Lagrange Interpolation Research Articles

Triangular finite differences using bivariate Lagrange polynomials with applications to elliptic equations

This paper proposes finite-difference schemes based on triangular stencils to approximate partial derivatives using bivariate Lagrange polynomials. We use first-order partial derivative approximations on triangles to introduce a novel hexagonal scheme for the second-order partial derivative on any rotated parallelogram grid. Numerical analysis of the local truncation errors shows that first-order partial derivative approximations depend strongly on the triangle vertices getting at least a first-order method. On the other hand, we prove that the proposed hexagonal scheme is always second-order accurate. Simulations performed at different triangular configurations reveal that numerical errors agree with our theoretical results. Results demonstrate that the proposed method is second-order accurate for the Poisson and Helmholtz equation. Furthermore, this paper shows that the hexagonal scheme with equilateral triangles results in a fourth-order accurate method to the Laplace equation. Finally, we study two-dimensional elliptic differential equations on different triangular grids and domains.

AUTOMATED SYNTHESIS OF OPTIMAL SCHEMES LOCATION OF PRODUCTION FACILITIES ON RUGGED TERRAIN

An automated method has been developed to synthesize layouts of industrial facilities on rugged terrains, aimed at minimizing costs associated with utilities routes development (construction and operation). Additional conditions of optimization synthesis include constraints on the regulatory minimum permissible clear distances between facilities and requirements that prevent facilities from being placed in restricted zones. Three characteristic types of terrain surface relief sufficient to solve the optimization problem presented under automated design conditions have been identified. The terrain allocated for enterprise placement is specified in the conditions of the optimization problem being solved by a regular grid created based on Delaunay triangulation. To calculate the elevation (applicate) of the center of the geometric image of the facility’s base at any position, a secondary approximation of a local small area is applied using a second-order Lagrange interpolation polynomial. Due to the multimodal nature of the objective function, partly caused by the complexity of the topography of the territory designated for facility location, the use of one of the stochastic population methods, specifically the modified method of differential evolution, is proposed to ensure a search for the global minimum of the objective function. A numerical example of synthesizing facilities layouts is provided, assuming their location on different slopes of a depression. The results of the numerical experiment confirmed the possibility of using the method presented in the paper for automated search of optimal (close to optimal) layouts for facilities on rugged terrains.

Threshold Changeable Secret Sharing (TCSS) via Skew Polynomials

Secret sharing is a tool to divide a secret into multiple shares, so that to reconstruct the secret, it is necessary to collect several shares under certain conditions. Secret sharing was first introduced by Adi Shamir, the scheme is based on polynomials. The secret is represented as a constant value polynomial, and the points on the polynomial graph serve as shares. Secret reconstruction is performed through Lagrange interpolation at a minimum of k out of n points, known as a threshold scheme (k, n). In 2010, Zhang Y. designed a secret sharing scheme through skew polynomials. Involving the role of the automorphism σ in the skew polynomial ring increases the complexity in share distribution and secret reconstruction. In 2012, Zhang Z. designed a secret sharing scheme with a threshold that can change according to the participants present during the reconstruction process, known as Threshold Changeable Secret Sharing (TCSS). This aims to prevent external parties from pretending to be valid participants in order to learn the secret. In this research, a TCSS scheme will be designed using skew polynomials. The aim is to make the TCSS scheme's calculations more complex, making it harder for adversaries to access the secret.

Full C(N)D-study of computational capabilities of Lagrange polynomials

In the article is determined the exact order of limiting error of inaccurate information in the problem of recovery functions from Sobolev classes according to the information received from all possible linear functionals. The speed of recovery is the same as for accurate information, although this property is lost when we multiply the limiting error for the any increasing sequence. As a consequence of this result, in the context of the Computational (numerical) diameter, it is shown that Lagrange spline interpolation is the most effective among all possible computing methods, according to the information by value at points. Computational experiments confirm this conclusion.

