Linear Solver Research Articles

The Power of Randomly Concrete Examples and Numerical Demonstrations

Householder reflector is a must-taught special unitary matrix in numerical linear algebra. It has plentiful theoretical as well as numerical properties, and it plays a significant role in practical algorithms. We use Householder reflectors as the linking thread to teach techniques and concepts on three topics: Deriving needed formulas, avoiding numerical cancelation errors, and implementing efficient code. Each topic is based on simple yet “randomly” concrete numerical examples, which are easy to construct and demonstrate via common MATLAB commands. We use the examples to bring forth salient as well as less-known features of a Householder reflector and related concepts. We also construct an extended version of a Householder reflector to address the perplexing question on the number of complex signs allowed in the reflector. Our numerical demonstrations, although simple in nature, can help students see the big picture of certain methods and concepts. The demonstrations are also designed to arouse students' interest so that they become more attentive in learning the more subtle technical details.

CoReSi: a GPU-based software for Compton camera reconstruction and simulation in collimator-free SPECT.

Compton cameras are imaging devices that may improve observation of sources of γ photons. We present CoReSi, a Compton Reconstruction and Simulation software implemented in Python and powered by PyTorch to leverage multi-threading and for easy interfacing with image processing and deep learning algorithms. The code is mainly dedicated to medical imaging and for near-field experiments where the images are reconstructed in 3D. It includes state-of-the-art mathematical models from the literature, from the simplest, which allow limited knowledge of the sources, to more sophisticated ones with a finer description of the physics involved. It offers flexibility in defining the geometry of the Compton camera and the detector materials. Several identical cameras can be considered at arbitrary positions in space. The main functions of the code are dedicated to the computation of the system matrix, leading to the forward and backward projector operators. These are the cornerstones of any image reconstruction algorithm. A simplified Monte Carlo data simulation function is provided to facilitate code development and fast prototyping. The code will be released under an open source licence after acceptance of the paper.

Combining a Standardized Growth Class Assessment, UAV Sensor Data, GIS Processing, and Machine Learning Classification to Derive a Correlation with the Vigour and Canopy Volume of Grapevines

Assessing vines’ vigour is essential for vineyard management and automatization of viticulture machines, including shaking adjustments of berry harvesters during grape harvest or leaf pruning applications. To address these problems, based on a standardized growth class assessment, labeled ground truth data of precisely located grapevines were predicted with specifically selected Machine Learning (ML) classifiers (Random Forest Classifier (RFC), Support Vector Machines (SVM)), utilizing multispectral UAV (Unmanned Aerial Vehicle) sensor data. The input features for ML model training comprise spectral, structural, and texture feature types generated from multispectral orthomosaics (spectral features), Digital Terrain and Surface Models (DTM/DSM- structural features), and Gray-Level Co-occurrence Matrix (GLCM) calculations (texture features). The specific features were selected based on extensive literature research, including especially the fields of precision agri- and viticulture. To integrate only vine canopy-exclusive features into ML classifications, different feature types were extracted and spatially aggregated (zonal statistics), based on a combined pixel- and object-based image-segmentation-technique-created vine row mask around each single grapevine position. The extracted canopy features were progressively grouped into seven input feature groups for model training. Model overall performance metrics were optimized with grid search-based hyperparameter tuning and repeated-k-fold-cross-validation. Finally, ML-based growth class prediction results were extensively discussed and evaluated for overall (accuracy, f1-weighted) and growth class specific- classification metrics (accuracy, user- and producer accuracy).

Expanded Generalized Needlet Internal Linear Combination (eGNILC) Framework for the 21 cm Foreground Removal

Abstract The Generalized Needlet Internal Linear Combination (GNILC) method is a nonparametric component separation algorithm to remove the foreground contamination of the 21 cm intensity mapping data. In this work, we perform the discrete cosine transform along the frequency axis in the expanded GNILC framework (denoted eGNILC), which helps reduce the power loss in low multipoles, and further demonstrates its performance. We also calculate the eGNILC bias to modify the criterion for determining the degrees of freedom (dof) of the foreground, and embed the robust principal component analysis in mixing matrix computation to obtain a blind component separation method. We find that the eGNILC bias is related to the averaged domain size and the dof of the foreground but not the underlying 21 cm signal. In the case of no beam effect, the eGNILC bias is negligible for simple power-law foregrounds outside the Galactic plane. We also examine the eGNILC performance in the SKA Phase-I in mid-frequency (SKA-MID) and Baryon Acoustic Oscillations from Integrated Neutral Gas Observations (BINGO) simulations. We show that if the adjacent frequency channels are not highly correlated, eGNILC can recover the underlying 21 cm signal with good accuracy. With the varying Airy-disk beam applied to both SKA-MID and BINGO, the power spectra of 21 cm can be effectively recovered at the multipoles ℓ ∈ [20, 250] and [20, 300], respectively. With no instrumental noise, the SKA-MID exhibits ≲20% power loss, and BINGO exhibits ~10% power loss. The varying Airy-disk beam only causes significant errors at large multipoles.

