Matrix Multiplication Research Articles

Analysis of Factors Influencing Experimental Teaching Satisfaction Based on ISM-MICMAC

The satisfaction with experimental teaching is of utmost importance for improving educational quality, promoting student development, and fostering the professional growth of teachers. Researching the factors influencing satisfaction with experimental teaching, as well as the hierarchical relationships and mechanisms of action among these factors, provides significant theoretical foundations and practical guidance for conducting scientific and targeted reforms in experimental teaching. Based on literature research and questionnaire surveys, fifteen factors affecting experimental teaching satisfaction were selected across five dimensions: teachers, curriculum, students, environment, and interaction. The Interpretive Structural Modeling (ISM) was used to reveal the hierarchical relationships among these factors, and the Cross-Impact Matrix Multiplication Applied to a Classification (MICMAC) method was employed to validate the model’s scientific validity and assess the driving forces and dependencies of each factor. The research results indicate that self-efficacy, academic emotions, perceived course value, teacher-student interaction, and peer interaction are direct factors influencing experimental teaching satisfaction and exhibit a high level of interdependence. Teacher’s teaching competence, laboratory management, course content, course resources, and laboratory environment are deeper influencing factors with strong driving forces, positively impacting both the direct factors and intermediary factors such as teaching models and course assessment. Based on the analysis results, improvement strategies, and recommendations are proposed to enhance the quality of experimental teaching and the effectiveness of talent development.

An In-Memory-Computing Structure with Quantum-Dot Transistor Toward Neural Network Applications: From Analog Circuits to Memory Arrays

The rapid advancements in artificial intelligence (AI) have demonstrated great success in various applications, such as cloud computing, deep learning, and neural networks, among others. However, the majority of these applications rely on fast computation and large storage, which poses significant challenges to the hardware platform. Thus, there is a growing interest in exploring new computation architectures to address these challenges. Compute-in-memory (CIM) has emerged as a promising solution to overcome the challenges posed by traditional computer architecture in terms of data transfer frequency and energy consumption. Non-volatile memory, such as Quantum-dot transistors, has been widely used in CIM to provide high-speed processing, low power consumption, and large storage capacity. Matrix-vector multiplication (MVM) or dot product operation is a primary computational kernel in neural networks. CIM offers an effective way to optimize the performance of the dot product operation by performing it through an intertwining of processing and memory elements. In this paper, we present a novel design and analysis of a Quantum-dot transistor (QDT) based CIM that offers efficient MVM or dot product operation by performing computations inside the memory array itself. Our proposed approach offers energy-efficient and high-speed data processing capabilities that are critical for implementing AI applications on resource-limited platforms such as portable devices.

Reservoir computing with a random memristor crossbar array.

Physical implementations of reservoir computing (RC) based on the emerging memristors have become promising candidates of unconventional computing paradigms. Traditionally, sequential approaches by time-multiplexing volatile memristors have been prevalent because of their low hardware overhead. However, they suffer from the problem of speed degradation and fall short of capturing the spatial relationship between the time-domain inputs. Here, we explore a new avenue for RC using memristor crossbar arrays (MCAs) with device-to-device variations, which serve as physical random weight matrices of the reservoir layers, enabling faster computation thanks to the parallelism of matrix-vector multiplication as an intensive operation in RC. To achieve this new RC architecture, ultralow-current, self-selective memristors are fabricated and integrated without the need of transistors, showing greater potential of high scalability and three-dimensional integrability compared to the previous realizations. The information processing ability of our RC system is demonstrated in tasks of recognizing digit images and waveforms. This work indicates that the "nonidealities" of the emerging memristor devices and circuits are a useful source of inspiration for new computing paradigms.

