Barycentric Lagrange Interpolation Research Articles

A generalized fuzzy barycentric Lagrange interpolation method for solving two-dimensional fuzzy fractional Volterra integral equations

A generalized fuzzy barycentric Lagrange interpolation method for solving two-dimensional fuzzy fractional Volterra integral equations

A Numerical Method Based on Operator Splitting Collocation Scheme for Nonlinear Schrödinger Equation

In this paper, a second-order operator splitting method combined with the barycentric Lagrange interpolation collocation method is proposed for the nonlinear Schrödinger equation. The equation is split into linear and nonlinear parts: the linear part is solved by the barycentric Lagrange interpolation collocation method in space combined with the Crank–Nicolson scheme in time; the nonlinear part is solved analytically due to the availability of a closed-form solution, which avoids solving the nonlinear algebraic equation. Moreover, the consistency of the fully discretized scheme for the linear subproblem and error estimates of the operator splitting scheme are provided. The proposed numerical scheme is of spectral accuracy in space and of second-order accuracy in time, which greatly improves the computational efficiency. Numerical experiments are presented to confirm the accuracy, mass and energy conservation of the proposed method.

Costate estimation for a multiple-interval pseudospectral method using collocation at the flipped legendre-Gauss-Radau points

A multiple-interval pseudospectral method is presented for direct trajectory optimization and costate estimation of optimal control problems using collocation at the flipped Legendre-Gauss-Radau points. The recently developed global flipped Radau pseudospectral method is one of the special case. The necessary optimality conditions of the multiple-interval differential scaled optimal control problem are rigorously derived for the first time by applying the calculus of variations. Then, the costate and constraint multiplier estimates for the proposed method are derived in detail by comparing the discretized optimality conditions with the Karush-Kuhn-Tucker conditions of the nonlinear programming problem resulting from collocation. Furthermore, in order to improve the numerical stability and alleviate the computational error when calculating the state pseudospectral differentiation matrix, the barycentric Lagrange interpolation is adopted in the state approximation instead of the classic Lagrange interpolation without losing the computational efficiency. The method presented in this paper is applied to two benchmark optimal control problems and is compared with the de-facto standard pseudospectral methods. Extensive numerical results indicate that it has the ability to obtain highly accurate solutions to a wide variety of complex constrained optimal control problems.

Barycentric Lagrange interpolation method for solving Love’s integral equations

In this paper, we present a new simple method for solving two integral equations of Love’s type that have many applications, especially in electrostatic systems. The approach of the solution is based on an innovative technique using matrix algebra for the barycentric Lagrange interpolation. The unknown function is expressed through the product of four matrices. The kernel is interpolated twice, so we get it in the product of five matrices. Additionally, we derive an equivalent linear algebraic system to the solution by substituting the matrix-vector barycentric interpolated unknown function together with the double interpolated kernel into both sides of the integral equation. Thus, there was no need to employ the collocation method. The obtained results converge strongly with the approximate analytical solutions, in addition to being uniformly approximated, continuous, and even, which proves the validity of the solution by the presented method.

Barycentric Lagrange interpolation collocation method for solving the Sine–Gordon equation

This paper mainly discusses the numerical solution of Sine–Gordon (SG) equation which is widely used in engineering field. Different from the classical numerical calculation method, we choose the barycentric Lagrange interpolation collocation method (BLICM) to solve the SG equation. Firstly, the barycentric Lagrange interpolation is introduced and its differential matrix is given. Secondly, two linearized iterative schemes namely direct linearized iterative scheme and Newton linearized iterative scheme to solve SG equation are constructed, the matrix equation of the two iterative scheme is obtained. Thirdly, the Newton–Raphson iterative scheme for SG equation is also presented and the detailed solution process is given. Finally, several numerical examples with exact solution are given to show the accuracy of BLICM, the results of three iterative schemes with equidistant nodes and Chebyshev nodes are compared. Further, some examples with line and ring solitons are given to simulate by BLICM, the accurate numerical results are obtained to support our experiments.