Piecewise linear value function approximations in nonlinear dynamic scheduling problems with VTOLs

Modern vertical take-off and landing vehicles (VTOLs) could significantly affect the future of mobility. The broad range of their application fields encompasses urban air mobility, transportation and logistics as well as reconnaissance and observation missions in a military context. This article presents an efficient numerical framework for dynamic scheduling of multiple VTOLs. It combines a scheduling problem with optimal trajectory planning. The whole setting is formulated as a mixed-integer bilevel optimization problem. At the upper level, VTOLs are scheduled, and their starting times are computed. The solution of the lower level problem involves the computation of a value function and yields optimal trajectories for every aerial vehicle. In order to solve the bilevel problem, it is recast into a single-level one. The resulting mixed-integer nonlinear program is then piecewise linearized and solved numerically by a mixed-integer linear solver based on the Branch-and-Bound algorithm. Numerical results prove the feasibility of the approach proposed herein.

Hybrid affine cipher and eigenvector methods for cryptography

This paper systematically presents a specific application of linear algebra in information security and cryptography, highlighting the crucial role of matrix operations and linear techniques in the design and analysis of encryption algorithms. Specifically, we focus on a method that utilizes linear algebra to enhance the security and efficiency of modern cryptographic systems. A detailed illustrative example is provided to help readers better understand how these mathematical tools are applied in practice. Furthermore, we review existing encryption techniques and propose a new matrix-based scheme that leverages linear algebra to improve data security, optimize the encryption-decryption process, and ensure the integrity and confidentiality of information throughout transmission and storage.

Realizing linear synaptic plasticity in electric double layer-gated transistors for improved predictive accuracy and efficiency in neuromorphic computing

Abstract Neuromorphic computing offers a low-power, parallel alternative to traditional von Neumann architectures by addressing the sequential data processing bottlenecks. Electric double layer-gated transistors (EDLTs) resemble biological synapses with their ionic response and offer low power operations, making them suitable for neuromorphic applications. A critical consideration for artificial neural networks (ANNs) is achieving linear and symmetric plasticity (i.e. weight updates) during training, as this directly affects accuracy and efficiency. This study uses finite element modeling to explore EDLTs as artificial synapses in ANNs and investigates the underlying mechanisms behind the nonlinear weight updates observed experimentally in previous studies. By solving modified Poisson–Nernst–Planck equations, we examined ion dynamics within an EDL capacitor and their effects on plasticity, revealing that the rates of EDL formation and dissipation are concentration-dependent. Fixed-magnitude pulse inputs result in decreased formation and increased dissipation rates, leading to nonlinear weight updates. For a pulse magnitude of 1 V, both 1 ms 500 Hz and 5 ms 100 Hz pulse inputs saturated at less than half of the steady state EDL concentration, limiting the number of accessible states and operating range of devices. To address this, we developed a predictive linear ionic weight update solver (LIWUS) in Python to predict voltage pulse inputs that achieve linear plasticity. We then evaluated an ANN with linear and nonlinear weight updates on the Modified National Institute of Standards and Technology classification task. The ANN with LIWUS-provided linear weight updates required 19% fewer (i.e. 5) epochs in both training and validation than the network with nonlinear weight updates to reach optimal performance. It achieved a 97.6% recognition accuracy, 1.5–4.2% higher than with nonlinear updates, and a low standard deviation of 0.02%. The network model is amenable to future spiking neural network applications, and the performance with linear weight updates is expected to improve for complex networks with multiple hidden layers.

An AI dose-influence matrix engine for robust pencil beam scanning protons therapy.