Quadratic-attraction subdivision with contraction-ratio [formula omitted]

Classic generalized subdivision, such as Catmull–Clark subdivision, as well as recent subdivision algorithms for high-quality surfaces, rely on slower convergence towards extraordinary points for mesh nodes surrounded by n>4 quadrilaterals. Slow convergence corresponds to a contraction-ratio of λ>0.5. To improve shape, prevent parameterization discordant with surface growth, or to improve convergence in isogeometric analysis near extraordinary points, a number of algorithms explicitly adjust λ by altering refinement rules. However, such tuning of λ has so far led to poorer surface quality, visible as uneven distribution or oscillation of highlight lines. The recent Quadratic-Attraction Subdivision (QAS) generates high-quality, bounded curvature surfaces based on a careful choice of quadratic expansion at the central point and, just like Catmull–Clark subdivision, creates the control points of the next subdivision ring by matrix multiplication. But QAS shares the contraction-ratio λCC>1/2 of Catmull–Clark subdivision when n>4. For n=5,…,10, QAS+ improves the convergence to the uniform λ=12 of binary domain refinement and without sacrificing surface quality compared to QAS.

Interaction Effects of Vegetation and Soil Factors on Microbial Communities in Alpine Steppe Under Degradation

To clarify the regulating effect of vegetation and soil factors on microbial communities in the alpine steppe under degradation on the Qinghai-Xizang Plateau, the alpine steppe in the Sanjiangyuan area of the Qinghai-Tibet Plateau was chosen. We analyzed the differences in vegetation and soil factors in different stages of degradation （non-degradation, moderate degradation, and severe degradation） and detected the variations in microbial community characteristics in the alpine steppe under different degradation stages using high-throughput sequencing technology. Eventually, redundancy analysis （RDA） and multiple regression matrixes （MRM） based on the similarity or dissimilarity matrix were used to identify key environmental factors regulating microbial （bacterial and fungal） community changes under degradation. The results showed that the degradation of the alpine steppe significantly changed the community coverage, height, biomass, and important value of graminae； significantly reduced the contents of soil organic matter, total nitrogen, total phosphorus, and silt； and increased the soil bulk density and sand content. Degradation did not change the composition of bacteria and fungi, but their composition proportions changed and also resulted in the loss of microbial richness （Chao1 index and Richness index） but did not significantly change the microbial diversity （Shannon index）. With the occurrence of degradation, the vegetation characteristics, soil physicochemical properties, and microbial diversity showed a consistent change trend. Combined with the characteristics of the network topology changes （the number of nodes and clustering coefficient significantly decreased）, it was found that degradation of the alpine steppe led to the decline of interspecies interactions, decentralization of network, and homogenization of microorganisms, but the cooperation relations among the species were maintained （positive correlation connections accounted for more than 90% in all degradation stages）. Under the alpine steppe degradation, the vegetation-soil interaction had the greatest effect on soil bacterial community, whereas soil physicochemical properties had the greatest influence on soil fungal community. Specifically, vegetation community height, biomass, and soil bulk density were the mutual factors regulating soil microorganisms, whereas the vegetation Simpson index, important value of graminae, soil total phosphorus, total potassium, and silt content were the unique factors affecting the soil bacterial community, and soil pH and total nitrogen content were the particular factors affecting the soil fungal community.

Optimization of Large-Scale Sparse Matrix-Vector Multiplication on Multi-GPU Systems

Sparse matrix-vector multiplication (SpMV) is one of the important kernels of many iterative algorithms for solving sparse linear systems. The limited storage and computational resources of individual GPUs restrict both the scale and speed of SpMV computing in problem-solving. As real-world engineering problems continue to increase in complexity, the imperative for collaborative execution of iterative solving algorithms across multiple GPUs is increasingly apparent. Although the multi-GPU-based SpMV takes less kernel execution time, it also introduces additional data transmission overhead, which diminishes the performance gains derived from parallelization across multi-GPUs. Based on the non-zero elements distribution characteristics of sparse matrices and the trade-off between redundant computations and data transfer overhead, this paper introduces a series of SpMV optimization techniques tailored for multi-GPU environments and effectively enhances the execution efficiency of iterative algorithms on multiple GPUs. Firstly, we propose a two-level non-zero elements-based matrix partitioning method to increase the overlap of kernel execution and data transmission. Then, considering the irregular non-zero elements distribution in sparse matrices, a long-row-aware matrix partitioning method is proposed to hide more data transmissions. Finally, an optimization using redundant and inexpensive short-row execution to exchange costly data transmission is proposed. Our experimental evaluation demonstrates that, compared with the SpMV on a single GPU, the proposed method achieves an average speedup of 2.00x and 1.85x on platforms equipped with two RTX 3090 and two Tesla V100-SXM2, respectively. The average speedup of 2.65x is achieved on a platform equipped with four Tesla V100-SXM2.

