Linear Solver Research Articles

Zhang neural networks: an introduction to predictive computations for discretized time-varying matrix problems

This paper wants to increase our understanding and computational know-how for time-varying matrix problems and Zhang Neural Networks. These neural networks were invented for time or single parameter-varying matrix problems around 2001 in China and almost all of their advances have been made in and most still come from its birthplace. Zhang Neural Network methods have become a backbone for solving discretized sensor driven time-varying matrix problems in real-time, in theory and in on-chip applications for robots, in control theory and other engineering applications in China. They have become the method of choice for many time-varying matrix problems that benefit from or require efficient, accurate and predictive real-time computations. A typical discretized Zhang Neural Network algorithm needs seven distinct steps in its initial set-up. The construction of discretized Zhang Neural Network algorithms starts from a model equation with its associated error equation and the stipulation that the error function decrease exponentially fast. The error function differential equation is then mated with a convergent look-ahead finite difference formula to create a distinctly new multi-step style solver that predicts the future state of the system reliably from current and earlier state and solution data. Matlab codes of discretized Zhang Neural Network algorithms for time varying matrix problems typically consist of one linear equations solve and one recursion of already available data per time step. This makes discretized Zhang Neural network based algorithms highly competitive with ordinary differential equation initial value analytic continuation methods for function given data that are designed to work adaptively. Discretized Zhang Neural Network methods have different characteristics and applicabilities than multi-step ordinary differential equations (ODEs) initial value solvers. These new time-varying matrix methods can solve matrix-given problems from sensor data with constant sampling gaps or from functional equations. To illustrate the adaptability of discretized Zhang Neural Networks and further the understanding of this method, this paper details the seven step set-up process for Zhang Neural Networks and twelve separate time-varying matrix models. It supplies new codes for seven of these. Open problems are mentioned as well as detailed references to recent work on discretized Zhang Neural Networks and time-varying matrix computations. Comparisons are given to standard non-predictive multi-step methods that use initial value problems ODE solvers and analytic continuation methods.

Quantum radio astronomy: Quantum linear solvers for redundant baseline calibration

The computational requirements of future large scale radio telescopes are expected to scale well beyond the capabilities of conventional digital resources. Current and planned telescopes are generally limited in their scientific potential by their ability to efficiently process the vast volumes of generated data. To mitigate this problem, we investigate the viability of emerging quantum computers for radio astronomy applications. In this a paper we demonstrate the potential use of variational quantum linear solvers in Noisy Intermediate Scale Quantum (NISQ) computers and combinatorial solvers in quantum annealers for a radio astronomy calibration pipeline. While we demonstrate that these approaches can lead to satisfying results when integrated in calibration pipelines, we show that current restrictions of quantum hardware limit their applicability and performance.

A numerical and closed-form analytical solution for the global buckling critical load of tall buildings considering soil flexibility: A coupled shear-flexural model

The critical global buckling load is a fundamental parameter governing the performance of tall buildings, however, the existing literature lacks an efficient solution that comprehensively captures the behavior of tall structures with uniform or variable properties. In response to this gap, this study introduces a novel and generalized solution, employing the coupled shear-flexural model. This solution is designed for determining the critical global buckling load in tall buildings subjected to vertical loads with diverse profiles, while also incorporating soil flexibility. The analytical solution, designed for the uniform continuous model, offers a breakthrough by providing graphical representations facilitating the direct determination of eigenvalues based on a single dimensionless parameter. Addressing the unique challenge posed by tall buildings with variable properties, a numerical transfer matrix method based on a Laplacian approach is proposed. This method enables the direct calculation of the transfer matrix for each level, displaying exceptional convergence in just three iterations. The study sheds light on a crucial finding that emphasizes the adverse impact of rotational flexibility on soil, leading to a decrease in eigenvalues as soil flexibility increases. Importantly, the proposed solution methods not only show excellent accuracy but also serve as invaluable tools for performing parametric analyses. Furthermore, they present cost-effective alternatives to exact methods, providing a solid foundation for studying the overall stability of tall buildings. This research contributes significantly to the existing literature by introducing efficient methodologies that improve our understanding and analysis of the behavior of tall buildings under vertical loads.