An image encryption algorithm based on heat flow cryptosystems

<abstract><p>Image encryption has been an important research topic in information security. Different from traditional encryption methods, heat flow cryptosystem is a new encryption method. This paper proposes an image encryption algorithm based on heat flow cryptosystem. First, a class of heat flow cryptosystem based on nonlinear pseudo-parabolic equations are given in this paper. Second, a numerical method with high precision namely barycentric Lagrange interpolation collocation method is proposed to solve the nonlinear pseudo-parabolic equation. Third, an image encryption algorithm based on the heat flow cryptosystem is designed, the detailed process of encryption and decryption algorithm is given, the flow diagram of algorithm is showed. Finally, the proposed encryption algorithm is applied to various image with gray and RGB format and compared with the current popular chaotic encryption algorithm. Many indicators such as histograms, information entropy and correlation are used to objectively evaluate the image encryption algorithm. The experimental results show that the proposed image encryption algorithm is better in most indicators and the algorithm is sensitive to the change of key and plaintext.</p></abstract>

Barycentric Lagrange Interpolation Methods for Evaluating Singular Integrals

We investigate new, straightforward methods for interpolating and integrating discontinuous single and two-variable real valued functions. The method is based on some modified matrix–vector barycentric Lagrange interpolation formulas. We developed seven rules for the optimum distribution of nodes inside the domain of integration: five for single-valued discontinuous functions, and two rules for the two independent variables of discontinuous functions. We designed these rules to depend on the length of the integration domain and the degree of the interpolant polynomials. Thus, we obtained uniform interpolation and minimum roundoff errors. Based on these rules with the application of the modified matrix–vector barycentric formulas, we easily isolated the singularities of the interpolant integrands and evaluated the corresponding interpolant integral values with super accuracy. The obtained numerical solutions of the five given examples show the high accuracy and efficiency of the presented method compared with the exact solutions and with the cited method.

TABI-PB 2.0: An Improved Version of the Treecode-Accelerated Boundary Integral Poisson-Boltzmann Solver.

This work describes TABI-PB 2.0, an improved version of the treecode-accelerated boundary integral Poisson-Boltzmann solver. The code computes the electrostatic potential on the molecular surface of a solvated biomolecule, and further processing yields the electrostatic solvation energy. The new implementation utilizes the NanoShaper surface triangulation code, node-patch boundary integral discretization, a block preconditioner, and a fast multipole method based on barycentric Lagrange interpolation and dual tree traversal. Performance-critical portions of the code were implemented on a GPU. Numerical results for protein 1A63 and two viral capsids (Zika, H1N1) demonstrate the code's accuracy and efficiency.

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.

Computational method for solving weakly singular Fredholm integral equations of the second kind using an advanced barycentric Lagrange interpolation formula

In this study, we applied an advanced barycentric Lagrange interpolation formula to find the interpolate solutions of weakly singular Fredholm integral equations of the second kind. The kernel is interpolated twice concerning both variables and then is transformed into the product of five matrices; two of them are monomial basis matrices. To isolate the singularity of the kernel, we developed two techniques based on a good choice of different two sets of nodes to be distributed over the integration domain. Each set is specific to one of the kernel arguments so that the kernel values never become zero or imaginary. The significant advantage of thetwo presented techniques is the ability to gain access to an algebraic linear system equivalent to the interpolant solution without applying the collocation method. Moreover, the convergence in the mean of the interpolant solution and the maximum error norm estimation are studied. The interpolate solutions of the illustrated four examples are found strongly converging uniformly to the exact solutions.

