Multi-node higher order expansions of a function

We study continuous finite element dicretizations for one dimensional hyperbolic partial differential equations. The main contribution of the paper is to provide a fully discrete spectral analysis, which is used to suggest optimal values of the CFL number and of the stabilization parameters involved in different types of stabilization operators. In particular, we analyze the streamline-upwind Petrov-Galerkin (SUPG) stabilization technique, the continuous interior penalty (CIP) stabilization method and the local projection stabilization (LPS). Three different choices for the continuous finite element space are compared: Bernstein polynomials, Lagrangian polynomials on equispaced nodes, and Lagrangian polynomials on Gauss-Lobatto cubature nodes. For the last choice, we only consider inexact quadrature based on the formulas corresponding to the degrees of freedom of the element, which allows to obtain a fully diagonal mass matrix. We also compare different time stepping strategies, namely Runge-Kutta (RK), strong stability preserving RK (SSPRK) and deferred correction time integration methods. The latter allows to alleviate the computational cost as the mass matrix inversion is replaced by the high order correction iterations. To understand the effects of these choices, both time-continuous and fully discrete Fourier analysis are performed. These allow to compare all the different combinations in terms of accuracy and stability, as well as to provide suggestions for optimal values discretization parameters involved. The results are thoroughly verified numerically both on linear and non-linear problems, and error-CPU time curves are provided. Our final conclusions suggest that cubature elements combined with SSPRK and CIP or LPS stabilization are the most promising combinations.

Sharing the secret among the partners is an interesting technique that distributes the secret information in several shares for enhancing security. Nowadays, the reversible watermarking scheme becomes very popular due to increasing privacy and security issues in the day to day life. Here, a new watermarking scheme has been proposed using Integer Wavelet Transform (IWT) and Lagrange Interpolation Polynomial (LIP). The reversibility can meet with the help of the IWT scheme and LIP provides the flavor of secret sharing. Moreover, experimental results with comparison have been drawn at the end of the paper. The experimental shows that the proposed scheme can achieve 58.9 dB PSNR with a high payload of 1.5 bpp. From the experimental results, it can be concluded that the proposed scheme outperforms the other existing scheme.KeywordsWatermarkingLagrange Interpolation Polynomial (LIP)Integer Wavelet Transform (IWT)Reversibility

In this brief, numerical techniques for an efficient implementation of the spline transcription (ST) method by using the pseudospectral framework are proposed. It intends to develop a new approach to the ST method, analyses with which can be carried out with nearly the same level of computational burden without resorting to a specific type of collocation points as in the pseudospectral method. Therefore, the piecewise continuous polynomial interpolation is converted into a global spline interpolation, which has the same forms as the Lagrange interpolation and is used to derive the formulas for the partial integration and the differentiation. Using these formulas, the nonlinear optimal control problem can be transcribed in a similar manner as in the pseudospectral method and analyzed with numerical techniques and mathematical theories established well for the pseudospectral method. Also, this brief shows the Lagrange and tension spline interpolation can be used for the proposed methods. Numerical accuracy and efficiency of the proposed method are accessed through its applications to the solution of two nonlinear optimal control problems on uniform nodes. The predicted results of states, controls, and costates are well correlated with the optimal solutions. Finally, this brief recommends the use of advanced grid topologies to enhance both accuracy and convergence by using the flexibility in node selections with the proposed methods.

Uniform approximation and Bernstein polynomials with coefficients in the unit interval

This paper presents a unified framework for constructing transfinite finite elements with arbitrary node distributions, using either Lagrange or Bernstein polynomial bases. Three distinct classes of elements are considered. The first includes elements with structured internal node layouts and arbitrarily positioned boundary nodes. The second comprises elements with internal nodes arranged to allow smooth transitions in a single direction. The third class consists of elements defined on structured T-meshes with selectively omitted internal nodes, resulting in sparsely populated or incomplete grids. For all three classes, new macro-element (global interpolation) formulations are introduced, enabling flexible node configurations. Each formulation supports representations based on either Lagrange or Bernstein polynomials. In the latter case, two alternative Bernstein-based models are developed as follows: one that is numerically equivalent to its Lagrange counterpart, and another that offers modest improvements in numerical performance.

