Lagrange Interpolation Technique Research Articles

Verifiable quantum secret sharing scheme based on orthogonal product states

Abstract In the domain of quantum cryptography, the implementation of quantum secret sharing stands as a pivotal element. In this paper, we propose a novel verifiable quantum secret sharing protocol using the $d$-dimensional product state and Lagrange interpolation techniques. This protocol is initiated by the dealer Alice, who initially prepares a quantum product state, selected from a predefined set of orthogonal product states within the $\mathbb{C}^d \otimes \mathbb{C}^d$ framework. Subsequently, the participants execute unitary operations on this product state to recover the underlying secret. Furthermore, we subject the protocol to a rigorous security analysis, considering both eavesdropping attacks and potential dishonesty from the participants. Finally, we conduct a comparative analysis of our protocol against existing schemes. Our scheme exhibits economies of scale by exclusively employing quantum product states, thereby realizing significant cost-efficiency advantages. In terms of access structure, we adopt an $(t,n)$-threshold architecture, a strategic choice that augments the protocol's practicality and suitability for diverse applications. Furthermore, our protocol includes a rigorous integrity verification mechanism to ensure the honesty and reliability of participants throughout the execution of the protocol.

A three step seventh order iterative method for solution nonlinear equation using Lagrange interpolation technique

This research paper comprehensively presents the derivation of a seventh-order iteration scheme designed to obtain simple roots of nonlinear equations through the utilization of Lagrange interpolation technique. The scheme is characterized by the requirement for three function evaluations and one evaluation of the first derivative in each iteration. A detailed convergence analysis is also carried out to assess the efficacy of the proposed method. Additionally, the paper includes comprehensive numerical experiments aimed at confirming the theoretical results and illustrating the competitive performance of the derived iteration scheme.

Coupled thermal-mechanical analysis of power electronic modules with finite element method and parametric model order reduction

This work presents a new approach for performing a parametric study and examining nonlinear material behaviours of a coupled thermal-mechanical model of a Power Electronics Module (PEM) by integrating the Finite Element Method (ANSYS-FEM) with Parametric Model Order Reduction (pMOR). The considered coupling method solves the thermal and structural models concurrently compared to the widely practised sequential coupling method. Instead of constant parameter values, which are generally regarded for pMOR studies, the temperature-dependent material properties of the wire material have been parametrised in the work using the pMOR method. A generalised 2D model has been regarded here for thermal-mechanical analysis with the pMOR approach, parametrising temperature-dependent coefficient of thermal expansion (CTE) and Young’s modulus (E) of the wire material to explore their impact on wire bonds. The matrix interpolation method has been applied here for the pMOR study, and PRIMA, a Krylov subspace-based model order reduction (MOR) technique, has been exercised for local model order reductions. A new efficient process of matrix interpolation, based on the Lagrange interpolation technique, has been developed to implement matrix interpolation in the parametric reduced order model (pROM). The local reduced order models (ROMs) have a degree of freedom (DOF) of just 8, compared to the full-order models’ (FOMs) of 50,602. The pROM provides an excellent solution and reduces computational time by 84% for the presented case.

A nonlinear system of hybrid fractional differential equations with application to fixed time sliding mode control for Leukemia therapy

In this article, a coupled nonlinear problem of hybrid fractional differential equations (HFDEs) is presented for the qualitative work and numerical results. Two types of operators are involved in the research problem. One of them is DβrR which represent Riemann–Liouville's (RL) fractional derivatives while the operator L(DϱrR) is a series operator and DϱrR's are RL operators such that βr,ϱr∈(0,1]. These operators are joined by Φp operator. As a result, we have a nonlinear coupled system of FDEs. The newly established nonlinear system is studied for the existence, uniqueness criteria, stability of the solutions, and numerical computations. For the theoretical results, we take help from the available literature about the fixed point (FP) techniques. Then a computational scheme is developed with the help of Lagrange's interpolation technique. An application of the problem as a particular case is presented in the sense of the Leukemia mathematical model. The model presents the infection propagation. Leukemia can be managed by providing a chemotherapeutic treatment generally accepted to be safe, and a fractional-order fixed-time terminal sliding mode control has been developed to achieve this goal of removing Leukemic cells while keeping a sufficient number of normal cells. In order to evaluate the proposed controller stability, the fixed-time Lyapunov stability theory is employed. To better illustrate the study, comparison simulations are shown, demonstrating that the suggested control approach has higher tracking and convergence performance.