Phasor-based analysis of a neuromorphic architecture for microwave sensing

This article presents a design procedure for implementing artificial neural networks (ANNs) using conventional microwave components at the hardware level with potential applications in radar and remote sensing. The main objective is to develop structured hardware design methods for implementing artificial neurons, utilizing microwave devices to create neuromorphic devices compatible with high-frequency electromagnetic waves. The research aims to address the challenge of encoding and modulating information in electromagnetic waves into a format suitable for the neuromorphic device by using frequency-modulated information instead of intensity-modulated information. It also proposes a method for integrating principal component analysis as a dimensionality reduction technique with the implementation of ANNs on a single hardware. As a dummy task, the process outlined here is used to implement an artificial neural network at the hardware level, with a specific emphasis on creating hardware that is capable of performing matrix multiplications in the form of dot products while also being able to extract the resulting data in an interpretable manner. The proposed implementation involves the use of directional couplers to implement weights and sample the resulting signal at specific intervals to obtain the multiplication result.

Control-oriented multiphysics model of a lithium-ion battery for thermal runaway estimation under operational and abuse conditions

This study proposes a control-oriented multiphysics model for lithium-ion batteries (LIBs) that can estimate electrochemical-thermal responses in real time under normal operation and abuse conditions. The proposed model integrates the simplified electrochemical model, the thermal resistance network, and the adaptive time-stepping method to ensure computational efficiency without sacrificing the accuracy. Specifically, the diffusion equation of the electrochemical model is simplified by addressing Padé approximation. The thermal resistance network estimates 3D temperature distribution through simple matrix multiplication to account for the entropic, ohmic, and chemical reaction during thermal runaway. The adaptive time-stepping method further secures accurate yet fast explicit calculation. Quantitative experimental validation reveals the high accuracy and robustness, and fast inference time of the proposed model to estimate the electrochemical-thermal responses with the average inference time of 0.0047 s per step. The application of the proposed model on the 26650 LFP cell also demonstrates the versatility on various shapes and types of LIBs. The systematic analysis on the 3D temperature distribution in the LIB of interest not only confirms the effectiveness of the internal temperature monitoring but also ensures the virtual sensing capability. The versatility of the proposed model underscores both design- and control- enabling solutions for battery thermal management.

Insights into the Impact of Irrigation Agriculture on the Economy of the Limpopo Province, South Africa: A Social Accounting Matrix Multiplier Analysis

Insights into the Impact of Irrigation Agriculture on the Economy of the Limpopo Province, South Africa: A Social Accounting Matrix Multiplier Analysis

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.

Hyperspectral in-memory computing with optical frequency combs and programmable optical memories

The rapid rise of machine learning drives demand for extensive matrix-vector multiplication operations, thereby challenging the capacities of traditional von Neumann computing systems. Researchers explore alternatives, such as in-memory computing architecture, to find energy-efficient solutions. In particular, there is renewed interest in optical computing systems, which could potentially handle matrix-vector multiplication in a more energy-efficient way. Despite promising initial results, developing high-throughput optical computing systems to rival electronic hardware remains a challenge. Here, we propose and demonstrate a hyperspectral in-memory computing architecture, which simultaneously utilizes space and frequency multiplexing, using optical frequency combs and programmable optical memories. Our carefully designed three-dimensional opto-electronic computing system offers remarkable parallelism, programmability, and scalability, overcoming typical limitations of optical computing. We have experimentally demonstrated highly parallel, single-shot multiply-accumulate operations with precision exceeding 4 bits in both matrix-vector and matrix-matrix multiplications, suggesting the system’s potential for a wide variety of deep learning and optimization tasks. Our approach presents a realistic pathway to scale beyond peta operations per second, a major stride towards high-throughput, energy-efficient optical computing.

Kaczmarz-type methods for solving matrix equations

In this paper, several Kaczmarz-type numerical methods for solving the matrix equations AX = B and XA = C are proposed, where the coefficient matrix A may be full rank or deficient rank. These methods are iterative methods without matrix multiplication. Theoretically, the convergence of these methods is proved. The numerical results show that these methods are more efficient than iterative methods involving matrix multiplication for high-dimensional matrices.