Research on Molten Iron Quality Prediction Based on Machine Learning

The quality of molten iron not only has a significant impact on the strength, toughness, smelting cost and service life of cast iron but also directly affects the satisfaction of users. The establishment of timely and accurate blast furnace molten iron quality prediction models is of great significance for the improvement of the production efficiency of blast furnace. In this paper, Si, S and P content in molten iron is taken as the important index to measure the quality of molten iron, and the 989 sets of production data from a No.1 blast furnace from August to October 2020 are selected as the experimental data source, predicting the quality of molten iron by the I-GWO-CNN-BiLSTM model. First of all, on the basis of the traditional data processing method, the missing data values are classified into correlation data, temporal data, periodic data and manual input data, and random forest, the Lagrangian interpolation method, the KNN algorithm and the SVD algorithm are used to complete them, so as to obtain a more practical data set. Secondly, CNN and BiLSTM models are integrated and I-GWO optimized hyperparameters are used to form the I-GWO-CNN-BiLSTM model, which is used to predict Si, S and P content in molten iron. Then, it is concluded that using the I-GWO-CNN-BiLSTM model to predict the molten iron quality can obtain high prediction accuracy, which can provide data support for the regulation of blast furnace parameters. Finally, the MCMC algorithm is used to analyze the influence of the input variables on the Si, S and P content in molten iron, which helps the steel staff control the quality of molten iron in a timely manner, which is conducive to the smooth running of blast furnace production.

Analysis of COVID-19 epidemic with intervention impacts by a fractional operator

This study introduces an innovative fractional methodology for analyzing the dynamics of COVID-19 outbreak, examining the impact of intervention strategies like lockdown, quarantine, and isolation on disease transmission. The analysis incorporates the Caputo fractional derivative to grasp long-term memory effects and non-local behavior in the advancement of the infection. Emphasis is placed on assessing the boundedness and non-negativity of the solutions. Additionally, the Lipschitz and Banach contraction theorem are utilized to validate the existence and uniqueness of the solution. We determine the basic reproduction number associated with the model utilizing the next generation matrix technique. Subsequently, by employing the normalized sensitivity index, we perform a sensitivity analysis of the basic reproduction number to effectively identify the controlling parameters of the model. To validate our theoretical findings, numerical simulations are conducted for various fractional order values, utilizing a two-step Lagrange interpolation technique. Furthermore, the numerical algorithms of the model are represented graphically to illustrate the effectiveness of the proposed methodology and to analyze the effect of arbitrary order derivatives on disease dynamics.

Use of the integration by part to derive beam/plate equilibrium equations from elasticity with applications to beams

This article proposes a rigorous mathematical justification of the derivation of the governing equations of any beam and plate theory by the three three-dimensional (3D) elasticity equations. This is done by the application of the integration by part theorem via weight residual-like methods. The Fis polynomials (subscripts i and s denote the i-direction and the generic s-term of the displacement expansion uis, respectively) related to a given beam/plate theory are used as weight functions for each of the displacement unknown. To show how the method works a number of simple problems are discussed. These are related to the equilibrium equations and boundary conditions of bending beams according to shear deformation theory. Three methods for obtaining these equations are compared in some detail. The first two are the well-known methods used in mechanics of deformable bodies: (1) use of the Principle of Virtual Displacements; (2) use of the six cardinal ‘Newton’ equations of statics. The third method coincides with the one proposed and rigorously justified in this paper which is (3) the direct integration of the equations of 3D elasticity. Beam theories based on Lagrange polynomials are used as well. The latter makes it possible to immediately develop theories for layered beams and obtain so-called ‘layer-wise’ models, which are only applied here to the case of sandwich structures. Three appendixes discuss: A – The problem in which the derivative of one of the unknowns appears among the unknowns, as is the case with the Euler–Bernoulli theory; B – The extension to dynamics; C-Simple derivation of constitutive equations and related closed-form solution via Navier method.

Parameterization of nonlinear aeroelastic reduced order models via direct interpolation of Taylor partial derivatives

The identification of optimally sparse Taylor partial derivatives presents a new opportunity in efficient nonlinear model reduction for complex aeroelastic systems. Unfortunately, for this class of reduced order model (ROM), the robustness that is observed in the linear regime to parameters including; dynamic pressure, control hinge linear stiffness, or even freeplay, can be quickly compromised in the nonlinear regime. In this paper, the nonlinear sensitivity of selected critical parameters is addressed by interpolating a library of nonlinear unsteady aerodynamic ROMs across a compact subspace in dynamic pressure and freeplay magnitude. The ROM, based on Lagrange interpolation of sparse higher-order Taylor partial derivatives, demonstrates excellent precision in modelling high amplitude transonic limit cycle oscillations for an all-movable wing with freeplay, capturing the LCO region (up to 96% of the linear flutter boundary), and for a range of freeplay values.