Discrete variable topology optimization for maximizing single/multiple natural frequencies and frequency gaps considering the topological constraint

AbstractFinding optimized structural topology design for maximizing natural frequencies and frequency gaps of continuum structures is crucial for engineering applications. However, two significant numerical issues must be addressed: non‐smoothness caused by multiple frequencies and Artificial Localized Rigid Motion (ALRM) modes due to the violation of the topological constraint related to isolated islands and point‐connections. The above two issues are solved by employing the discrete variable topology optimization method based on Sequential Approximate Integer Programming (SAIP). First, the directional differentiability and multiple frequencies preservation constraints are formulated as the linear integer constraints. And then, the integer programming with these linear integer constraints is established and solved by the discrete variable linear or quadratic programming solver with multi‐constraint Canonical Relaxation Algorithm (CRA). We also prove that the popular Average Modal Frequencies (AMF) strategy, like the Kreisselmeier‐Steinhauser (KS) function, cannot rigorously tackle this non‐smoothness caused by multiple frequencies. Furthermore, to eliminate the ALRM modes and concentrate on the real structural global modes, the burning method is employed to impose the topological constraint of the first Betti number that represents the number of isolated islands and point‐connections. Numerical examples, including 2D and 3D, two‐fold and three‐fold multiple frequencies, natural frequencies and frequency gaps, are presented to show the effectiveness of the proposed method.

The Application of Geographic Information System in Urban Forest Ecological Compensation and Sustainable Development Evaluation

Urban forests can alleviate the urban heat island effect, improve air quality, and improve residents’ mental health. By studying urban forests, these resources can be better used and managed to create more livable urban environments. Therefore, the urban forest in the Taishan region is taken as the research object, and the ecological compensation and sustainable development of urban forest in Tai’an City are deeply analyzed by GIS. It divided the area into forest land, water bodies, wetlands, grasslands, and shrubs as the basic ecosystem types. And through secondary interpretation and combination, a complete urban forest information database was established. To evaluate the comprehensive benefits of urban forests, the analytic hierarchy process was utilized to establish a corresponding evaluation index system. Based on the assessment outcomes of the comprehensive benefits of urban forests in the area, a standard accounting method for urban forest ecological compensation was proposed. The results showed that each index of the comprehensive benefits of urban forests and the random consistency ratio were both less than 0.1. This indicated that the matrix calculation results of various indicators of urban forest comprehensive benefits had good consistency. At the target level, the comprehensive evaluation score of urban forests in the study area was 7.69. At the factor level, the weight value of the urban forest landscape structure was 0.675, and the comprehensive score was 7.62. The weight value of urban forest comprehensive benefits was 0.325, and the comprehensive score was 7.82. The quantitative weight value of urban forest greening in the study area was 0.6138, with a comprehensive score of 7.57. Based on the analysis of the issues in urban forests and ecological compensation in the research area of Tai’an City, corresponding ecological compensation strategies have been proposed. It is of great value to study the urban forest of Tai’an city, which can help to formulate more effective urban planning and sustainable development strategies. The research results can also provide a valuable reference and inspiration for the improvement of urban forest ecological environment and biodiversity protection in other areas.

Solving differential‐algebraic equations in power system dynamic analysis with quantum computing

AbstractPower system dynamics are generally modeled by high dimensional nonlinear differential‐algebraic equations (DAEs) given a large number of components forming the network. These DAEs' complexity can grow exponentially due to the increasing penetration of distributed energy resources, whereas their computation time becomes sensitive due to the increasing interconnection of the power grid with other energy systems. This paper demonstrates the use of quantum computing algorithms to solve DAEs for power system dynamic analysis. We leverage a symbolic programming framework to equivalently convert the power system's DAEs into ordinary differential equations (ODEs) using index reduction methods and then encode their data into qubits using amplitude encoding. The system nonlinearity is captured by Hamiltonian simulation with truncated Taylor expansion so that state variables can be updated by a quantum linear equation solver. Our results show that quantum computing can solve the power system's DAEs accurately with a computational complexity polynomial in the logarithm of the system dimension. We also illustrate the use of recent advanced tools in scientific machine learning for implementing complex computing concepts, that is, Taylor expansion, DAEs/ODEs transformation, and quantum computing solver with abstract representation for power engineering applications.