Numerical investigation of convection–diffusion model by using the new upwind finite volume approach

In this study, we constructed a numerical technique for the simulation of the convection–diffusion problem under convection dominancy. Lagrange interpolation technique is applied to obtain new expressions for the approximation of variable at the interfaces of control volume. Moreover, based on these interface approximations new numerical scheme is developed to approximate convection–diffusion phenomena. The Crank–Nicolson approach is applied for the temporal approximation. This newly constructed numerical scheme is unconditionally stable with second-order accuracy in time and space both. Numerical tests are carried out for the justification of the new algorithm. A comparison of numerical results produced by proposed technique and some other numerical approaches is presented. This comparison indicates that for convection dominant phenomena, the numerical solution of conventional finite volume method contains with non-physical oscillations which analyze that proposed numerical technique results in a high accurate and stable solution.

Integrity Protection for Data Aggregation in Smart Grid

To evaluate the working state of the smart grid, users need to count the status of devices collected by multiple smart meters. To avoid network congestion during data collection processes in the smart grid, an aggregation protocol is required to aggregate messages from multiple smart meters into one short message. However, since these messages may contain sensitive information of smart meters, privacy is a fundamental requirement for such a data aggregation protocol. At the same time, to avoid tampering of transmitted messages, data integrity is another important requirement during data aggregation. Currently, the privacy issue has been well addressed by using homomorphic encryption algorithms such as the Paillier cryptographic system, where multiple smart meters use the same encryption key for encrypting different messages. However, the integrity issue is much harder than the privacy issue since multiple smart meters use different private keys for signing different messages. To address the integrity issue, we propose a novel data aggregation protocol for the smart grid, called DASG. In DASG, we shall show how to aggregate signatures from multiple smart meters using the Chinese Remainder theorem and the Lagrange interpolation techniques, respectively. Since these two techniques can aggregate multiple messages into one, DASG is quite light-weight. Therefore, our newly designed data aggregation protocol for the smart grid can achieve both security and efficiency goals in the smart grid environment. Experimental results show that DASG is feasible for real world applications.

High-order finite element method on a Vulanović–Bakhvalov mesh for a singularly perturbed convection–diffusion problem

A kth order (k≥1) finite element method on a Vulanović–Bakhvalov mesh is proposed for a singularly perturbed convection–diffusion problem. Some properties of the Vulanović–Bakhvalov mesh are derived. Based on these estimations and a new Lagrange interpolation technique, an optimal order of uniform convergence with respect to the perturbation parameter is proved. Numerical results are provided to demonstrate the theoretical findings.

Interpolation based formulation of the oscillatory finite Hilbert transforms

A novel quadrature based on interpolation is proposed for approximation of the finite Hilbert transforms with high-frequency parameter. The transforms are arduous to evaluate due to the existence of Cauchy-type singularity and oscillatory kernel. Accordingly, we fixate the Lagrange interpolation technique with asymptotic-meshfree collocation method to tackle the two challenges. Applying Lagrange’s interpolation and root-coefficient relation of a polynomial to avoid Cauchy-type singularity in the domain. After a large manipulation, the finite Hilbert transform is reduced to a sum of finite Fourier transforms and a singular part integral. The meshfree collocation method is implemented to evaluate the oscillatory Fourier transforms, while the singular part integral is evaluated analytically. Numerical examples confirm the accuracy and efficiency of the proposed algorithm. Results of the new procedure are compared with some state of the art methods and assure better performance of the proposed procedure.

