Lagrange Interpolation Research Articles

Modern and generalized analysis of exogenous growth models

The work presents a critical analysis of the model that could replicate the dynamic of long-run economic growth by regarding the capital accumulation, the labor or the population growth and increases in productivity that depends on technological progress. In this case, we calculated the equilibrium point and gave the model’s existence and uniqueness conditions. We introduced an analysis of the Lyapunov function for the first and second derivatives. Infections point analysis was presented in detail, and a numerical scheme based on the Lagrange polynomial interpolation was used to derive the numerical solution of the classical model. To introduce into mathematical formulation nonlocal behaviors, four cases were considered, including the effect of the power-law, the impact of fading memory, the development of crossover from stretched exponential to power-law and crossover behaviors from power law to stochastic. For each case, Linear growth and Lipschitz conditions were used to establish the uniqueness of the exact solution. Further, different numerical techniques were used to prove convergence of the used numerical scheme. Finally, numerical simulations were performed for other cases.

Spectral element method for 3-D controlled-source electromagnetic forward modelling using unstructured hexahedral meshes

SUMMARYFor forward modelling of realistic 3-D land-based controlled-source electromagnetic (EM) problems, we develop a parallel spectral element approach, blending the flexibility and versatility of the finite element method in using unstructured grids with the accuracy of the spectral method. Complex-shaped structures and topography are accommodated by using unstructured hexahedral meshes, in which the elements can have curved edges and non-planar faces. Our code is the first spectral element algorithm in EM geophysics that uses the total field formulation (here that of the electric field). Combining unstructured grids and a total field formulation provides advantages in dealing with topography, in particular, when the transmitter is located on rough surface topography. As a further improvement over existing spectral element methods, our approach does not only allow for arbitrary distributions of conductivity, but also of magnetic permeability and dielectric permittivity. The total electric field on the elements is expanded in terms of high-order Lagrangian interpolants, and element-wise integration in the weak form of the boundary value problem is accomplished by Gauss–Legendre–Lobatto quadrature. The resulting complex-valued linear system of equations is solved using the direct solver MUMPS, and, subsequently, the magnetic field is computed at the points of interest by Faraday’s law. Five numerical examples comprehensively study the benefits of this algorithm. Comparisons to semi-analytical and finite element results confirm accurate representation of the EM responses and indicate low dependency on mesh discretization for the spectral element method. A convergence study illuminates the relation between high order polynomial approximation and mesh size and their effects on accuracy and computational cost revealing that high-order approximation yields accurate modelling results for very coarse meshes but is accompanied by high computational cost. The presented numerical experiments give evidence that 2nd and 3rd degree polynomials in combination with moderately discretized meshes provide better trade-offs in terms of computational resources and accuracy than lowest and higher order spectral element methods. To our knowledge, our final example that includes pronounced surface topography and two geometrically complicated conductive anomalies represents the first successful attempt at using 2nd order hexahedral elements supporting curved edges and non-planar faces in controlled-source EM geophysics.

Stability analysis of milling process for multi-hardness splicing die

The tool bears the impact load and the alternating force load caused by the sudden change of hardness in the milling process of automobile panel splicing die, which aggravates the milling vibration and reduces the accuracy of die surface. According to the machining characteristics of automobile panel splicing die, the dynamic equation considering the impact of through-seam is established. The state item, the time-delay item and the periodic coefficient item in the state space form of dynamic equation were discretized by multi-order Lagrangian interpolation, and a fourth-order complete discretization method is proposed. Based on the proposed fourth-order complete discretization method, the multi-period milling stability lobe was obtained, and the milling stability lobe band in the splicing region was obtained by combining the maximum and minimum envelope method. The correctness of the dynamic model was verified by the time-frequency domain method such as Fourier transform and milling vibration and milling force data collected from the machining process. The research results can provide theoretical support for the selection and optimization of milling process parameters of splicing die.

Approximate Hermite Interpolations for Compactly Supported Linear Canonical Transforms

There has been several Lagrange and Hermite type interpolations of entire functions whose linear canonical transforms have compact supports inℝ. There interpolation representations are called sampling theorems for band-limited signals in signal analysis. The truncation, amplitude, and jitter errors associated with the Lagrange type interpolations are investigated rigorously. Nevertheless, the amplitude and jitter errors arising from perturbing samples and nodes are not studied before. The aim of this work is to establish rigorous analysis of their types of perturbation errors, which is important from both practical and theoretical points of view. We derive precise estimates for both types of errors and expose various numerical examples.