Operator splitting scheme based on barycentric Lagrange interpolation collocation method for the Allen-Cahn equation

Operator splitting scheme based on barycentric Lagrange interpolation collocation method for the Allen-Cahn equation

Interpolation method for evaluating weakly singular kernels

The solution of initial, boundary, and mixed value problems through the integral equation method yields certain boundary singular integral equations. In many scientific and industrial applications in artificial intelligence, biological systems, scattering, radiation, and image processing, there is a need to evaluate such types of singular integrals, especially the weakly singular kernels. This study presents a new numerical method for evaluating weakly singular kernels based on some advanced matrix-vector barycentric Lagrange interpolation formulas. We developed these formulas to be applied to numerically evaluating singular integrals. At the same time, we created three computational rules to determine the optimal locations for the distribution of the interpolation nodes to be within the integration domain and never be outside for any value of the interpolant degree. These rules are devised so that the equidistant nodes depend on the step-sizes, which are defined as functions of the interpolant degree by some small real number greater than or equal to zero. Thus, we overcame the singularity of the kernels on the whole integration domain and obtained uniform interpolation. Moreover, the presented method gives the kernel's values and the kernel's integral values at the singular points, whereas the numerical or exact values do not exist. The solutions to the illustrated four examples are shown in the given tables and figures. The interpolant solutions which we obtain by low-degree interpolants are faster to converge to the numerical or exact ones (if they exist). This confirms the originality of the presented method and its effectiveness in obtaining high-precision results.

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.

Numerical solution of coupled 1D Burgers’ equation by employing Barycentric Lagrange interpolation basis function based differential quadrature method

The aim of present study is to develop a numerical scheme by using the notion of Barycentric Lagrange interpolation based Differential quadrature method to solve the coupled 1D Burgers’ equation. This method reduced the mentioned partial differential equation into the set of ordinary differential equations, which can be dealt by the SSP-RK43 scheme. The proposed method has been implemented upon the different numerical examples in order to test the accuracy and effectiveness of the proposed method. It is observed that obtained results are in good compatibility with the exact solution and are better than the previous results. The stability of the proposed method is also discussed by using the matrix stability analysis method, which represents that the proposed method is unconditionally stable.

Numerical Approximation of One- and Two-Dimensional Coupled Nonlinear Schrödinger Equation by Implementing Barycentric Lagrange Interpolation Polynomial DQM

In this paper, a new numerical method named Barycentric Lagrange interpolation-based differential quadrature method is implemented to get numerical solution of 1D and 2D coupled nonlinear Schrödinger equations. In the present study, spatial discretization is done with the aid of Barycentric Lagrange interpolation basis function. After that, a reduced system of ordinary differential equations is solved using strong stability, preserving the Runge-Kutta 43 method. In order to check the accuracy of the proposed scheme, we have used the formula ofL∞error norm. The matrix stability analysis method is implemented to test the proposed method’s stability, which confirms that the proposed scheme is unconditionally stable. The present scheme produces better results, and it is easy to implement to obtain numerical solutions of a class of partial differential equations.

Solution of the 3D Helmholtz equation using barycentric Lagrange interpolation collocation method

PurposeThis meshless collocation method is applicable not only to the Helmholtz equation with Dirichlet boundary condition but also mixed boundary conditions. It can calculate not only the high wavenumber problems, but also the variable wave number problems.Design/methodology/approachIn this paper, the authors developed a meshless collocation method by using barycentric Lagrange interpolation basis function based on the Chebyshev nodes to deduce the scheme for solving the three-dimensional Helmholtz equation. First, the spatial variables and their partial derivatives are treated by interpolation basis functions, and the collocation method is established for solving second order differential equations. Then the differential matrix is employed to simplify the differential equations which is on a given test node. Finally, numerical experiments show the accuracy and effectiveness of the proposed method.FindingsThe numerical experiments show the advantages of the present method, such as less number of collocation nodes needed, shorter calculation time, higher precision, smaller error and higher efficiency. What is more, the numerical solutions agree well with the exact solutions.Research limitations/implicationsCompared with finite element method, finite difference method and other traditional numerical methods based on grid solution, meshless method can reduce or eliminate the dependence on grid and make the numerical implementation more flexible.Practical implicationsThe Helmholtz equation has a wide application background in many fields, such as physics, mechanics, engineering and so on.Originality/valueThis meshless method is first time applied for solving the 3D Helmholtz equation. What is more the present work not only gives the relationship of interpolation nodes but also the test nodes.

