Multiquadric Quasi-interpolation Research Articles

A kind of bivariate Bernoulli-type multiquadric quasi-interpolation operator with higher approximation order

In this paper, a kind of bivariate Bernoulli-type multiquadric quasi-interpolation operator is studied by combining the known multiquadric quasi-interpolation operator with the generalized Taylor polynomial as the expansion in the bivariate Bernoulli polynomials. Some error bounds and convergence rates of the combined operators are studied. A selection of numerical examples is presented to compare the performances of the obtained scheme. Furthermore, our method can be applied to time-dependent differential equations. Its advantage is that the algorithm is very simple and easy to implement.

A meshless multiquadric quasi-interpolation method for time fractional Black–Scholes model

Based on multiquadric quasi-interpolation, this study presents a meshless numerical method for time fractional Black–Scholes (B–S) model. The method is highly accurate and flexible. The stability and convergence of the proposed scheme are discussed in detail. Numerical results show the accuracy and efficiency of the proposed method.

A meshless quasi-interpolation method for solving hyperbolic conservation laws based on the essenstially non-oscillatory reconstruction

In this paper, a quasi-interpolation method is proposed for solving hyperbolic conservation laws based on the essentially non-oscillatory (ENO) scheme. The hyperbolic equation is discretized in space with the finite difference ENO method and then the semi-discrete system is integrated by the strong stability preserving Runge-Kutta scheme. However, in the step of ENO reconstruction, the finite difference method is replaced by several quasi-interpolation schemes, including the multiquadric quasi-interpolation, the integral-type multiquadric quasi-interpolation and the cubic B-spline quasi-interpolation. Our quasi-interpolation method is simple and easy to implement since it doesn't need to solve any linear system. Moreover, it is also suitable for nonuniform grids and noisy sampling data. Nonlinear hyperbolic problems we target include one-dimensional and two-dimensional Burgers' equations and also the Euler equation. Numerical results demonstrate that the proposed quasi-interpolation ENO method is stable and has good accuracy.

Numerical study of nonlinear generalized Burgers–Huxley equation by multiquadric quasi-interpolation and pseudospectral method

Numerical study of nonlinear generalized Burgers–Huxley equation by multiquadric quasi-interpolation and pseudospectral method

Abel–Goncharov Type Multiquadric Quasi-Interpolation Operators with Higher Approximation Order

A kind of Abel–Goncharov type operators is surveyed. The presented method is studied by combining the known multiquadric quasi-interpolant with univariate Abel–Goncharov interpolation polynomials. The construction of new quasi-interpolants ℒ m AG f has the property of m m ∈ ℤ , m > 0 degree polynomial reproducing and converges up to a rate of m + 1 . In this study, some error bounds and convergence rates of the combined operators are studied. Error estimates indicate that our operators could provide the desired precision by choosing the suitable shape-preserving parameter c and a nonnegative integer m. Several numerical comparisons are carried out to verify a higher degree of accuracy based on the obtained scheme. Furthermore, the advantage of our method is that the associated algorithm is very simple and easy to implement.

Numerical investigation of the generalized Burgers-Huxley equation using combination of multiquadric quasi-interpolation and method of lines

In this article, an efficient method for approximate the solution of the generalized Burgers-Huxley (gB-H) equation using multiquadric quasi-interpolation approach is considered. This method consists of two phases. First, the spatial derivatives are evaluated by MQ quasi-interpolation, So the gB-H equation is reduces to a nonlinear system of ordinary differential equations. In phase two, the obtained system is solved by using ODE solvers. Numerical examples demonstrate the validity and applicability of the method.