Rapid planning is of tremendous value in proton pencil beam scanning (PBS) therapy in overcoming range uncertainty. However, the dose calculation of the dose influence matrix (Dij) in robust PBS plan optimization is time-consuming and requires substantial acceleration to enhance efficiency. To accelerate the Dij calculations in PBS therapy, we developed an AI-Dij engine integrated into our in-house treatment planning system (TPS). The AI-Dij engine calculates spot dose using a transformer-based spot dose calculation model (SDM), which takes CT volumes (CT-bars, 256 16 16 voxels, 3mm resolution) and energy (a float value) as inputs and outputs the spot dose distribution (256 16 16). The SDM was trained on over 200 000 CT-bars and Monte Carlo (MC) spot dose (spanning energy levels from 70 to 225 MeV). Clinical-implemented treatment plans for the head, lung, and liver, initially created on Raystation, were replanned using our AI-Dij engine under identical gantry angles and uncertainties settings. After optimizing the spot weight, each in-house plan was recalculated using MCsquare for MC dose evaluation. The dose-volume histogram (DVH) metrics from the in-house TPS and Raystation were compared, evaluating both the optimized and MC doses. In optimization, the differences of DVH metrics (%, Valuein-house-ValueRaystation) across all uncertainty scenarios between the in-house and Raystation plans were 0.93±2.04% for clinical target volume (CTV) and -5.94±12.19% for organ at risks (OARs). For the MC doses, the differences were 2.48±2.78% for CTV and -5.47±14.16% for OARs. The time cost of a robust AI-Dij calculation can be within 2s on an RTX3090 GPU. We conducted a feasibility study on AI-Dij engine-based robust PBS plan optimization, demonstrating both high planning speed and quality.

Linear Algebra Education in University: A Literature Review

The objective of this study is to conduct a thorough literature review in order to give a comprehensive understanding of the opportunities and challenges associated with the teaching and learning of linear algebra in university environments. The review assesses the pedagogical strategies used by academic institutions to enhance student engagement and comprehension, along with the application of linear algebra. We conducted a four-stage process to analyze five articles indexed in Scopus. The review emphasizes the importance of incorporating technology and real-world applications into linear algebra instruction to promote student motivation and active learning. It underscores the necessity of pedagogical approaches that promote student agency, particularly in online learning environments, where traditional teaching practices frequently persist. Exploring the correlation between learning outcomes and semantic networks, evaluating the impact of online platforms on student participation and agency, and evaluating real-world applications for teaching linear algebra are all potential future research directions.

Fast linear solvers for incompressible CFD simulations with compatible discrete operator schemes

Fast linear solvers for incompressible CFD simulations with compatible discrete operator schemes

A SIMPLIFIED FREQUENCY DOMAIN BEAMFORMING APPROACH FOR ACTIVE AND PASSIVE SURFACE WAVE SURVEYS

Frequency domain beamforming (FDBF) and cylindrical frequency domain beamforming (CFDBF) techniques are among the most robust methods for extracting experimental phase velocity dispersion data from wavefields recorded during either active- or passive-source surface wave testing. Both FDBF and CFDBF transformations rely on calculating the spatiospectral correlation matrix to compute the steered power response spectrum, which is essential for extracting surface wave dispersion data. The dimensions of the spatiospectral correlation matrix are directly influenced by the frequency sampling and the number of sensors employed in the survey setup. When the number of sensors is large, and/or the frequency range is broad, and/or the frequency sampling is dense, the spatiospectral correlation matrix size leads to a greater number of required computations to calculate the steered power response spectrum for both the FDBF and CFDBF techniques. Therefore, we propose a simplified and more computationally efficient frequency domain beamforming algorithm for both active and passive surface wave surveys. This method avoids the calculation of the spatiospectral correlation matrix and its multiplication with the Hermitian transpose of the steering vector, thereby reducing the computational complexity by an order. Furthermore, by leveraging vectorization techniques, the proposed algorithm employs matrix multiplications instead of loops, significantly heightening computational efficiency. We rigorously tested the algorithm using surface wave datasets collected across diverse subsurface conditions with varying sensor arrays and acquisition parameters (e.g., sampling frequencies and window lengths) to evaluate its efficacy. The results demonstrate that our proposed algorithm outperforms the conventional FDBF method in the processing time, highlighting its practical effectiveness. This method shows promise for integration into surface wave data processing software, offering substantial improvements in computational efficiency without compromising accuracy

A Review on Mathematical Methods to Approximate µ-Values

In this article, we review a number of iterative and analytical techniques to approximate structured singular values or µ-values which is a straightforward generalization of singular values for square and rectangular matrices. The µ-value is a well-known tool which acts as a strong link between numerical linear algebra and control theory. The computation of structured singular value provides a platform to study and discuss stability, performance, and robustness of the system. Furthermore, we review some very important literature that discusses the applications of structured singular values in different areas of engineering.