A GPU-accelerated fast multipole method based on barycentric Lagrange interpolation and dual tree traversal

We present a GPU-accelerated fast multipole method (FMM) called BLDTT, which uses barycentric Lagrange interpolation for the near-field and far-field approximations, and dual tree traversal to construct the interaction lists. The scheme replaces well-separated particle-particle interactions by adaptively chosen particle-cluster, cluster-particle, and cluster-cluster approximations given by barycentric Lagrange interpolation on a Chebyshev grid of proxy particles in each cluster. The BLDTT employs FMM-type upward and downward passes, although here they are adapted to interlevel polynomial interpolation. The BLDTT is kernel-independent, and the approximations have a direct sum form that efficiently maps onto GPUs, where targets provide an outer level of parallelism and sources provide an inner level of parallelism. The code uses OpenACC directives for GPU acceleration and MPI remote memory access for distributed memory parallelization. Computations are presented for different particle distributions, domains, and interaction kernels, and for unequal targets and sources. The BLDTT consistently outperforms our earlier particle-cluster barycentric Lagrange treecode (BLTC). On a single GPU for problem size ranging from N=1E5 to 1E8, the BLTC scales like O(Nlog⁡N) and the BLDTT scales like O(N). We also present MPI strong scaling results for the BLDTT and BLTC with N=64E6 particles running on 1 to 32 GPUs.

Numerical Solution of Euler-Bernoulli Beam Equation by Using Barycentric Lagrange Interpolation Collocation Method

Euler-Bernoulli beam equation is very important that can be applied in the field of mechanics, science and technology. Some authors have put forward many different numerical methods, but the precision is not enough high. In this paper, we will illustrate the high-precision numerical method to solve Euler-Bernoulli beam equation. Three numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by our method indicate new algorithm has the following advantages: small computational work, fast convergence speed and high precision.

Interpolation Method for Solving Weakly Singular Integral Equations of the Second Kind

We establish a new straightforward interpolation method for solving linear Volterra integral ‎‎equations with weakly singular kernels. The proposed method is fundamentally different from all other published methods for solving this type of equations. We have modified some vector-matrix barycentric Lagrange interpolation formulas to be convenient for interpolating the kernel twice concerning the two variables of the kernel and introducing new ideas for selecting interpolation nodes that ensure isolation of the singularity of the kernel. We create two rules for selecting the distribution nodes of ‎‎the two kernel variables that do not allow the ‎‎denominator of the kernel to contain an imaginary value. We interpolate the unknown and data functions ‎‎into the corresponding interpolant polynomial; each of the same degree via three matrices, one of ‎‎which is a monomial. By applying the presented method based on the two created rules, we transformed the ‎kernel into a double ‎interpolant polynomial with a degree equal to that of the unknown ‎function via five matrices, two of ‎which are monomials. We substitute the interpolate unknown ‎function twice; on the left side and on the ‎right side of the integral equation to get an ‎algebraic linear system without applying the ‎collocation method. The solution of this system yields ‎the unknown coefficients matrix that is necessary to find the interpolant solution. We ‎solve three ‎different examples for different values of the upper integration variable. The obtained ‎results as ‎shown in tables and figures prove that the obtained interpolate solutions are extraordinarily faster ‎‎to converge to the exact ones using interpolants of lowest degrees and give better results than those obtained by ‎other ‎methods. This confirms the originality and the potential of the presented method.‎

A Meshless Collocation Method with Barycentric Lagrange Interpolation for Solving the Helmholtz Equation

In this paper, Chebyshev interpolation nodes and barycentric Lagrange interpolation basis function are used to deduce the scheme for solving the Helmholtz equation. First of all, the interpolation basis function is applied to treat the spatial variables and their partial derivatives, and the collocation method for solving the second order differential equations is established. Secondly, the differential equations on a given test node. Finally, based on three kinds of test nodes, numerical experiments show that the present scheme can not only calculate the high wave numbers problems, but also calculate the variable wave numbers problems. In addition, the algorithm has the advantages of high calculation accuracy, good numerical stability and less time consuming.