Multiquadric quasi-interpolation for integral functionals

Multiquadric (MQ) quasi-interpolation has been extensively studied in approximation theory and its applications. However, most of them only focus on the case that the sampling data are point-evaluation functionals (discrete function values). This restricts its applications since a more general form of data in real applications is integral functionals (integration of an approximand over subdomains that form a partition of the considering region). In addition, integral functionals also demonstrate characteristics of sampling devices. Therefore it is more meaningful to extend multiquadric quasi-interpolation to integral functionals. The purpose of the paper is to construct such a new multiquadric quasi-interpolation scheme. The scheme possesses the constant reproduction property and thus preserves positivity of an approximand. Moreover, if the approximand is a density function of a random variable, then the resulting quasi-interpolant is also a density function. Thus, as a byproduct, we also provide a scheme for nonparametric kernel density estimation. However, unlike classical kernel density estimation that usually has a boundary bias problem over a bounded domain, our scheme circumvents the problem. Numerical simulations including posteriori error estimates and density estimation are demonstrated at the end of the paper to show efficiency and robustness of the scheme.

Solving one dimensional nonlinear coupled Burger’s equations using high accuracy multiquadric quasi-interpolation

In this paper a multiquadric quasi-interpolation (MQQI) scheme for solving the system of 1-D coupled nonlinear Burger’s equations (CNBE) is presented. The scheme utilizes the derivative of the quasi-interpolation for approximating the spatial derivative and the Taylor series expansion for temporal derivatives. Simulations are presented to demonstrate the efficiency and applicability of the scheme. Also, we have shown that our scheme is superior to some numerical schemes already done.

An iterated quasi-interpolation approach for derivative approximation

Given discrete function values sampled at uniform centers, the iterated quasi-interpolation approach for approximating the m th derivative consists of two steps. The first step adopts m successive applications of the operator DQ (the quasi-interpolation operator Q first, and then the differentiation operator D) to get approximated values of the m th derivative at uniform centers. Then, by one further application of the quasi-interpolation operator Q to corresponding approximated derivative values gives the final approximation of the m th derivative. The most salient feature of the approach is that it approximates all derivatives with the same convergence rate. In addition, it is valid for a general multivariate function, compared with the existing iterated interpolation approaches that are only valid for periodic functions, so far. Numerical examples of approximating high-order derivatives using both the iterated and direct approach based on B-spline quasi-interpolation and multiquadric quasi-interpolation are presented at the end of the paper, which demonstrate that the iterated quasi-interpolation approach provides higher approximation orders than the corresponding direct approach.

Algorithms for numerical solution of the equal width wave equation using multi-quadric quasi-interpolation method

In this paper, we analyze equal width wave equation using multi-quadric quasi-interpolation (MQQI) scheme. The traveling wave solutions called solitary waves: Single solitary wave motion, interaction of two and three solitary waves and evolution of solitons have been investigated computationally. Several comparisons are made to demonstrate the effectiveness of the proposed method. Error norms are used to observe the accuracy and efficiency of this scheme.

Numerical Solution of High-Dimensional Shockwave Equations by Bivariate Multi-Quadric Quasi-Interpolation

Radial basis function-based quasi-interpolation performs efficiently in high-dimensional approximation and its applications, which can attain the approximant and its derivatives directly without solving any large-scale linear system. In this paper, the bivariate multi-quadrics (MQ) quasi-interpolation is used to simulate two-dimensional (2-D) Burgers’ equation. Specifically, the spatial derivatives are approximated by using the quasi-interpolation, and the time derivatives are approximated by forward finite difference method. One advantage of the proposed scheme is its simplicity and easy implementation. More importantly, the proposed scheme opens the gate to meshless adaptive moving knots methods for the high-dimensional partial differential equations (PDEs) with shock or soliton waves. The scheme is also applicable to other non-linear high-dimensional PDEs. Two numerical examples of Burgers’ equation (shock wave equation) and one example of the Sine–Gordon equation (soliton wave equation) are presented to verify the high accuracy and efficiency of this method.