Current Sensor-Free Output-Feedback Voltage Control for DC/DC Converters via Critical Damping Injection Technique for Converter-Fed Servo System Applications

This study proposes an observer-based multi-loop output-feedback system that regulates the output DC-link voltage of DC/DC converters to the desired value to lower the system model dependence level. There are a few features: (a) the outer loop injecting the nonlinear active damping to ensure the critically damped output voltage response, (b) a model-independent output voltage derivative observer facilitates the inner loop and eliminates the need for complex matrix calculations for tuning the estimation performance, and (c) the use of a pole-zero cancellation acceleration error stabilizer for the inner loop helps maintain the targeted closed-loop performance. The experimental study adopts a 420 W brushless DC motor dynamo system as the load for the prototype 600 W bi-directional converter controlled using the proposed solution.

Energy demand forecasting of remote areas using linear regression and inverse matrix analysis

Efficient energy demand forecasting is pivotal for addressing energy challenges in remote areas of Bangladesh, where reliable access to energy resources remains a concern. This study proposes an innovative approach that combines linear regression analysis (LRA) and inverse matrix calculation (IMC) to forecast energy demand accurately in these underserved regions. By leveraging historical energy consumption data and pertinent predictors, such as meteorological conditions, population dynamics, economic indicators, and seasonal patterns, the model provides reliable forecasts. The application of the proposed methodology is demonstrated through a case study focused on remote regions of Bangladesh. The results showcase the approach's effectiveness in capturing the intricate dynamics of energy demand and its potential to inform sustainable energy management strategies in these remote areas. This research contributes to the advancement of energy planning and resource allocation in regions facing energy scarcity, fostering a path towards improved energy efficiency and development. These techniques can be applied to estimate short-term electricity demand for any rural or isolated region worldwide.

Event-Variant and Time-Variant (max,+) Systems

This paper deals with the input-output representation of a class of timed Discrete Event Systems. The systems considered are those that can be described using Timed Event Graphs extended with weights on the arcs and clock rate modifiers or time varying delays. The model relies on periodic expressions using six elementary operators: shift, multiplication and division of events and time. In this context, we show how to develop the transfer matrix computation based on a matrix decomposition called core decomposition.

A Conv‐GEMM reconfigurable accelerator with WS‐RS dataflow for high throughput processing

AbstractConvolution and matrix operations are both important computations in Deep Neural Networks (DNNs). However, the significant differences between convolution and matrix computation patterns have posed a challenge in efficiently supporting both convolution (Conv) and general matrix multiplication (GEMM) on hardware design. This paper proposes a Conv‐GEMM reconfigurable accelerator architecture for high throughput edge processing. A weight stationary‐row streaming (WS‐RS) dataflow scheme is proposed, which maximizes data reuse through hierarchical memory structures and flexible PE connections, and supports high throughput edge‐based deep learning algorithms. Based on the proposed dataflow, multi‐scale memory access network (MMAN), reconfigurable accumulator array (RAA), and configurable instruction set architecture (ISA) are designed to optimize computation throughput and energy efficiency. The accelerator is designed under 65 nm technology, achieves peak performance of 1.15 TOPS at 250 MHz, with an energy efficiency of 1.14 TOPS/W. The GEMM computation achieves 85.7% latency improvement and the Mobilenet‐V1 processing achieves a throughput of 529 fps under a 256 × 224 image size and an 87.15% (top‐5) accuracy on the ImageNet dataset.