Approximation of the Hilbert transform on the half–line

The paper concerns the weighted Hilbert transform of locally continuous functions on the semiaxis. By using a filtered de la Vallée Poussin type approximation polynomial recently introduced by the authors, it is proposed a new “truncated” product quadrature rule (VP- rule). Several error estimates are given for different smoothness degrees of the integrand ensuring the uniform convergence in Zygmund and Sobolev spaces. Moreover, new estimates are proved for the weighted Hilbert transform and for its approximation (L-rule) by means of the truncated Lagrange interpolation at the same Laguerre zeros. The theoretical results are validated by the numerical experiments that show a better performance of the VP-rule versus the L-rule.

Numerical study and dynamics analysis of diabetes mellitus with co-infection of COVID-19 virus by using fractal fractional operator

COVID-19 is linked to diabetes, increasing the likelihood and severity of outcomes due to hyperglycemia, immune system impairment, vascular problems, and comorbidities like hypertension, obesity, and cardiovascular disease, which can lead to catastrophic outcomes. The study presents a novel COVID-19 management approach for diabetic patients using a fractal fractional operator and Mittag-Leffler kernel. It uses the Lipschitz criterion and linear growth to identify the solution singularity and analyzes the global derivative impact, confirming unique solutions and demonstrating the bounded nature of the proposed system. The study examines the impact of COVID-19 on individuals with diabetes, using global stability analysis and quantitative examination of equilibrium states. Sensitivity analysis is conducted using reproductive numbers to determine the disease’s status in society and the impact of control strategies, highlighting the importance of understanding epidemic problems and their properties. This study uses two-step Lagrange polynomial to analyze the impact of the fractional operator on a proposed model. Numerical simulations using MATLAB validate the effects of COVID-19 on diabetic patients and allow predictions based on the established theoretical framework, supporting the theoretical findings. This study will help to observe and understand how COVID-19 affects people with diabetes. This will help with control plans in the future to lessen the effects of COVID-19.

Uni/multi variate polynomial embeddings for zkSNARKs

AbstractA zero-knowledge proof is a cryptographic primitive that enables a prover to convince a verifier the validity of a mathematical statement (an NP statement) without revealing any secret inputs to the verifier. A special case, called zero-knowledge Succinct Non-interactive ARgument of Knowledge (zkSNARK) is particularly designed for arithmetic circuit proof systems which have important applications in blockchain privacy. The major computations in this type of zkSNARK proofs with post-quantum security are polynomial evaluations and Lagrange interpolations over finite fields. Given a sequence over a finite field, in the field of coding and sequences research, we understand that there are two representations of the sequence, one is a univariate polynomial and the other, a multivariate polynomial. This is exactly what is done in those zero-knowledge proof systems to transform the proof of a R1CS relation to evaluate uni/multi variate polynomials at some random points in the finite field. In this paper, we present a comparative analysis on how to convert a rank 1 constrained satisfiability (R1CS) system (more general than a circuit system) into a polynomial equality and provide analysis on the concrete complexities of provers, proof sizes and verifiers. We use two concrete zkSNARK schemes, i.e., Polaris, univariate polynomial encodings and Spartan, multivariate polynomial encodings, as examples to show our analysis. Secondly, we propose to select interpolating sets as subfields instead of affine spaces of a large field for Lagrange interpolation. This new method has improved the performance of R1CS encodings largely. We comment that post-quantum secure zkSNARKs yield post-quantum digital signatures with security only depending on symmetric-key schemes. Some open problems are proposed at the end of the paper.

A projection‐based quaternion discretization of the geometrically exact beam model

AbstractIn the present work the geometrically exact beam model is formulated in terms of unit quaternions. A projection‐based discretization approach is proposed which is based on a normalization of the quaternion approximation. The discretization relies on NURBS shape functions and, alternatively, on Lagrangian interpolation. The redundancy of the quaternions is resolved by applying the method of Lagrange multipliers. In a second step the Lagrange multipliers are eliminated circumventing the need to solve saddle point systems. The resulting finite elements retain the objectivity of the underlying beam formulation. Optimal rates of convergence are observed in representative numerical examples.