Mahalanobis distance may be used as a measure of the disparity between an individual’s profile of scores and the average profile of a population of controls. The degree to which the individual’s profile is unusual can then be equated to the proportion of the population who would have a larger Mahalanobis distance than the individual. Several estimators of this proportion are examined. These include plug-in maximum likelihood estimators, medians, the posterior mean from a Bayesian probability matching prior, an estimator derived from a Taylor expansion, and two forms of polynomial approximation, one based on Bernstein polynomial and one on a quadrature method. Simulations show that some estimators, including the commonly-used plug-in maximum likelihood estimators, can have substantial bias for small or moderate sample sizes. The polynomial approximations yield estimators that have low bias, with the quadrature method marginally to be preferred over Bernstein polynomials. However, the polynomial estimators sometimes yield infeasible estimates that are outside the 0–1 range. While none of the estimators are perfectly unbiased, the median estimators match their definition; in simulations their estimates of the proportion have a median error close to zero. The standard median estimator can give unrealistically small estimates (including 0) and an adjustment is proposed that ensures estimates are always credible. This latter estimator has much to recommend it when unbiasedness is not of paramount importance, while the quadrature method is recommended when bias is the dominant issue.

The complexity of interpolation attacks on block ciphers depends on the degree of the polynomial approximation and/or on the number of terms in the polynomial approximation expression. In some situations, the round function or the S-boxes of the block cipher are expressed explicitly in terms of algebraic function, yet in many other occasions the S-boxes are expressed in terms of their Boolean function representation. In this case, the cryptanalyst has to evaluate the algebraic description of the S-boxes or the round function using the Lagrange interpolation formula. A natural question is what is the effect of the choice of the irreducible polynomial used to construct the finite field on the degree of the resulting polynomial. Another question is whether or not there exists a simple linear transformation on the input or output bits of the S-boxes (or the round function) such that the resulting polynomial has a less degree or smaller number of non-zero coefficients. In this paper we give an answer to these questions. We also present an explicit relation between the Lagrange interpolation formula and the Galois Field Fourier Transform.

Secure reversible watermarking schemes are essential for image authentication and tamper detection in medical, military and government applications. In this paper, a secure reversible color image watermarking scheme has been proposed considering the merits of three individual tools namely Local Binary Pattern (LBP), Lagrange Interpolation Polynomial (LIP) and Weighted Matrix (WM). The proposed reversible scheme has been designed to address the issues related to image authentication and tamper detection. Here, LBP operator is employed to generate Tamper Detection Code (TDC), LIP and WM are used to embed watermark within the sub-sampled images. Authentication is accomplished by utilizing an Authentication Code (AC), which is generated from binary string of watermark logo using a Secure Hash Algorithm (SHA-512). Repeated entry-wise-multiplication operations have been performed with each sub-sampled image block to increase the payload without affecting visual quality of the watermarked image significantly. Some standard NIST recommended steganalysis and attacks are conducted to evaluate its robustness and imperceptibility. It is observed that the developed scheme is secure and robust against several attacks as well as can detect tampered regions. Experimental results are compared with the state-of-the-art schemes to present the merits of our scheme.

In the present note, we extend some univariate uniform approximation results by means of Lagrange interpolating polynomials [Ivan, M., Elements of Interpolation Theory, Mediamira Science Publisher, Cluj-Napoca (2004)] to the bivariate case. It is well known that generally, in the univariate case, the sequence of Lagrange interpolation polynomials does’t converges to the approximated function. This fact was first observed by G. Faber (see [9]), which constructed an example when the sequence of Lagrange interpolation polynomials diverges. The result of G. Faber was more generalized by I. Muntean (see [12]). M. Ivan established first sufficient conditions for the uniform convergence of the sequence of Lagrange interpolation polynomials associated to a univariate real valued function. First, we represent the remainder term of bivariate Lagrange interpolation formula in terms of bivariate divided difference. Using this representation we establish sufficient conditions for the uniform convergence of the sequence of bivariate Lagrange interpolation polynomials to the approximated function.

Lagrange interpolation is extended to the complex plane in this paper. It turns out to be composed of two parts: polynomial and rational interpolations of an analytical function. Based on Lagrange interpolation in the complex plane, a complex variable boundary collocation approach is constructed. The method is truly meshless and singularity free. It features high accuracy obtained by use of a small number of nodes as well as dimensionality advantage, that is, a two-dimensional problem is reduced to a one-dimensional one. The method is applied to two-dimensional problems in the theory of plane elasticity. Numerical examples are in very good agreement with analytical ones. The method is easy to be implemented and capable to be able to give the stress states at any point within the solution domain.

