion coefficients of the monic orthogonal polynomials associated with the measure, while the nonsymmetric Lanczos process determines a nonsymmetric tridiagonal matrix, whose entries are recursion coefficients for a pair of sets of bi-orthogonal polynomials. Recently, it has been shown, by applying the results of Peherstorfer, that optimal averaged Gauss quadrature rules, which are associated with a nonnegative measure with support on the real axis, can be expressed as a weighted sum of two quadrature rules. This decomposition allows faster evaluation of optimal averaged Gauss quadrature rules than the previously available representation. The present paper provides a new self-contained proof of this decomposition that is based on linear algebra techniques. Moreover, these techniques are generalized to determine a decomposition of the optimal averaged quadrature rules that are associated with the tridiagonal matrices determined by the nonsymmetric Lanczos process. Also, the splitting of complex symmetric tridiagonal matrices is discussed. The new splittings allow faster evaluation of optimal averaged Gauss quadrature rules than the previously available representations. Computational aspects are discussed.

A measurement-based approach to voltage-constrained hosting capacity analysis with controllable reactive power behind-the-meter

We propose an efficient method for the assessment of localized voltage-constrained distributed generation hosting capacity given an engineer’s choice or estimate of a power factor control setpoint for each customer. The advantage of the proposed method is that it extends recent methods to perform HCA using only advanced metering infrastructure (AMI) measurements with no prior knowledge of the network or its topology. This is achieved by developing efficient decision-making algorithms based on linear algebra and optimization techniques. The performance of the method is benchmarked against conventional model-based hosting capacity analysis (HCA) techniques, and we demonstrate the success of the method on several real-world AMI datasets. We find that comparable performance is achieved to model-based methods. Calculating HC from AMI datasets has the potential to avoid modeling errors common in circuit models.