Dengue dynamics in Nepal: A Caputo fractional model with optimal control strategies

An infectious disease called dengue is a significant health concern nowadays. The dengue outbreak occurred with a single serotype all over Nepal in 2023. In the tropical and subtropical regions, dengue fever is a leading cause of sickness and death. Currently, there is no specified treatment for dengue fever. Avoiding mosquito bites is strongly advised to reduce the likelihood of controlling this disease. In underdeveloped countries like Nepal, the implementation of appropriate control measures is the most important factor in preventing and controlling the spread of dengue illness. The Caputo fractional dengue model with optimum control variables, including mosquito repellent and insecticide use, investigates the impact of alternative control strategies to minimize dengue prevalence. Using the fixed point theorem, the existence and uniqueness of a solution will be demonstrated for the problem. Ulam-Hyers stability, disease-free equilibrium point stability, and basic reproduction number are studied for the proposed model. The model is simulated using a two-step Lagrange interpolation technique, and the least squares method is used to estimate parameter values using real monthly infected data. We then analyze the sensitivity analysis to determine influencing parameters and the control measure effects on the basic reproduction number. The Pontryagin Maximum Principle is used to determine the optimal control variable in the dengue model for control strategies. The present study suggests that the deployment of control measures is extremely successful in lowering infectious disease incidences. Which facilitates the decision-makers to practice rigorous evaluation of such an epidemiological scenario while implementing appropriate control measures to prevent dengue disease transmission in Nepal.

Establishment of fog droplet distribution model and study on canopy deposition uniformity

In plant protection operations, the distribution of droplets affects the atomization effect. To make the distribution of fog droplets more uniform in the air field, a fog droplet distribution model was established based on a three-dimensional motion model of droplets and the particle size distribution function, combined with a two-dimensional normal distribution function. The effects of the initial incidence angle and the additional wind speed on the distribution of fog droplets were analyzed. The fog droplet distribution was simulated to analyze the droplet distribution in the spatial layer, which was compared with the experimental results. To investigate the impacts of different factors on the atomization distribution, the Lagrangian interpolation method was employed, and the optimal initial incidence angle and external wind speed were found. When the initial angle of incidence was 17°, the slope of the fitted curve was the smallest, with a coefficient of determination of 0.9622 and a relative error of 3.12%. With an additional wind speed of 0.1 m/s, the coefficient of determination was 0.9782, with an average error of 4.61%. The simulation results were consistent with the experimental findings, and the accuracy of the fog droplet distribution model was verified. In summary, this research provides a novel method to improve the uniformity of the droplet distribution, which can provide a theoretical basis for determining the operating parameters.

Exterior Orientation Parameter Refinement of the First Chinese Airborne Three-Line Scanner Mapping System AMS-3000

The exterior orientation parameters (EOPs) provided by the self-developed position and orientation system (POS) of the first Chinese airborne three-line scanner mapping system, AMS-3000, are impacted by jitter, resulting in waveform distortions in rectified images. This study introduces a Gaussian Markov EOP refinement method enhanced by cubic spline interpolation to mitigate stochastic jitter errors. Our method first projects tri-view images onto a mean elevation plane using POS-provided EOPs to generate Level 1 images for dense matching. Matched points are then back-projected to the original Level 0 images for the bundle adjustment based on the Gaussian Markov model. Finally, cubic spline interpolation is employed to obtain EOPs for lines without observations. Experimental comparisons with the piecewise polynomial model (PPM) and Lagrange interpolation model (LIM) demonstrate that our method outperformed these models in terms of geo-referencing accuracy, EOP refinement metric, and visual performance. Specifically, the line fitting accuracies of four linear features on Level 1 images were evaluated to assess EOP refinement performance. The refinement performance of our method showed improvements of 50%, 45.1%, 29.9%, and 44.6% over the LIM, and 12.9%, 69.2%, 69.6%, and 49.3% over the PPM. Additionally, our method exhibited the best visual performance on these linear features.

Broadcast signcryption scheme with equality test in smart transportation system

With the generation of massive traffic information in the smart transportation system, the traffic control center efficiently utilizes broadcast communication to send multiple messages to multiple vehicles. Besides, diversified privacy disclosure and security attack issues also emerged spontaneously. To achieve secure communication between the traffic control center and vehicles in the smart transportation system, we design a broadcast signcryption scheme with equality test in the smart transportation system based on the certificateless cryptosystem and equality test. The scheme realizes message confidentiality and vehicle privacy by using the Lagrange interpolation theorem to encrypt messages and vehicle identities, while also achieving classify ciphertext by using the equality test and facilitate road traffic information management. Through numerical experiment analysis, the proposed work has higher operation efficiency and is more suitable for application in smart transportation systems.