Pulse RFI Mitigation in Synthetic Aperture Radar Data via a Three-Step Approach: Location, Notch, and Recovery

In complex electromagnetic environments, synthetic aperture radar (SAR) is severely affected by radio frequency interference (RFI) from other systems, such as ground-based radar, cellular networks, and global positioning systems, and this interference cannot be neglected. Pulse RFI (PRFI), a common form of RFI, can hinder SAR signal processing and image interpretation to varying degrees. The time-domain notch filtering method designed for mitigating PRFI can locate and mitigate the evident PRFI covered in SAR echo data, but it is helpless against PRFI hidden in a strong echo signal. In this article, a three-step approach is proposed to tackle the PRFI problem. In the proposed approach, the first step is to detect and locate PRFI; this is based on eigenvalue decomposition (EVD) and the short-time Fourier transform (STFT). The second step is to notch PRFI, and this is based on a time-domain notch filter. The third step is to recover the notched signal using a novel matrix completion strategy, which integrates with a robust low-rank matrix completion (LRMC) technique—i.e.,the singular value thresholding (SVT) algorithm—and a well-known Lagrange interpolation technique. Experimental results via simulated SAR data, Sentinel-1 level-0 raw data, and L-band airborne SAR raw data demonstrate the performance of the proposed approach.

A radial hydraulic fracture driven by a Herschel–Bulkley fluid

We analyse the influence of fluid yield stress on the propagation of a radial (penny-shaped) hydraulic fracture in a permeable reservoir. In particular, the Herschel–Bulkley rheological model is adopted that includes yield stress and non-linearity of the shear stress. The rock is assumed to be linear elastic, and the fracture is driven by the point source fluid injection with a constant volumetric rate. The fracture propagation condition follows the theory of linear elastic fracture mechanics, and Carter’s leak-off law is selected to govern the fluid exchange process between the fracture and formation. The numerical solution of the problem is found using the algorithm based on Gauss–Chebyshev quadrature and Barycentric Lagrange interpolation techniques. We also construct an approximate solution with the help of the global fluid balance equation and the near-tip region asymptote. The latter approximation is computationally efficient, and we estimate its accuracy by comparing the primary crack characteristics such as opening, pressure, and radius with those provided by the full numerical solution. We present examples corresponding to typical field cases and demonstrate that the addition of yield stress can lead to a shorter radius and wider opening compared to the corresponding case with simpler power-law fluid rheology. Further, we quantify the limiting propagation regimes (or vertex solutions) characterised by the dominance of a particular physical phenomenon. Relative to the power-law results, there are two new vertices that are associated with the domination of yield stress: storage-yield-stress and leak-off-yield-stress. To understand the influence of various problem parameters, we utilise the constructed approximate solution to investigate the dimensionless parametric space of the problem, in which the applicability domains of the limiting solutions are quantified. This enables one to quickly determine whether yield stress provides a strong influence for given problem parameters. • Analysis of the impact of fluid yield stress on radial fracture propagation. • Comparison with the power-law rheological model. • Numerical and rapid semi-analytical approximate solutions for the problem. • Construction of the dimensionless parameter space for the problem. • Quantitative identification of parameters for which yield stress is important.

Propagation of Smallness in Elliptic Periodic Homogenization

The paper is mainly concerned with an approximate three-ball inequality for solutions in elliptic periodic homogenization. We consider a family of second order operators $\mathcal{L}_\epsilon$ in divergence form with rapidly oscillating and periodic coefficients. It is the first time such an approximate three-ball inequality for homogenization theory is obtained. It implies an approximate quantitative propagation of smallness. The proof relies on a representation of the solution by the Poisson kernel and the Lagrange interpolation technique. Another full propagation of smallness result is also shown.