Analisis Daya Saing, Mutu dan Pemasaran Kopi di Desa Karang Lantang Kecamatan Muara Jaya Kabupaten Oku

The competitiveness of coffee makes coffee quality the main requirement in marketing coffee products. This research aims to analyze the competitiveness, quality and marketing of coffee in Karang Lantang Village, Muara Jaya District, OKU Regency. The research method used is a survey method and data collection using simple random sampling. The number of respondents used in this research was 30 samples. Analysis of coffee competitiveness data uses Police Analysis Matrix (PAM) calculations, quality analysis uses perception questionnaires so that the results are in percentage form, and for marketing it is analyzed using marketing margins. The results of this research are that coffee farmers in Karang Lantang Village, Muara Jaya District, OKU Regency have private profits of Rp. 48,378,333 and social benefits of Rp. 39,518,333. Coffee farmers in Karang Lantang Village, Muara Jaya District, OKU Regency have strong competitiveness because they have comparative advantages and competitive advantages with a DRC value of 0.023 and competitive advantages with a PCR value of 0.021. The results of the quality of coffee in Karang Lantang village, Muara Jaya District, OKU Regency, are the picking process, sorting processing, packaging and warehousing with a number of frequently carried out as many as 342.86 percent and very often carried out as much as 57.14 percent. Coffee farmers in Karang Lantang Village, Muara Jaya District, OKU Regency have haphazard marketing of coffee sales by selling to collectors. Coffee marketing in Karang Lantang Village shows results of around 3.03%. The convenience of a cash economy and no special treatment for the coffee being sold are the main reasons why farmers sell coffee to collectors.

Review of Matrix Rank Constraint Model for Impulse Interference Image Inpainting

Camera failure or loss of storage components in imaging equipment may result in the loss of important image information or random pulse noise interference. The low-rank prior is one of the most important priors in image optimization processing. This paper reviews and compares some low-rank constraint models for image matrices. Firstly, an overview of image-inpainting models based on nuclear norm, truncated nuclear norm, weighted nuclear norm, and matrix-factorization-based F norm is presented, and corresponding optimization iterative algorithms are provided. Then, we use different image matrix low-order constraint models to recover satellite images from three types of pulse interference and provide our experimental visual and numerical results. Finally, it can be concluded that the method based on the weighted nuclear norm can achieve the best image restoration effect. The F norm method based on matrix factorization has the shortest computational time and can be used for large-scale low-rank matrix calculations. Compared with nuclear norm-based methods, weighted nuclear norm-based methods and truncated nuclear norm-based methods can significantly improve repair performance.

Structure-exploiting interior-point solver for high-dimensional entropy-sparsified regression learning

The solution of high-dimensional nonlinear regression problems through standard machine learning approaches often relies on first-order information, due to the numerical and memory challenges arising from the computation of the Hessian matrix and of the higher-order derivatives. While this scenario seems not favorable to second-order methods, here we show that an efficient and modular structure-exploiting interior-point solver can be successfully applied to the recently introduced class of entropy-based methods for regression learning. Specifically, by exploiting the favorable structure of the problem and of the Hessian matrix, we suggest a robust solution strategy based on explicit low-rank updates combined with an iterative Symmetric Quasi-Minimal Residual (SQMR) algorithm to solve the underlying system of linear equations. The results show that the proposed structure-exploiting solver – which relies on the hybrid parallelism and distributed-memory computing paradigm – allows a significant solution time speed-up with respect to a naive solution strategy. Furthermore, through an adequate use of the Message Passing Interface (MPI) and of Open Multi-Processing (OpenMP), the proposed solver enables the solution of large-scale problems on high-performance computing architectures consisting of thousands of compute nodes. The accompanying detailed convergence and performance analyses demonstrate both numerical robustness and high-performance capabilities for increasingly high-dimensional problems.

A benchmark of optimization solvers for genome-scale metabolic modeling of organisms and communities.