On the qualitative behavior and numerical analysis of a calcium oscillation model through Atangana–Baleanu operator in Caputo sense

In this research, we regard a miniature significant for many cell types and activates stimulants against external interventions, thereby causing calcium oscillation. We stage the dependent variables regarding the Atangana–Balenau derivative operator in terms of Caputo so that the model can be physically interpreted better. The study investigates the existence and uniqueness of solutions using fixed point theory, stability in terms of Hyers–Ulam, positivity and numerical solutions. In addition, spacetime, phase diagrams and three-dimensional graphic simulations are presented due to the Lagrange interpolation technique for parameter selections that match the existence and uniqueness condition of the solution.

A modified lattice Boltzmann approach based on radial basis function approximation for the non‐uniform rectangular mesh

AbstractWe have presented a novel lattice Boltzmann approach for the non‐uniform rectangular mesh based on the radial basis function approximation (RBF‐LBM). The non‐uniform rectangular mesh is a good option for local grid refinement, especially for the wall boundaries and flow areas with intensive change of flow quantities. Which allows, the total number of grid cells to be reduced and so the computational cost, therefore improving the computational efficiency. But the grid structure of the non‐uniform rectangular mesh is no longer applicable to the classic lattice Boltzmann method (CLBM), which is based on the famous BGK collision‐streaming evolution. This is why the present study is inspired by the idea of the interpolation‐supplemented LBM (ISLBM) methodology. The ISLBM algorithm is improved in the present manuscript and developed into a novel LBM approach through the radial basis function approximation instead of the Lagrangian interpolation scheme. The new approach is validated for both steady states and unsteady periodic solutions. The comparison between the radial basis function approximation and the Lagrangian interpolation is discussed. It is found that the novel approach has a good performance on computational accuracy and efficiency. Proving that the non‐uniform rectangular mesh allows grid refinement while obtaining precise flow predictions.

A test of meta-heuristic algorithms for parameter extraction of next-generation solar cells with S-shaped current–voltage curves

Identifying parameters of photovoltaic (PV) models based on measured current–voltage (IV) characteristic curves is critical for simulating, evaluating, and controlling PV systems. IV characteristics of the latest-generation solar cells (SCs) often display an S-shaped deformation. In this paper, we explore the potential of meta-heuristic algorithms to address the parameter estimation problems associated with PV cells that exhibit S-shaped IV characteristics. This estimation is performed within the framework of the opposed two-diode model. We implemented a total of 14 algorithms from various classes to extract the SC parameters from synthetic IV curves, which were generated using a range of parameter values. The results were compared by using nonparametric statistical methods. These methods include the Wilcoxon signed-rank test for pairwise comparisons, and the Friedman, Friedman Aligned, and Quade tests for multiple comparisons. Comprehensive results and analyses show that the STLBO (Simplified teaching–learning based optimization algorithm) and ADELI (Adaptive differential evolution with the Lagrange interpolation argument) algorithms demonstrate highly competitive performance in terms of accuracy and reliability. This paper underscores the efficacy of advanced meta-heuristic algorithms in solving complex non-linear optimization problems in the domain of photovoltaic research, particularly concerning the unique challenges posed by S-shaped IV characteristics of new-generation solar cells.

Solving Nonlinear Wave Equations Based on Barycentric Lagrange Interpolation

In this paper, we deeply study the high-precision barycentric Lagrange interpolation collocation method to solve nonlinear wave equations. Firstly, we introduce the barycentric Lagrange interpolation and provide the differential matrix. Secondly, we construct a direct linearization iteration scheme to solve nonlinear wave equations. Once again, we use the barycentric Lagrange interpolation to approximate the (2+1) dimensional nonlinear wave equations and (3+1) dimensional nonlinear wave equations, and describe the matrix format for direct linearization iteration of the nonlinear wave equations. Finally, the comparative experiments show that the barycentric Lagrange interpolation collocation method for solving nonlinear wave equations have higher calculation accuracy and convergence rate.