Correction of High-Order $$L_k$$ Approximation for Subdiffusion

The subdiffusion equations with a Caputo fractional derivative of order \(\alpha \in (0,1)\) arise in a wide variety of practical problems, which describe the transport processes, in the force-free limit, slower than Brownian diffusion. In this work, we derive the correction schemes of the Lagrange interpolation with degree k (\(k\le 6\)) convolution quadrature, called \(L_k\) approximation, for the subdiffusion. The key step of designing correction algorithm is to calculate the explicit form of the coefficients of \(L_k\) approximation by the polylogarithm function or Bose-Einstein integral. To construct a \(\tau _8\) approximation of Bose-Einstein integral, the desired \((k+1-\alpha )\)th-order convergence rate can be proved for the correction \(L_k\) scheme with nonsmooth data, which is higher than kth-order BDFk method in [Jin, Li, and Zhou, SIAM J. Sci. Comput., 39 (2017), A3129–A3152; Shi and Chen, J. Sci. Comput., (2020) 85:28]. The numerical experiments with spectral method are given to illustrate theoretical results.

Extended Powell–Sabin finite element scheme for linear elastic fracture mechanics

Powell–Sabin B-splines, which are based on triangles, are employed in the framework of the extended finite element method (XFEM) for fracture analysis. This avoids the necessity of remeshing in discrete fracture models and increases the solution accuracy around the crack tip. Powell–Sabin B-splines are C1-continuous throughout the whole domain. The stresses around crack tips are captured more accurately than when using elements with a standard Lagrangian interpolation. Although Powell–Sabin B-splines do not hold the Kronecker-delta property, the Heaviside function and the tip enrichment function are confined to the cracked elements only, similar to the traditional XFEM but different from the extended isogeometric method. In addition, Powell–Sabin B-splines still hold C1-continuous throughout cracked elements. There is no need to lower the continuity at element boundaries, to confine basis function support in cracked elements. Shifting is used to ensure compatibility with the surrounding discretization. The sub-triangle technique is employed for the numerical integration over crack elements. The versatility and accuracy of the approach to simulate crack problems are assessed in case studies, featuring mode-I and mixed-mode crack problems.

A New Fuzzy Vault based Biometric System robust to Brute-Force Attack

Fuzzy vault based biometric systems use fuzzy vaults during the coding that occurs within the enrollment stage inside a biometric system. In that stage the biometric system creates a vault from the biometric data, user’s key and chaff points. In the verification stage, a new biometric data is introduced and user’s key can be recovered through the use of the Lagrange polynomial interpolation method. As a consequence, in this kind of systems brute force attack would be successful because fuzzy vaults are finite. Most of the related works focus on designing secure fuzzy vault systems through the use of a password or using hybrid systems. This leads to security falling on the same user or increasing the number of security elements such as chaff points, the degree of the polynomial, or multiple biometric samples. This paper proposes a new system that considers cryptography to achieve a fuzzy vault biometric system robust against brute-force attacks. It is important to say that our system does not need a higher number of chaff points or even a higher polynomial degree. Obtained results show that this new fuzzy vault biometric system not only would be secure for current times but also would be for the future time.

Combining Interpolation Schemes and Lagrange Interpolation on the Unit Sphere in ℝN+1

Combining Interpolation Schemes and Lagrange Interpolation on the Unit Sphere in ℝN+1

Interpolation method for solving Volterra integral equations with weakly singular kernel using an advanced barycentric Lagrange formula

We presented an interpolation method for solving weakly singular Volterra integral equations of the second kind (SVK2). The method based on the barycentric Lagrange interpolation.. For the chosen nodes of the two singular kernel variables, we created two rules that confirm that the denominator of the kernel will never become zero or have an imaginary value. The product of four matrices, one of which is the square coefficients matrix of the barycentric functions, expresses both the unknown and data functions. The singular kernel is interpolated twice with to its two variables. Furthermore, the kernel is represented by five matrices: two of which are the monomial basis of the kernel's variables, and one matrix expresses the kernel's values at the two sets of nodes for the kernel's variables. Without using the collocation approach, we get the matrix of the unknown coefficients by substituting the interpolated solution into the integral equation to obtain an equivalent algebraic system. By solving the algebraic system, we get the interpolated solution. We solved five examples, two for non-singular equations and three for weakly singular equations. By adopting lower interpolation degrees, the interpolated solutions for non-singular equations are shown to be equal to the exact ones, whereas, for singular equations, they are found to strongly converge to the exact ones and are superior to those obtained by other cited methods. This ensures the proposed method's uniqueness and high accuracy.

