Showing papers on "Legendre polynomials published in 2023"

The Legendre Pseudorandom Function as a Multivariate Quadratic Cryptosystem: Security and Applications

Abstract: Abstract Sequences of consecutive Legendre and Jacobi symbols as pseudorandom bit generators were proposed for cryptographic use in 1988. Major interest has been shown towards pseudorandom functions (PRF) recently, based on the Legendre and power residue symbols, due to their efficiency in the multi-party setting. The security of these PRFs is not known to be reducible to standard cryptographic assumptions. In this work, we show that key-recovery attacks against the Legendre PRF are equivalent to solving a specific family of multivariate quadratic (MQ) equation system over a finite prime field. This new perspective sheds some light on the complexity of key-recovery attacks against the Legendre PRF. We conduct algebraic cryptanalysis on the resulting MQ instance. We show that the currently known techniques and attacks fall short in solving these sparse quadratic equation systems. Furthermore, we build novel cryptographic applications of the Legendre PRF, e.g., verifiable random function and (verifiable) oblivious (programmable) PRFs.

Legendre-meshfree vibration analysis of cross-ply laminated elliptical shell of revolution considering the effect of drop-off ply

Abstract: In this study, the vibration investigations on cross-ply laminated closed elliptical shell introducing the effect of symmetrical and unsymmetrical drop-off ply are performed by using a Legendre moving least squares (Legendre-MLS) meshfree method. Several uniformly subshells of uniform thickness without drop-off plies are assembled rigidly in order to consider the drop-off ply included in the elliptical shell, and the corresponding theoretical formulations is derived following the first-order laminated shell theory (FSDT). The boundary conditions of elliptical shell and continuity relationship between adjacent interfaces of substructures are imposed by artificial springs having variable stiffness. On the basis of meshfree theory, Legendre-MLS shape function in conjunction with circumferential Fourier series is developed to approximate the displacement variables. Thus, boundary equations and the governing equations are discretized to obtain the vibration solutions of cross-ply laminated elliptical shell of revolution considering the effect of symmetrical and unsymmetrical drop-off ply. The applicability of the Legendre-meshfree model is clarified by carrying out some comparative evaluations of current results and those from archived literature and finite element simulation models. And then, some parametric studies for the influence of structural boundaries, geometric dimensions, lamination schemes etc., on the vibration properties are performed. • A Legendre-moving least squares meshfree method is proposed. • Accurate vibration prediction for cross-ply laminated elliptical shell of revolution with drop-off ply is achieved. • Remarkable effect of laminated parameters and drop-off ply etc., on the vibration behaviors is revealed.

Solving fractional time-delay diffusion equation with variable-order derivative based on shifted Legendre–Laguerre operational matrices

Abstract: Abstract This article adopts a novel technique to numerical solution for fractional time-delay diffusion equation with variable-order derivative (VFDDEs). As a matter of fact, the variable-order fractional derivative (VFD) that has been used is in the Caputo sense. The first step of this technique is constructive on the construction of the solution using the shifted Legendre–Laguerre polynomials with unknown coefficients. The second step involves using a combination of the collocation method and the operational matrices (OMs) of the shifted Legendre–Laguerre polynomials, as well as the Newton–Cotes nodal points, to find the unknown coefficients. The final step focuses on solving the resulting algebraic equations by employing Newton’s iterative method. To illustrate and demonstrate the technique’s efficacy and applicability, two examples have been provided.

Linear Jacobi-Legendre expansion of the charge density for machine learning-accelerated electronic structure calculations

Abstract: Kohn-Sham density functional theory (KS-DFT) is a powerful method to obtain key materials' properties, but the iterative solution of the KS equations is a numerically intensive task, which limits its application to complex systems. To address this issue, machine learning (ML) models can be used as surrogates to find the ground-state charge density and reduce the computational overheads. We develop a grid-centred structural representation, based on Jacobi and Legendre polynomials combined with a linear regression, to accurately learn the converged DFT charge density. This integrates into a ML pipeline that can return any density-dependent observable, including energy and forces, at the quality of a converged DFT calculation, but at a fraction of the computational cost. Fast scanning of energy landscapes and producing starting densities for the DFT self-consistent cycle are among the applications of our scheme.