Deep Learning-Driven Compiler Enhancements for Efficient Matrix Multiplication

Matrix multiplication is a fundamental operation in many computational fields, requiring optimization to handle increasing data sizes efficiently. In this paper, the implementation of Deep Learning in Matrix multiplication is reviewed, which is considered important nowadays due to the growing complexity of matrix multiplication for gaming and complex programs. The current standard matrix multiplication and the time taken by it on different matrix sizes are described. The Tiled Matrix multiplication, which trims the matrix into various pieces and calculates the product for each piece, and thereafter combines the result, is also described. The times taken by both methods for different matrix sizes were compared. The main idea was to use Deep Neural Networks (DNN) to compare and rank code variants that are obtained in pieces and determine their relative performance. A tournament-based ranking system is used for assigning ranks to the code versions. The effectiveness of these techniques was evaluated on various matrix multiplication operations commonly found in deep learning workloads. Up to 8.844x speedup over the naive implementation for a matrix size of 1024 is achieved by this approach. The results demonstrate the effectiveness of combining compiler optimization techniques and deep learning models in optimizing matrix multiplication.

Solving Sparse Linear Systems Faster than Matrix Multiplication

Can linear systems be solved faster than matrix multiplication? While there has been remarkable progress for the special cases of graph-structured linear systems, in the general setting, the bit complexity of solving an n × n linear system Ax = b is Õ ( n ω ), where ω is the matrix multiplication exponent. Improving on this has been an open problem even for sparse linear systems with poly( n ) condition number. In this paper, we present an algorithm that solves linear systems with sparse coefficient matrices asymptotically faster than matrix multiplication for any ω > 2. This speedup holds for any input matrix A with o ( n ω−1 / log (κ( A ))) non-zeros, where κ( A ) is the condition number of A . Our algorithm can be viewed as an efficient, randomized implementation of the block Krylov method via recursive low displacement rank factorization. It is inspired by an algorithm of Eberly et al. for inverting matrices over finite fields. In our analysis of numerical stability, we develop matrix anti-concentration techniques to bound the smallest eigenvalue and the smallest gap in the eigenvalues of semi-random matrices.

SDIM: A Subtly Designed Invertible Matrix for Enhanced Privacy-Preserving Outsourcing Matrix Multiplication and Related Tasks

SDIM: A Subtly Designed Invertible Matrix for Enhanced Privacy-Preserving Outsourcing Matrix Multiplication and Related Tasks

Dynamic behavior of a wheeled rocket launcher using transfer matrix method of multibody system

Abstract One of the most sophisticated causes of unguided rocket dispersion is the uncertainties in launch conditions. This arises due to the fact that a launcher, in reality, shows dynamic behavior as it is a multibody system. This paper investigates the dynamic behavior of a wheeled rocket launcher that is loaded on a vehicle. The launcher and vehicle system are modeled as a multibody system using the transfer matrix method of multibody system (TMM). TMM is chosen due to its suitability for linear systems and its ability to obtain the overall transfer matrix for the entire system by performing successive matrix multiplications of the individual body’s matrix, thereby avoiding the complexity of solving the overall dynamic equations of the system. The developed dynamic model is then validated by comparing its results with those in the open literature using the Lagrange approach. The study considers dynamic loads that result from the rocket motion on the launcher. A parametric study is conducted to address the impact of launcher structural aspects. By investigating the dynamic behavior of the wheeled rocket launcher, this study aims to enhance the understanding of its performance.