A study on the Application of Lagrange Interpolation to the Estimations of Gas Production Based over Time

In this study, Lagrange interpolation which could be an alternative modeling passing throughout exactly all data points with respect to linear or nonlinear models was applied to the experimental data, the findings of gas production based over time. After getting Lagrange interpolation, extremum points, turning points (if exists) could be found. Also for some intermediate level, its estimate values could be found. Furthermore, for a desired result value, its intermediate level could be estimated. The graphs of Lagrange interpolation were given. The standard errors with confidence level 95% of the values mentioned above could be estimated by using Lagrange interpolation technique. The graphs of standard errors were given. In addition, missing data values can be estimated and compared with observation value.

A Machine Learning Model for Population Analysis among Different States in India which Influences the Socio, Demographic and Economic Needs of Society

In this work Data from 2011 census is taken to identify the state which influences more in Population census among the different states identified. The data is considered from Madhya Pradesh, followed with Utter Pradesh, then to Bihar, Bengal and Orissa. Similarly other case studies are also done for Southern Indian states and North Eastern States. Genetic algorithm will be tried to find the optimal location for the given study. A fitting function is calculated for the population data of 2011 using Lagrange Interpolation technique. This fitting function is given as input to Genetic algorithm to find the optimal state which have maximum influence in the population growth among different states of India as per the Case studies done.

Propagation of a fluid-driven fracture with fracture length dependent apparent toughness

We investigate the growth of a hydraulic fracture assuming a power-law dependence of material toughness with fracture length for plane strain and radial geometries. Such a toughness fracture length dependence models in a simple manner a toughening mechanism for rocks. We develop an efficient numerical method for the hydraulic fracture growth problem combining Gauss-Chebyshev quadrature and Barycentric Lagrange interpolation techniques. Scaling and numerical results demonstrate that the transition from the viscosity to the toughness dominated regime occurs earlier. The toughness dominated regime always governs growth at large time for both geometries. In all cases, larger net pressure and shorter length are obtained. The solution is very well approximated by the existing constant toughness solutions using the instantaneous value of toughness. If the apparent fracture toughness saturates beyond a given length scale, the solution transitions back to the constant toughness solutions.

A steady barycentric Lagrange interpolation method for the 2D higher‐order time‐fractional telegraph equation with nonlocal boundary condition with error analysis

In this paper, we consider the numerical solution of the time‐fractional telegraph equation with a nonlocal boundary condition. A novel barycentric Lagrange interpolation collocation method is developed to solve this equation. Two difficulties have been sorted: the singularity of the integration and the higher accuracy. At the same, we put forward a steady barycentric Lagrange interpolation technique to overcome the new “Runge” phenomenon in computation. Error estimates of the barycentric Lagrange interpolation and the time‐fractional telegraph system for the present method are presented in Sobolev spaces. High convergence rates of the proposed method are obtained and are consisted with the numerical values. Especially in the time dimension, we get the error bound, for h‐refinement and for nt‐density in the L2 norms. The numerical results obtained show that the proposed numerical algorithm is accurate and computationally efficient for solving time‐fractional telegraph equation. Experiments demonstrate the high convergence rates of the proposed method are consisted with the theoretical values.

Split-step theta method for stochastic delay integro-differential equations with mean square exponential stability

In this paper, we propose the split-step theta method for stochastic delay integro-differential equations by the Lagrange interpolation technique and investigate the mean square exponential stability of the proposed scheme. It is shown that the split-step theta method can inherit the mean square exponential stability of the continuous model under the linear growth condition and the proposed stability condition by the delayed differential and difference inequalities established in the paper. A numerical example is given at the end of the paper to illustrate the method and conclusion of the paper. In addition, the convergence of the split-step theta method is proved in the Appendix.

Meshfree approach for solving multi-dimensional systems of Fredholm integral equations via barycentric Lagrange interpolation

A highly accurate meshfree approach based on the barycentric Lagrange basis functions is reported for solving the linear and nonlinear multi-dimensional systems of Fredholm integral equations (FIEs) of the second kind. The method is an improved Lagrange interpolation technique which possesses high precision and inexpensive procedure. The systems of FIEs are handled with the barycentric formula and transformed into the corresponding linear and nonlinear systems of algebraic equations. The stability of the solved integral equation systems is guaranteed by the numerical stability of the barycentric Lagrange formula, which is dominated by barycentric weights. The convergence analysis and error estimation are included. Some computational results indicate that the method is simple and efficient.