Spectral Solutions of Even-Order BVPs Based on New Operational Matrix of Derivatives of Generalized Jacobi Polynomials

Abstract: The primary focus of this article is on applying specific generalized Jacobi polynomials (GJPs) as basis functions to obtain the solution of linear and non-linear even-order two-point BVPs. These GJPs are orthogonal polynomials that are expressed as Legendre polynomial combinations. The linear even-order BVPs are treated using the Petrov–Galerkin method. In addition, a formula for the first-order derivative of these polynomials is expressed in terms of their original ones. This relation is the key to constructing an operational matrix of the GJPs that can be used to treat the non-linear two-point BVPs. In fact, a numerical approach is proposed using this operational matrix of derivatives to convert the non-linear differential equations into effectively solvable non-linear systems of equations. The convergence of the proposed generalized Jacobi expansion is investigated. To show the precision and viability of our suggested algorithms, some examples are given.

Accelerations of public transport vehicles: A method to derive representative generic pulses for passenger safety testing

Abstract: Investigating the postural balance and stability of standing passengers of public transport in laboratory or numerical tests requires generic test pulses, which replicate the acceleration/deceleration characteristics of common public transport vehicles such as buses or trams. We propose a method to generate such test pulses based on measured acceleration time series. The method consists of an automated splitting algorithm, an expansion in Legendre polynomials and a weighted mean to obtain average pulses which are not dominated by the events of highest magnitude. As a demonstration, the method is applied to acceleration time series obtained on public buses in normal operation, resulting in scalable generic pulse shapes. These can be used as the basis of a standardised framework for physical and virtual testing addressing the standing passenger problem.

Implementation of the ELECTR module in NJOY

Abstract: The ELECTR module of NJOY is designed to produce complete and accurate multigroup electroatomic cross sections from ENDF/B-VII data[1, 2]. electr produces restricted cross sections consistent with a solution of the multigroup Boltzmann-Fokker-Planck (BFP) equation. Total, elastic, inelastic (collision and bremsstrahlung) cross sections can be averaged using a variety of group structures and weighting functions. The Legendre components of the within-group elastic and group-to-group inelastic collision cross sections are calculated using tabulated data in energy and analytic expressions of the angular deviation recovered from the CEPXS code[3]. Here, we propose an Open-Source implementation of this module, named electr in NJOY2012 and NJOY-2016.[4] electr also computes partial energy deposition and charge deposition cross sections for each reaction and sum these partial contributions. The resulting multigroup constants are written on an intermediate gendf file for later conversion to any desired format.

Spectral Quasilinearization Method for the Numerical Solution of the Non-standard Volterra Integral Equations

Abstract: In this paper, an iterative scheme is proposed through quasilinearization and the Legendre spectral collocation method to approximate the solution of the non-standard Volterra integral equations (NVIEs). At first, the process of quasilinearization serves to turn a NVIE into a sequence of linear Volterra integral equations. Then, in each iteration, the linear problem is solved by the Legendre spectral collocation method. After that, a convergence analysis is performed. The efficiency and accuracy of the proposed scheme are also verified through several illustrative examples.

The Legendre Polynomial Axial Expansion Method

Abstract: Abstract This work presents a new formulation of the axial expansion transport method explicitly using Legendre polynomials for arbitrarily high-order expansions. This new formulation also features an alternative method of axial leakage calculation to allow for nonextruded flat source region meshes. This alternative axial leakage is introduced alongside a balance equation requirement to ensure that neutron balance is preserved in the coarse mesh for a given axial leakage formulation, which allows for effective coarse mesh finite difference acceleration. A matrix exponential table method is derived to allow for fast computations of arbitrarily high-order matrix exponentials for this work and precludes the need for further research into matrix exponential calculations for this method. Numerical results are presented that demonstrate the stability of the axial expansion method in systems with voidlike regions, showcase the speedup from matrix exponential tables, and investigate the axial convergence of the method in terms of both expansion order and mesh size.