An implicit tailored finite point method for the Burgers’ equation: leveraging the Cole-Hopf transformation

Any expedition in designing numerical methods besides aiming at accuracy, also equally steers for simplicity and ease in implementation. This paper brings in one such algorithm the tailored finite point method (TFPM) in tandem with the Cole Hopf transformation. At the initiation, the non-linear Burgers’ equation is transformed into a linear heat equation to which TFPM is applied. The proffered TFPM functions on an explicit stencil on the left boundary of the domain and on an implicit stair stencil throughout the rest of the domain. On these stencils, the nodal solutions at the advanced temporal level are written as a linear combination of the solutions at the remaining nodes within the stencil. The scalars involved in the linear combination are identified by the application of fundamental solutions into the stencil resultantly infusing the essential nature of the local exact solutions into the approximations. The foundation of such a linear combination avoids the need for complex computations involving matrix multiplication and inversion. The numerical accuracy of the method is established through comparisons of TFPM solutions of classical examples with the exact solutions and solutions from other contemporary methodologies. The theoretical correctness of the method is established through analyses of consistency, stability, and convergence. Furthermore, the method exhibits the potential for extension to higher dimensions and other complex modalities.

The tensor as an informational resource.

A tensor is a multidimensional array of numbers that can be used to store data, encode a computational relation, and represent quantum entanglement. In this sense, a tensor can be viewed as valuable resource whose transformation can lead to an understanding of structure in data, computational complexity, and quantum information. In order to facilitate the understanding of this resource, we propose a family of information-theoretically constructed preorders on tensors, which can be used to compare tensors with each other and to assess the existence of transformations between them. The construction places copies of a given tensor at the edges of a hypergraph and allows transformations at the vertices. A preorder is then induced by the transformations possible in a given growing sequence of hypergraphs. The new family of preorders generalizes the asymptotic restriction preorder which Strassen defined in order to study the computational complexity of matrix multiplication. We derive general properties of the preorders and their associated asymptotic notions of tensor rank and view recent results on tensor rank nonadditivity, tensor networks, and algebraic complexity in this unifying frame. We hope that this work will provide a useful vantage point for exploring tensors in applied mathematics, physics, and computer science but also from a purely mathematical point of view.

A Retina-inspired Sampling Method for Coding and Compression for Enhancement Image Based on Discrete Double Density Wavelet Transformer

The retina, a layer of light-sensitive tissue on the inner surface of the eye, is what allows for vision. That propose to design and analytically examine an image quantizer coding that is inspired by the way the retina manages and compresses visual features based on past studies. Retina-inspired coding is a new cipher algorithm which is very promising. Comparing the Double-Density Discrete Wavelet Transform (DDDWT) to earlier traditional approaches, it is more difficult but also produces excellent results. This efficient coding is credited to a streamlined account that, by the way, deals with 2-D and 3-D pictures, transformation matrices, and the conversion of the number DDWT as if via matrix multiplication. Naturally, this code makes a lot of interesting points of view available. In this proposal, we'll try to show how various theoretical model extensions can be applied to applications in video surveillance, information technology, and neuroscience. Therefore, that believe that using a multidisciplinary approach will help system achieve objectives. This strategy would apply signal processing methods with data from neurophysiology. There are hoping that our efforts will result in some innovative new coding algorithms. The proposed codes will be implemented in MATLAB for ease of experimentation. To implement this code more successfully, a low-level programming language is required. Index Terms— DDDWT, Retina-inspired, Coding.

A novel perspective of calvarial development: the cranial morphogenesis and differentiation regulated by dura mater.

There are lasting concerns on calvarial development because cranium not only accommodates the growing brain, but also safeguards it from exogenous strikes. In the past decades, most studies attributed the dynamic expansion and remodeling of cranium to the proliferation of osteoprecursors in cranial primordium, and the proliferation of osteoprogenitors at the osteogenic front of cranial suture mesenchyme. Further investigations identified series genes expressed in suture mesenchymal cells as the markers of the progenitors, precursors and postnatal stem cells in cranium. However, similar to many other organs, it is suggested that the reciprocal interactions among different tissues also play essential roles in calvarial development. Actually, there are increasing evidence indicating that dura mater (DM) is indispensable for the calvarial morphogenesis and osteogenesis by secreting multiple growth factors, cytokines and extracellular matrix (ECM). Thus, in this review, we first briefly introduce the development of cranium, suture and DM, and then, comprehensively summarize the latest studies exploring the involvement of ECM in DM and cranium development. Eventually, we discussed the reciprocal interactions between calvarium and DM in calvarial development. Actually, our review provides a novel perspective for cranium development by integrating previous classical researches with a spotlight on the mutual interplay between the developing DM and cranium.