Genome-scale metabolic modeling is a powerful framework for predicting metabolic phenotypes of any organism with an annotated genome. For two decades, this framework has been used for the rational design of microbial cell factories. In the last decade, the range of applications has exploded, and new frontiers have emerged, including the study of the gut microbiome and its health implications and the role of microbial communities in global ecosystems. However, all the critical steps in this framework, from model construction to simulation, require the use of powerful linear optimization solvers, with the choice often relying on commercial solvers for their well-known computational efficiency. In this work, I benchmark a total of six solvers (two commercial and four open source) and measure their performance to solve linear and mixed-integer linear problems of increasing complexity. Although commercial solvers are still the fastest, at least two open-source solvers show comparable performance. These results show that genome-scale metabolic modeling does not need to be hindered by commercial licensing schemes and can become a truly open science framework for solving urgent societal challenges.IMPORTANCEModeling the metabolism of organisms and communities allows for computational exploration of their metabolic capabilities and testing their response to genetic and environmental perturbations. This holds the potential to address multiple societal issues related to human health and the environment. One of the current limitations is the use of commercial optimization solvers with restrictive licenses for academic and non-academic use. This work compares the performance of several commercial and open-source solvers to solve some of the most complex problems in the field. Benchmarking results show that, although commercial solvers are indeed faster, some of the open-source options can also efficiently tackle the hardest problems, showing great promise for the development of open science applications.

Mod2VQLS: A Variational Quantum Algorithm for Solving Systems of Linear Equations Modulo 2

This paper presents a system for solving binary-valued linear equations using quantum computers. The system is called Mod2VQLS, which stands for Modulo 2 Variational Quantum Linear Solver. As far as we know, this is the first such proposal. The design is a classical–quantum hybrid. The quantum components are a new circuit design for implementing matrix multiplication modulo 2, and a variational circuit to be optimized. The classical components are the optimizer, which measures the cost function and updates the quantum parameters for each iteration, and the controller that runs the quantum job and classical optimizer iterations. We propose two alternative ansatze or templates for the variational circuit and present results showing that the rotation ansatz designed specifically for this problem provides the most direct path to a valid solution. Numerical experiments in low dimensions indicate that Mod2VQLS, using the custom rotations ansatz, is on par with the block Wiedemann algorithm, which is the best-known to date solution for this problem.

Cross trust: A decentralized MA-ABE mechanism for cross-border identity authentication

With an increasing demand for authenticated data exchange between jurisdictions, ensuring the privacy and security of data interactions is crucial for national security, public health, and economic vitality, becoming a fundamental national infrastructure. Current solutions can be categorized into two types: fully decentralized autonomous systems based on blockchains or centralized solutions that rely on authoritative centers such as certification authorities (CAs). In reality, a balance needs to be struck between guaranteed authority and privacy independence. A certain authority is needed as an authorization guarantee, and decentralization is required to ensure privacy and the independence of the authority. This paper proposes a novel scheme, CT-MA-ABE (Cross-Trust Multiple Authorization Attribute-Based Encryption), to address these issues by implementing MA-ABE for cross-border institutional authorization interactions, utilize blockchain certification authority (BCA) for credibility and encryption-based authorization to protect attribute data privacy. This solution integrates the role of 'notary' in cross-border interactions, addressing the supervision problem in fully decentralized approaches while also considering the trust issue in centralized systems. This paper also introduces the Universal Certificate Authority Pool (UCAP), an innovative hybrid federated authorization method, creatively utilizing the implied authorization conditions of attributes to create a flexible and transitive authorization mechanism based on attribute relationships and extensions, enhancing privacy protection and improving the speed of authorization matrix calculation. The successful deployment of the system between the legal jurisdictions in South China, Zhuhai and Macau as a critical infrastructure component for securing data interactions further demonstrates its effectiveness as a reliable and secure solution.

HyLAC: Hybrid linear assignment solver in CUDA

The Linear Assignment Problem (LAP) is a fundamental combinatorial optimization problem with a wide range of applications. Over the years, significant progress has been made in developing efficient algorithms to solve the LAP, particularly in the realm of high-performance computing, leading to remarkable reductions in computation time. In recent years, hardware improvements in General Purpose Graphics Processing Units (GPGPUs) have shown promise in meeting the ever-increasing compute bandwidth requirements. This has attracted researchers to develop GPU-accelerated algorithms to solve the LAP.Recent work in the GPU domain has uncovered parallelism available in the problem structure to achieve significant performance improvements. However, each solution presented so far targets either sparse or dense instances of the problem and has some scope for improvement. The Hungarian algorithm is one of the most famous approaches to solving the LAP in polynomial time. Hungarian algorithm has classical O(N4) (Munkres') and tree based O(N3) (Lawler's) implementations. It is well established that the Munkres' implementation is faster for sparse LAP instances while the Lawler's implementation is faster for dense instances. In this work, we blend the GPU implementations of Munkres' and Lawler's to develop a Hybrid GPU-accelerated solver for LAP that switches between the two implementations based on available sparsity. Also, we improve the existing GPU implementations to reduce memory contention, minimize CPU-GPU synchronizations, and coalesced memory access. The resulting solver (HyLAC) works faster than existing CPU/GPU LAP solvers for sparse as well as dense problem instances. HyLAC achieves a speedup of up to 6.14× over existing state-of-the-art GPU implementation when run on the same hardware. We also develop an implementation to solve a list of small LAPs (tiled LAP), which is particularly useful in the optimization domain. This tiled LAP solver performs 22.59× faster than the existing implementation.