Based on the Lagrange interpolation polynomial algorithm, the error analysis is discussed in this paper. Firstly, we derive the Lagrange interpolation polynomial algorithm and introduce the shape function with the usage of related data figures. Secondly, how to derive the Lagrange interpolation function is also simply introduced. At last, some examples have been given to display the error analysis of the Lagrange interpolation polynomial algorithm and two conclusions are suggested to minimize the errors in Lagrange interpolation.

Derivative error bounds for Lagrange interpolation: An extension of Cauchy's bound for the error of Lagrange interpolation

The paper deals with an analysis of Gaussian interpolation function algorithm on elementary algebraic functions, the results have been compared with the Lagrange polynomial and the algorithm is improved by changing the variational coefficient α, that lead to decreasing interpolation error.          Set of mathematical tools with different functional content are used with working mathematical model. Interpolation is one of the types of work with experimental data. Set of tools of mathematical models has many different interpolation methods, but Lagrange polynomial is used most often on practice as classic method of polynomial interpolation.          The results of some interpolation algorithms are presented in the article. Results of interpolation of elementary algebraic functions by classical Lagrange method and three types of Gaussian interpolation functions with a small number of interpolation nodes and a steady step have been considered and analyzed. These methods have been tested with variable interpolation step. In addition, the results of interpolation with different variable parameter α are presented. It has been concluded that it is appropriate to change this parameter for different functions.          For the arbitrary function analysis, also consider the interpolation of experimental data using the Gauss interpolation function on the example of COVID-19 distribution in Ukraine. It has been concluded about the use of different values of the variable parameter for the interpolation of experimental data with minimal error.          Based on using Gauss interpolation function, computer system for analyzing a working principle of the Gauss function by the example of elementary mathematical functions with constant and variable steps and with ability to control the value of the variable parameter, which reduced the interpolation error has been created.          Keywords: interpolation, Gaussian interpolation function, interpolation error, variable coefficient.

This paper deals with an analysis of the free vibration of nonprismatic thin-walled beams, with a special focus on the effect of a change in the support point location on the eigenfrequencies of the systems. A change in the support point location is understood here as occurring within the same fixed cross-section of the beam where the latter is supported. The original elements of this study are a thin-walled beam model and a method of solving differential equations, not previously used by other authors. The equations describing the model used in this paper were derived using the momentless theory of shells and the Vlasov theory assumptions. The displacement equations were derived relative to an arbitrary rectilinear reference axis. In most works known to the authors the equations describing the vibration of the nonprismatic thin-walled beam do not take into account the effects due to the curvature of the axis formed by the shear centers. The recursive algorithm was used to solve the obtained differential equations with variable coefficients. The algorithm enables one to solve the analyzed displacement equations in the form of series relative to the orthogonal Gegenbauer polynomials. For special values of the parameter defining the order of the polynomials the Gegenbauer polynomials became Chebyshev or Lagrange polynomials. In the provided numerical example, the effect of a change in the support point location within the fixed cross-section is examined. It is also analyzed which of the approximation polynomials (Chebyshev or Lagrange polynomials) yield more precise results for a small approximation base. In order to verify the model and the effectiveness of the adopted solution method, the results obtained using this method are compared with the results yielded by FEM. The results obtained showed a significant effect of the position of the support point, within a fixed section, on the eigenfrequencies values of the thin-walled systems.

Hierarchical threshold secret sharing (HTSS) schemes can be thought as a generalization of classical threshold secret sharing schemes, and they have been extensively in the literature. In an HTSS, participants are classified into different security levels, and the threshold value of a higher level is smaller than that of a lower level. Participants in each level can recover the secrets, if the number of shares is equal to or more than the corresponding threshold value. Share of a higher level participant can be used to reconstruct the secret at lower level. In this paper, we proposed first hierarchical threshold multi-secret sharing scheme based on polynomial interpolation. Proposed scheme is a variation to HTSS schemes based on the CRT suggested by Singh et al. and Harn et al. Novelty of the proposed scheme is that each participant requires to keep only one secret share and multiple secrets can be shared separately without refreshing the secret share. Also, secrets are recovered in stage by stage. Our scheme which is unconditionally secure, is based on Lagrange interpolation polynomial and one-way function.