Efficient Solution of Sequences of Parametrized Lyapunov Equations With Applications

ABSTRACTSequences of parametrized Lyapunov equations can be encountered in many application settings. Moreover, solutions of such equations are often intermediate steps of an overall procedure whose main goal is the computation of , where denotes the solution of a Lyapunov equation and is a given matrix. We are interested in addressing problems where the parameter dependency of the coefficient matrix is encoded as a low‐rank modification to a seed, fixed matrix. We propose two novel numerical procedures that fully exploit such a common structure. The first one builds upon the Sherman‐Morrison‐Woodbury (SMW) formula and recycling Krylov techniques, and it is well‐suited for small dimensional problems as it makes use of dense numerical linear algebra tools. The second algorithm can instead address large‐scale problems by relying on state‐of‐the‐art projection techniques based on the extended Krylov subspace. We test the new algorithms on several problems arising in the study of damped vibrational systems and the analyses of output synchronization problems for multi‐agent systems. Our results show that the algorithms we propose are superior to state‐of‐the‐art techniques as they are able to remarkably speed up the computation of accurate solutions.

Reducing Numerical Precision Requirements in Quantum Chemistry Calculations.

The abundant demand for deep learning compute resources has created a renaissance in low-precision hardware. Going forward, it will be essential for simulation software to run on this new generation of machines without sacrificing scientific fidelity. In this paper, we examine the precision requirements of a representative kernel from quantum chemistry calculations: the calculation of the single-particle density matrix from a given mean-field Hamiltonian (i.e., Hartree-Fock or density functional theory) represented in an LCAO basis. We find that double precision affords an unnecessarily high level of precision, leading to optimization opportunities. We show how an approximation built from an error-free matrix multiplication transformation can be used to potentially accelerate this kernel on future hardware. Our results provide a roadmap for adapting quantum chemistry software for the next generation of high-performance computing platforms.

Marketing Strategy of the Samosir Lake Toba Regency Culture and Tourism Office in Increasing the Number of Tourists

The urgency of this research is that this research is important because it can provide an in-depth look at the strategies implemented by the Samosir Regency Culture and Tourism Office, so that concrete recommendations can be made to increase the attractiveness and popularity of these tourist destinations. This research aims to provide a deeper understanding of the challenges and opportunities faced by the Samosir Regency Culture and Tourism Office in expanding the number of tourists. The research methods used in this research are data collection obtained from observations, interviews with research subjects, and documentation studies, literature studies, identifying problems, preprocessing, analyzing data using SWOT analysis, BCG analysis, SPACE analysis, and QSPM analysis, data analysis results, final evaluation, research approach with mix method. The results of SWOT analysis on the evaluation of internal factors IFE and evaluation of external factors EFE show that the internal and external positions are in quadrant I, in a growing position and must continue to be improved. The results of the BCG matrix evaluation, Samosir Lake Toba Regency tourist destinations are classified in the Stars position. The results of the SPACE matrix calculation, the tourist position of Samosir Lake Toba Regency is in an aggressive position that occurs when all dimensions are positive. Based on the results of QSPM, 3 prioritized strategies can be taken and implemented as the main steps in developing tourist destinations.

Plug-and-play method for segmenting concrete bridge cracks using the segment anything model with a fractal dimension matrix prompt

This paper addresses the diverse scenarios of bridge crack segmentation, proposing a method for detecting cracks on land and underwater using the Segment Anything Model (SAM) prompted by a fractal dimension matrix. The proposed method does not require additional training and obtains fractal feature information of cracks through fractal dimension matrix calculation. These feature information serve as prompt information for SAM to establish a plug-and-play crack segmentation method. The method achieves high detection performance, with a mean accuracy, IoU, and F1-Score of 99.6 %, 0.89, and 0.95 for land cracks, and 97.6 %, 0.89, and 0.95 for underwater cracks, respectively. This represents a significant improvement over methods that do not use the fractal dimension matrix for SAM prompts. Additionally, the method requires no additional training, showcasing excellent generalizability and practical potential for real-world applications in diverse environments.

Strategies for Developing Hydroponic Lettuce (Lactuca sativa L.) Agribusiness in Samarinda City, Indonesia

This research aims to determine the internal and external factors that can influence hydroponic lettuce agribusiness and to find strategies for developing hydroponic lettuce agribusiness in Samarinda City. This research was carried out in several hydroponic lettuce agribusiness locations in Samarinda City from December 2023 to February 2024. The data collection method was carried out by observation, direct interviews with respondents, and a literature study. The sampling method in this research was carried out by purposive sampling, namely, as many as 10 market-oriented business actors. Data analysis uses the SWOT analysis method and internal and external environmental analysis (IFAS and EFAS). Based on the SWOT analysis, it shows that the matrix calculation results are located in quadrant I, namely the S-O strategy or supporting aggressive strategies, namely business actors can explore various potentials, maintain vegetable quality, optimize cultivation experience, create loyalty programs, and complete facilities.