Flutter analysis of delaminated composite box-beam using higher-order kinematics

The flutter analysis of delaminated box-beam has been performed using the Lagrange polynomial in Carrera Unified Formulation (CUF) framework. In this formulation, the cross-sectional displacement field of each layer can be defined by the arbitrary displacement variables using the Layerwise (LW) approach. The cross-section of the present structural model is discretized using the quadratic elements. The unsteady aerodynamic contribution has been obtained using Theodorsen’s lift function. Subsequently, the structural and aeroelastic model has been defined using the Principle of Virtual Displacement (PVD) in the CUF framework. For verification of the present model, free variation analysis has been performed for delaminated and healthy composite structures. Furthermore, a composite box-beam model with single and multiple delaminations has been developed to perform free vibration and flutter analysis. This study supports the capability of the CUF model to perform flutter analysis of the complex model with multi-delaminated composite box-beam.

Linear Pseudospectral Method with Chebyshev Collocation for Optimal Control Problems with Unspecified Terminal Time

In this paper, a linear Chebyshev pseudospectral method (LCPM) is proposed to solve the nonlinear optimal control problems (OCPs) with hard terminal constraints and unspecified final time, which uses Chebyshev collocation scheme and quasi-linearization. First, Taylor expansion around the nonlinear differential equations of the system is used to obtain a set of linear perturbation equations. Second, the first-order necessary conditions for OCPs with these linear equations and unspecified terminal time are derived, which provide the successive correction formulas of control and terminal time. Traditionally, these formulas are linear time varying and cannot be solved in an analytical manner. Third, Lagrange interpolation, whose supporting points are orthogonal Chebyshev–Gauss–Lobatto (CGL), is employed to discretize the resulting problem. Therefore, a series of analytical correction formulas are successfully derived in approximating polynomial space. It should be noted that Chebyshev approximation is close to the best polynomial approximation, and CGL points can be solved in closed form. Finally, LCPM is applied to the air-to-ground missile guidance problem. The simulation results show that it has high computational efficiency and convergence rate. A comparison with the other typical OCP solvers is provided to verify the optimality of the proposed algorithm. In addition, the results of Monte Carlo simulations are presented, which show that the proposed algorithm has strong robustness and stability. Therefore, the proposed method has potential to be onboard application.

Deep learning based correction of low performing pixel in computed tomography

Low Performing Pixel (LPP)/bad pixel in CT detectors cause ring and streaks artifacts, structured non-uniformities and deterioration of the image quality. These artifacts make the image unusable for diagnostic purposes. A missing/defective detector pixel translates to a channel missing across all views in sinogram domain and its effect gets spill over entire image in reconstruction domain as artifacts. Most of the existing ring and streak removal algorithms perform correction only in the reconstructed image domain. In this work, we propose a supervised deep learning algorithm that operates in sinogram domain to remove distortions cause by the LPP. This method leverages CT scan geometry, including conjugate ray information to learn the interpolation in sinogram domain. While the experiments are designed to cover the entire detector space, we emphasize on LPPs near detector iso-center as these have most adverse impact on image quality specially if the LPPs fall on the high frequency region (bone-tissue interface). We demonstrated efficacy of the proposed method using data acquired on GE RevACT multi-slice CT system with flat-panel detector. Experimental results on head scans show significant reduction in ring artifacts regardless of LPP location in the detector geometry. We have simulated isolated LPPs accounting for 5% and 10% of total channels. Detailed statistical analysis illustrates approximately 5dB improvement in SNR in both sinogram and reconstruction domain as compared to classical bicubic and Lagrange interpolation methods. Also, with reduction in ring and streak artifacts, the perceptual image quality is improved across all the test images.