Legendre-Petrov-Galerkin Chebyshev spectral collocation method for second-order nonlinear differential equations

Abstract: We propose a Legendre-Petrov-Galerkin Chebyshev spectral collocation method for initial value problems (IVPs) of second-order nonlinear ordinary differential equations (ODEs). The Legendre-Petrov-Galerkin method is applied to time discretization and the nonlinear term is dealt with Chebyshev spectral collocation method. The scheme results in a simple algebraic system by choosing appropriate basis functions. Optimal error estimates in $ L^2 $-norm for the single and multiple interval methods are given. As an application of the method, we construct the space-time spectral schemes for solving some nonlinear time-dependent partial differential equations (PDEs). Numerical experiments suggest the efficiency of the methods.

Data-Driven Intelligent Recognition of Flatness Control Efficiency for Cold Rolling Mills

Abstract: In the production process of strip tandem cold rolling mills, the flatness control system is important for improving the flatness quality. The control efficiency of actuators is a pivotal factor affecting the flatness control accuracy. At present, the data-driven methods to intelligently identify the flatness control efficiency have become a research hotspot. In this paper, a wavelet transform longitudinal denoising method, combined with a genetic algorithm (GA-WT), is proposed to handle the big noise of the measured data from each signal channel of the flatness meter, and Legendre orthogonal polynomial fitting is employed to extract the effective flatness features. Based on the preprocessed actual production data, the adaptive moment estimation (Adam) optimization algorithm is applied, to intelligently identify the flatness control efficiency. This paper takes the actual production data of a 1420 mm tandem cold mill as an example, to verify the performance of the new method. Compared with the control efficiency determined by the empirical method, the flatness residual MSE 0.035 is 5.4% lower. The test results indicate that the GA-WT-Legendre-Adam method can effectively reduce the noise, extract the flatness features, and achieve the intelligent determination of the flatness control efficiency.

A space-time spectral collocation method for solving the variable-order fractional fokker-planck equation

Abstract: A numerical approach for solving the variable-order fractional Fokker-Planck equation (VO-FFPE) is proposed. The computational scheme is based on the shifted Legendre Gauss-Lobatto and the shifted Chebyshev Gauss-Radau collocation methods. The VO-FFPE is written as a truncated series of shifted Legendre and shifted Chebyshev polynomials for space and time variables, respectively. The residuals of the VO-FFPE at the shifted Legendre Gauss-Lobatto and shifted Chebyshev Gauss-Radau quadrature points are estimated. The original problem is converted into a system of algebraic equations that can be solved easily. Several examples are presented to demonstrate the efficacy of the technique.

Pseudospectral Convex Optimization Based Model Predictive Static Programming for Constrained Guidance

Abstract: This article presents a pseudospectral convex optimization-based model predictive static programming (PCMPSP) for the constrained guidance problem. First, the sensitivity relation between the state increment and control correction is reformulated using Legendre–Gauss and Legendre–Gauss–Radau pseudospectral transcriptions. Second, the convex optimal control problem associated with the trajectory optimization is defined by introducing the quadratic performance index. Third, modifications to the initial guess solution and reference trajectory update are introduced to enhance the accuracy and robustness of the algorithm. Finally, a model predictive guidance law is designed based on the proposed PCMPSP algorithm for the air-to-surface missile guidance with impact angle constraint. The simulation results show that the PCMPSP has lower sensitivity to the initial guess trajectory, higher accuracy, as well as faster convergence speed than existing convex programming methods. Moreover, the robustness of the proposed guidance law to uncertainties is demonstrated through the Monte Carlo campaign.

Discrete Event-Triggered Fault-Tolerant Control of Underwater Vehicles Based on Takagi–Sugeno Fuzzy Model

Abstract: This article investigates the fault estimation and fault-tolerant control problem for underwater vehicles with the Takagi–Sugeno (T–S) fuzzy model. In order to deal with the disturbance of complex ocean environment, a stable fuzzy controller for underwater vehicle is proposed to realize efficient operation, which is on the basis of T–S fuzzy model with pitch angle membership function. Meanwhile, to reduce the waste of communication resources, a novel discrete event-triggered control scheme is proposed to use multiple historical sampled data to determine the next release instant. The discrete event-triggered fault-tolerant controller compensates for the influence of system faults by using state estimators and fault estimators. It is noted that the canonical Bessel–Legendre inequality and delay-dependent canonical orthogonal Legendre polynomials play an important role in dealing with the asymptotical stability of the T–S fuzzy delayed model with an $H_{\infty }$ performance. Finally, a simulation example is carried out to show the validity of the presented theorem.