Optimal local truncation error method on unfitted Cartesian meshes for solution of 3-D wave and heat equations for heterogeneous materials

In the paper we develop the optimal local truncation error method (OLTEM) with the non-diagonal and diagonal mass matrices on unfitted Cartesian meshes for the 3-D time-dependent wave and heat equations for heterogeneous materials with irregular interfaces. 27-point stencils that are similar to those for linear finite elements are used with OLTEM. There are no unknowns for OLTEM on interfaces between different materials; the structure of the global discrete equations is the same for homogeneous and heterogeneous materials. The calculation of the unknown stencil coefficients is based on the minimization of the local truncation error of the stencil equations, includes the use of the interface conditions and the entire PDE in derivations, and yields the optimal fourth order of accuracy of OLTEM with the non-diagonal mass matrix. This is two order higher than for linear finite elements with similar 27-point stencils. OLTEM with the diagonal mass matrix includes the rigorous calculation of the diagonal mass matrix and provides the optimal second order of accuracy. The 3-D numerical results for heterogeneous materials with irregular interfaces show that at the same number of degrees of freedom, OLTEM is even much more accurate than high-order (up to the 7-th order - the highest order in the commercial code COMSOL) finite elements with much wider stencils. Compared to linear finite elements with similar 27-point stencils, at an accuracy of 0.1% OLTEM decreases the number of degrees of freedom by 550–7300 times. This leads to a huge reduction in computation time.A new 3-D post-processing procedure has been developed with OLTEM for the calculation of the spatial derivatives of time-dependent numerical solutions. The spatial derivatives for each grid point are calculated with the help of one compact 27-point stencil (the same as that in basic computations). In contrast to known post-processing procedures, the new approach includes also the use of the time derivatives of the function, the interface conditions and the original PDE. The spatial derivatives of the OLTEM solutions calculated with the new post-processing procedure are much more accurate compared to those obtained by high-order (up to the 7-th order) finite elements with much wider stencils. At an accuracy of 0.1% for the spatial derivatives, OLTEM decreases the number of degrees of freedom by 1.9⋅106–6⋅109 times compared to linear finite elements. The new post-processing procedure can be equally applied to the calculation of the partial derivatives obtained by other numerical methods as well as to the numerical results for other PDEs.Due to the huge reduction in the computation time compared to existing methods and the use of trivial unfitted Cartesian meshes that are independent of irregular geometry, the proposed technique does not require remeshing for the shape change of irregular geometry and it will be effective for many design and optimization problems as well as for multiscale problems without the scale separation.

Generalized Stokes polarimetry design for a 3-D polarization field.

Optical polarization is three-dimensional (3-D). Its complete information is described by the nine-component generalized Stokes vector (GSV). However, existing Stokes polarimetry and its design theory are primarily based on the paraxial four-component Stokes vector and 4 × 4 Mueller matrices. In this Letter, we introduce a novel concept of generalized Stokes polarimetry (GSP), which can reconstruct nine generalized Stokes parameters through a series of non-paraxial polarized modulations and intensity projections. The reconstruction theory of GSP is based on the 9 × 9 generalized Mueller matrix (GMM) calculator we reported previously. In addition, to optimize the 9 × 9 analysis matrix of GSP, we developed an optimization algorithm combined Monte Carlo and gradient descent (GD) methods, finding the optimal configuration with CN = 3.7261 and EWV = 1.2405. The simulated results of noise sources and GSV reconstruction verified the significant improvement in accuracy and stability of optimized configuration.

Numerical Calculation for Form Function of Gun Propellant with Simple Mesh Generation

Abstract A numerical model was built for the form function of gun propellant. To simplify the heavy mesh work, the 3D-grain of propellant would be transformed into the porous medium in control volumes. Then, the iterative calculation of solid fraction matrix could be established to reflect the burn-back of burning surface. Finally, based on the weighting sum of matrix element, the percentage of the propellant mass burned (Ψ) varying with the relative propellant thickness (Z) could be obtained as the numerical results for the form function. The step size of Z is dependent on the size of control volume for the geometry simplification, and the accuracy of Ψ-Z relies on the modification factor connecting with the gradient of solid fraction in the iterative process. The present model could adapt to different geometric conditions, such as 7-pore grain and spherical propellant. The deviation magnitude of Ψ would be less than O(10−2).