Error-constant estimation under the maximum norm for linear Lagrange interpolation

For the linear Lagrange interpolation over a triangular domain, we propose an efficient algorithm to rigorously evaluate the interpolation error constant under the maximum norm by using the finite-element method (FEM). In solving the optimization problem corresponding to the interpolation error constant, the maximum norm in the constraint condition is the most difficult part to process. To handle this difficulty, a novel method is proposed by combining the orthogonality of the space decomposition using the Fujino–Morley FEM space and the convex-hull property of the Bernstein representation of functions in the FEM space. Numerical results for the lower and upper bounds of the interpolation error constant for triangles of various types are presented to verify the efficiency of the proposed method.

A novel isogeometric layerwise element for piezoelectric analysis of laminated plates with straight/curvilinear fibers

This study presents an isogeometric layerwise element, L-IGA based on the principle of virtual displacement theory to model the bending behavior of laminated smart composite plates integrated with piezoelectric layers. Instead of using Lagrangian or Hermitian type polynomials encountered in standard finite element technology, L-IGA utilizes high-order Non-Uniform Rational B-Splines (NURBS) functions for both in-plane and through-the-thickness discretization of the geometry and the kinematic variables. Additionally, it allows different numbers of NURBS degrees and elements to be used for each patch through the thickness of the plate. In this way, exact geometry, and highly accurate solutions with rapid convergence for the benchmark electromechanical problems in the literature have been guaranteed by using L-IGA analysis. The precision of the results is meticulously verified by 3-D Ansys SOLID226 finite elements and analytical solutions. Thus, the L-IGA element can be adopted for computationally efficient and accurate static analysis of laminated plates having straight/curvilinear fibers for piezoelectric actuator and sensor configurations.

Symmetry in multivariate ideal interpolation

An interpolation problem is defined by a set of linear forms on the (multivariate) polynomial ring and values to be achieved by an interpolant. For Lagrange interpolation the linear forms consist of evaluations at some nodes, while Hermite interpolation also considers the values of successive derivatives. Both are examples of ideal interpolation in that the kernels of the linear forms intersect into an ideal. For an ideal interpolation problem with symmetry, we address the simultaneous computation of a symmetry adapted basis of the least interpolation space and the symmetry adapted H-basis of the ideal. Beside its manifest presence in the output, symmetry is exploited computationally at all stages of the algorithm. For an ideal invariant, under a group action, defined by a Gröbner basis, the algorithm allows to obtain a symmetry adapted basis of the quotient and of the generators. We shall also note how it applies surprisingly but straightforwardly to compute fundamental invariants and equivariants of a reflection group.

Mathematical modeling of corona virus (COVID-19) and stability analysis

In this paper, the mathematical modeling of the novel corona virus (COVID-19) is considered. A brief relationship between the unknown hosts and bats is described. Then the interaction among the seafood market and peoples is studied. After that, the proposed model is reduced by assuming that the seafood market has an adequate source of infection that is capable of spreading infection among the people. The reproductive number is calculated and it is proved that the proposed model is locally asymptotically stable when the reproductive number is less than unity. Then, the stability results of the endemic equilibria are also discussed. To understand the complex dynamical behavior, fractal-fractional derivative is used. Therefore, the proposed model is then converted to fractal-fractional order model in Atangana-Baleanu (AB) derivative and solved numerically by using two different techniques. For numerical simulation Adam-Bash Forth method based on piece-wise Lagrangian interpolation is used. The infection cases for Jan-21, 2020, till Jan-28, 2020 are considered. Then graphical consequences are compared with real reported data of Wuhan city to demonstrate the efficiency of the method proposed by us.