A solution method of slowing-down distribution of energetic particles in tokamaks

Abstract: The eigen equation of pitch-angle distribution derived from the slowing-down distribution equation with an energetic particle source term is solved by using the Legendre series expansion method. An iteration matrix is established when pitch-angle scattering terms become important. The whole pitch-angle region is separated into three parts, two passing regions, and one trapped area. The slowing-down distribution for each region is finally obtained. The method is applied to solve the slowing-down equations with source terms that the pitch-angle distribution is Maxwellian-like, neutral beam injection, and radial drifts. The distribution functions are convergent for each source with different pitch-angle distribution. The method is suitable for solving a kinetic equation that pitch-angle scattering collision is important.

A hybrid approach for piecewise fractional reaction–diffusion equations

Abstract: In this paper, the Caputo and Atangana–Baleanu fractional derivatives are handled to introduce a type of piecewise fractional derivative. More precisely, a linear combination of the Caputo and Atangana–Baleanu fractional derivatives are considered in each sub-interval to define this fractional derivative. It is employed to generate another form of nonlinear reaction–diffusion equations. The orthonormal Legendre polynomials together with the orthonormal piecewise Legendre functions are used to make a hybrid algorithm for this new problem. In this way, an explicit formula for computing the piecewise fractional differentiation of the stated piecewise basis functions is obtained and applied in generating the method. The applicability and validity of the adopted procedure are examined through three examples.

Mollification of Fourier Spectral Methods with Polynomial Kernels

Abstract: Many attempts have been made in the past to regain the spectral accuracy of the spectral methods, which is lost drastically due to the presence of discontinuity. In this article, an attempt has been made to show that mollification using Legendre and Chebyshev polynomial based kernels improves the convergence rate of the Fourier spectral method. Numerical illustrations are provided with examples involving one or more discontinuities and compared with the existing Dirichlet kernel mollifier. Dependence of the efficiency of the polynomial mollifiers on the parameter P is analogous to that in the Dirichlet mollifier, which is detailed by analysing the numerical solution. Further, they are extended to linear scalar conservation law problems.

A Legendre-tau-Galerkin method in time for two-dimensional Sobolev equations

Abstract: This work is devoted to present the Legendre space-time spectral method for two-dimensional (2D) Sobolev equations. Considering the asymmetry of the first-order differential operator, the Legendre-tau-Galerkin method is employed in time discretization and its multi-interval form is also investigated. In the theoretical analysis, rigorous proof of the stability and $ L^2(\Sigma) $-error estimates is given for the fully discrete schemes in both single-interval and multi-interval forms. Being different from the general Legendre-Galerkin method, we specifically take the Fourier-like basis functions in space to save the computing time and memory in the algorithm of the proposed method. Numerical experiments were included to confirm that our method attains exponential convergence in both time and space and that the multi-interval form can achieve improved numerical results compared with the single interval form.

Visualization method for multidimentional random processes

Abstract: The article proposes a method for visualizing multidimensional random process realizations using the example of the concentrations of harmful gases emitted into the atmosphere from a thermal power plant. The method is based on the transformation of gas concentration values in one point of multidimensional space at the same time into a two-dimensional curve, which is described by the sum of products of normalized concentrations by orthogonal Legendre functions of the corresponding order. The combination of such curves on a two-dimensional plane at discrete times creates a characteristic image that can be used to visually detect features of gas concentrations over time by a human operator.

Adaptive energy-preserving algorithms for guiding center system

Abstract: We develop two types of adaptive energy preserving algorithms based on the averaged vector field for the guiding center dynamics, which plays a key role in magnetized plasmas. The adaptive scheme is applied to the Gauss Legendre's quadrature rules and time stepsize respectively to overcome the energy drift problem in traditional energy-preserving algorithms. These new adaptive algorithms are second order, and their algebraic order is carefully studied. Numerical results show that the global energy errors are bounded to the machine precision over long time using these adaptive algorithms without massive extra computation cost.