Norm-Attainable Operators in Hilbert Spaces: Properties and Applications

This research paper explores various properties and implications of norm-attainable operators on Hilbert spaces. We establish lemmas, propositions, and theorems that shed light on the characteristics of these operators and their relationship with the geometry and structure of the underlying Hilbert space. These results have applications in functional analysis, linear algebra, and operator theory.

Are offline ART decisions for NSCLC impacted by the type of dose calculation algorithm?

IntroductionDecisions for plan-adaptations may be impacted by a transitioning from one dose-calculation algorithm to another. This study examines the impact on dosimetric-triggered offline adaptation in LA-NSCLC in the context of a transition from superposition/convolution dose calculation algorithm (Type-B) to linear Boltzmann equation solver dose calculation algorithms (Type-C). Materials & MethodsTwo dosimetric-triggered offline adaptive treatment workflows are compared in a retrospective planning study on 30 LA-NSCLC patients. One workflow uses a Type-B dose calculation algorithm and the other uses Type-C. Treatment plans were re-calculated on the anatomy of a mid-treatment synthetic-CT utilizing the same algorithm utilized for pre-treatment planning. Assessment for plan-adaptation was evaluated through a decision model based on target coverage and OAR constraint violation. The impact of algorithm during treatment planning was controlled for by recalculating the Type-B plan with Type-C. ResultsIn the Type-B approach, 13 patients required adaptation due to OAR-constraint violations, while 15 patients required adaptation in the Type-C approach. For 8 out of 30 cases, the decision to adapt was opposite in both approaches. None of the patients in our dataset encountered CTV-target underdosage that necessitated plan-adaptation. Upon recalculating the Type-B approach with the Type-C algorithm, it was shown that 10 of the original Type-B plans revealed clinically relevant dose reductions (≥3%) on the CTV in their original plans. This re-calculation identified 21 plans in total that required ART. DiscussionIn our study, nearly one-third of the cases would have a different decision for plan-adaption when utilizing Type-C instead of Type-B. There was no substantial increase in the total number of plan-adaptations for LA-NSCLC. However, Type-C is more sensitive to altered anatomy during treatment compared to Type-B. Recalculating Type-B plans with the Type-C algorithm revealed an increase from 13 to 21 cases triggering ART.

Varying coefficients in parallel shared-memory variational splitting solvers for non-stationary Maxwell equations

Direction-splitting implicit solvers employ the regular structure of the computational domain augmented with the splitting of the partial differential operator to deliver linear computational cost solvers for time-dependent simulations. The finite difference community originallye mployed this method to deliver fast solvers for PDE-based formulations. Later, this method was generalized into so-called variationals plitting. The tensor product structure of basis functions over regular computational meshes allows us to employ the Kronecker product structureo f the matrix and obtain linear computational cost factorization for finite element method simulations. These solvers are traditionally usedf or fast simulations over the structures preserving the tensor product regularity. Their applications are limited to regular problems and regularm odel parameters. This paper presents a generalization of the method to deal with non-regular material data in the variational splitting method. Namely, we can vary the material data with test functions to obtain a linear computational cost solver over a tensor product grid with nonregularm aterial data. Furthermore, as described by the Maxwell equations, we show how to incorporate this method into finite element methods imulations of non-stationary electromagnetic wave propagation over the human head with material data based on the three-dimensional MRI scan.