Reliability Assessment of Power Distribution System Using Relative Customer Average Interruptions Duration Index Model

Reliability assessment of power distribution system deals with the adequacy of overall system supply and indicates the system behaviour and response at the customer end. In order to attain satisfactory degree of reliability in electric power distribution system, there is the need to develop a model that will improve the three major system reliability indices via Systems Average Interruptions Duration Index (SAIDI), Systems Average Interruptions Frequency Index (SAIFI) and Customer Average Interruptions Duration Index (CAIDI). This research paper therefore, developed a Relative CAIDI model for assessment of reliability indices of electrical power distribution system. In this paper, ten (10) years outage data from selected distribution feeders of Ibadan, Ikeja and Port-Harcourt distribution systems were obtained and analyzed using curve fitting tools in MATLAB. The system reliability indices served as input parameters for the development of Relative CAIDI model. The best curves that fitted the relationship between the Relative CAIDI and the number of feeders were obtained using Lagrange polynomial functions, Newton polynomial functions and Chebyshev polynomial functions as performance evaluation metrics. The results showed that Ibadan, Ikeja and Port-Harcourt distribution systems had Relative CAIDI of 0.718, 0.3976 and 0.5279 respectively and the validation results produced by Largrange polynomial function, Newton polynomial function and Chebyshev polynomial function were 0.717, 0.3979 and 0.5278 respectively. The model developed is a polynomial of order six which depends majorly on the power requirements of the distribution systems. The developed model can be used for effective reliability assessment of electrical power distribution systems as well as forming a base-line information for system planning and maintenance strategies. Keywords: Reliability Indices, SAIDI, SAIFI, CAIDI, Chebyshev Polynomial Function, Lagrange Polynomial Function, Newton Polynomial Function. DOI: 10.7176/NCS/14-01 Publication date: August 31 st 2022

Iterative Continuous Collocation Method for Solving Nonlinear Volterra Integral Equations

This paper is concerned with the numerical solution of nonlinear Volterra integral equations. The main purpose of this work is to provide a new numerical approach based on the use of continuous collocation Lagrange polynomials for the numerical solution of nonlinear Volterra integral equations. It is shown that this method is convergent. The results are compared with the results obtained by other well-known numerical methods to prove the effectiveness of the presented algorithm.

Efficient Encrypted Range Query on Cloud Platforms

In the Internet of Things (IoT) era, various IoT devices are equipped with sensing capabilities and employed to support clinical applications. The massive electronic health records (EHRs) are expected to be stored in the cloud, where the data are usually encrypted, and the encrypted data can be used for disease diagnosis. There exist some numeric health indicators, such as blood pressure and heart rate. These numeric indicators can be classified into multiple ranges, and each range may represent an indication of normality or abnormity. Once receiving encrypted IoT data, the CS maps it to one of the ranges, achieving timely monitoring and diagnosis of health indicators. This article presents a new approach to identify the range that an encrypted numeric value corresponds to without exposing the explicit value. We establish the sufficient and necessary condition to convert a range query to matchings of encrypted binary sequences with the minimum number of matching operations. We further apply the minimization of range queries to design and implement a secure range query system, where numeric health indicators encrypted independently by multiple IoT devices can be cohesively stored and efficiently queried by using Lagrange polynomial interpolation. Comprehensive performance studies show that the proposed approach can protect both the health records and range query against untrusted cloud platforms and requires less computational and communication cost than existing techniques.

Numerical Integration of Stiff Differential Systems Using Non-Fixed Step-Size Strategy

Over the years, researches have shown that fixed (constant) step-size methods have been efficient in integrating a stiff differential system. It has however been observed that for some stiff differential systems, non-fixed (variable) step-size methods are required for efficiency and for accuracy to be attained. This is because such systems have solution components that decay rapidly and/or slowly than others over a given integration interval. In order to curb this challenge, there is a need to propose a method that can vary the step size within a defined integration interval. This challenge motivated the development of Non-Fixed Step-Size Algorithm (NFSSA) using the Lagrange interpolation polynomial as a basis function via integration at selected limits. The NFSSA is capable of integrating highly stiff differential systems in both small and large intervals and is also efficient in terms of economy of computer time. The validation of properties of the proposed algorithm which include order, consistence, zero-stability, convergence, and region of absolute stability were further carried out. The algorithm was then applied to solve some samples mildly and highly stiff differential systems and the results generated were compared with those of some existing methods in terms of the total number of steps taken, number of function evaluation, number of failure/rejected steps, maximum errors, absolute errors, approximate solutions and execution time. The results obtained clearly showed that the NFSSA performed better than the existing ones with which we compared our results including the inbuilt MATLAB stiff solver, ode 15s. The results were also computationally reliable over long intervals and accurate on the abscissae points which they step on.