Application of Mathematical Interpolation Technique in Nano Based Insulating Mineral Oil

In general, insulating oil serves as an insulation and cooling medium of power transformers and different High Voltage (HV) application. The properties of insulating oil like breakdown voltage (BDV), viscosity, flash point and fire point are enhanced by adding of different additives in an optimal way. In this work, a mathematical model has been developed by using Lagrange interpolation technique for predicting the critical properties values in different volume concentrations (in Vol %). The predicted critical properties are compared with the set of measured data. Based on this approach, accuracy of predicted value has been obtained in between the two available measured data in two known different volume concentrations. From the volume concentrations (0-0.01%), the error seems to be negligible. Hence, Lagrange interpolation technique is considered as a reliable technique for predicting the intermediate value for different concentrations of nano powder doped insulating oil used in different HV power apparatus.

2-D High-Order DGTD Method and Its Application in Analysis of Sheath Propagation Characteristics

A 2-D discontinuous Galerkin time-domain (DGTD) scheme based on high-order hierarchical basis function is presented in this paper. Combined with the shift operator method solving the electromagnetic problem of dispersive media in finite-difference time domain, the iterative formula of DGTD method in dealing with plasma media is given. On this basis, taking the example of line element radiation, the computing ability of different basis functions is compared. To study the transmission characteristics of the electromagnetic (EM) wave in sheath, the DGTD calculation models of the plasma sheath of a blunt cone at different flight heights and velocities are established using Lagrange interpolation technique based on the calculation results of COMSOL (business software). The influences of antenna position, flight altitude, and flight speed on transmission characteristics of the EM wave are discussed through calculation and analyses.

High-resolution regional gravity field model of Pakistan based on best residual terrain model (RTM) of topography and interpolation techniques

This study focuses on the development of absolute gravity model for Pakistan based on best possible residual terrain model of gravity using residual terrain modeling technique. The datasets used for the development of model are observed gravity, global gravity models, and Shuttle Radar Topographic Mission (SRTM30) elevation data. The residual terrain modeling technique has been used in the remove-restore procedure for smoothing the observed gravity field. Different topographic elevation models were tested in the model selection and one best possible model with minimum mean and standard deviation was selected for residual terrain effects. Least square collocation technique has been used for quality control and error estimates. The best possible covariance model was established from residual gravity for onward prediction of gravity anomalies at the earth surface for error and prediction analysis. The residual terrain effect of gravity, value of free air anomaly from EGM96, and observed free air anomaly are added to normal gravity to compute the absolute gravity at earth surface. The prediction of these parameters is made by employing Lagrange interpolation with least square adjustment. The results are compared with ∼5% randomly selected data points not utilized for the development of covariance function and/or model development. Spline interpolation technique has also been used for the prediction of gravity field-related parameters. Lagrange interpolation exhibits relatively superior results over spline-based interpolation. This is as per expectation due to the reason that additional gridding for spline interpolation filters the signal part as well. This fact is evident from the results of spline interpolation of Grid-I and Grid-II with relatively better prediction results in Grid-I. This version of the model is capable of prediction having limiting error of 30 mGal. The predicted results show that 96.16% of prediction data falls within above-mentioned limit with Lagrange interpolation technique with least square adjustment for whole Pakistan area. The adverse effect of gridding is absent in case of Grid-I due to relatively flat areas and predicted data matches totally with control values for both spline as well as Lagrange interpolations. However, in case of Grid-II which includes high mountains of Himalaya, gridding effect is present and the accuracy of the predicted results falls to ∼92%. The computed results have been compared with absolute values predicted using EGM96 and EGM2008 models as well. The gravity field recovered with PAKGM model is much better, i.e., ∼ 96.16%, than both with EGM96 and EGM2008 which is about 85% only.