Adaptive moving knots meshless method for Degasperis-Procesi equation with conservation laws

The nonlinear Degasperis-Procesi (DP) equation allows shock wave solutions and possesses bi-Hamiltonian properties. Considering both of the two properties, we introduce a numerical scheme which involves three key features. First, a stable moving knots strategy is introduced to simulate the shock waves with less knots. Second, the moving knots equation together with the DP equation are simulated using the meshless multi-quadric (MQ) quasi-interpolation method, which allows scattered data approximation. Third, the proposed moving knots meshless scheme preserves the conservation laws. Both theoretical analysis and numerical examples demonstrate that the algorithm saves the computational time, improves the accuracy, and possesses a long-time tracking capability.

A flexible symplectic scheme for two-dimensional Schrödinger equation with highly accurate RBFS quasi-interpolation

Based on highly accurate multiquadric quasi-interpolation, this study suggests a meshless symplectic procedure for two-dimensional time-dependent Schr?dinger equation. The method is highorder accurate, flexible with respect to the geometry, computationally efficient and easy to implement. We also present a theoretical framework to show the conservativeness and convergence of the proposed method. As the numerical experiments show, it not only offers a high order accuracy but also has a good performance in the long time integration.

Adaptive moving knots meshless method for simulating time dependent partial differential equations

This paper presents an improvement of Huang’s moving knots method (Huang et al.) for solving the PDE on the contours of Equidistribution Principle (EP-contours). Firstly, a new moving knots strategy is generated by discovering a significant factor ignored (dropped) by Huang (Huang et al.). The proposed strategy ensures that arbitrary initial knots could asymptotically converge to the EP-contours. Then the moving knots strategy and PDE are simulated by using multi-quadric (MQ) quasi-interpolation step by Step (3.3, 3.4). Error estimates of the algorithm are given. At last, numerical experiments are provided to illustrate the validity of the algorithm. Both theoretical analysis and numerical results show that the proposed algorithm benefits the methods in Huang et al. and Wu.

A multiquadric quasi-interpolations method for CEV option pricing model

The pricing of option contracts when the underlying process follows the constant elasticity of variance (CEV) model is considered. For CEV European options, the closed-form solutions involve the non-central chi-square distribution, whose computations by the current literatures are rather unstable and extremely expensive. Based on multiquadric quasi-interpolation methods, this study suggests a stable and fast numerical algorithm for CEV option pricing model. The method is confirmed to be a multinomial tree, in which the underlying variable moves from its initial value to an infinity of possible values of the next time step. The probabilities in the associated tree are ensured to be positive, which is a sufficient condition for stability and convergence. The method is flexible, since it is simple to implement with the nonuniform knots. Moreover, the method is easy to value the Greek letters which are important parameters in financial engineering, as the multiquadric function is infinitely continuously differentiable. Besides, the method does not require solving a resultant full matrix, the ill-conditioning problem arising when using the radial basis functions as a global interpolant can be avoided. Numerical experiments imply that the method is highly effective to calculate the stock options and its Greeks under the CEV model.

Numerical solution of time fractional partial differential equations using multiquadric quasi-interpolation scheme

In this paper, a meshfree method is presented to solve time fractional partial differential equations. It is based on the multiquadric quasi-interpolation operator LW2 . In the present scheme, quadrature formula is used to discretise the temporal Caputo fractional derivative of order α ∈ (0, 1] and the quasiinterpolation is used to approximate the solution function and its spatial derivatives. Our numerical results are compared with the exact solutions as well as the results obtained from the other numerical schemes. It can be easily seen that the proposed method is a reliable and effective method to solve fractional partial differential equation. Furthermore, the stability analysis of the method is surveyed.

Adaptive Conservative Numerical Method of Sine-Gordon Equation Based on Multi-quadric Quasi-interpolation

Adaptive Conservative Numerical Method of Sine-Gordon Equation Based on Multi-quadric Quasi-interpolation

Symplectic multiquadric quasi-interpolation approximations of KdV equation

Radial basis functions quasi-interpolation is very useful tool for the numerical solution of differential equations, since it possesses shape-preserving and high-order approximation properties. Based on multiquadric quasi-interpolations, this study suggests a meshless symplectic procedure for KdV equation. The method has a number of advantages over existing approaches including no need to solve a resultant full matrix, accuracy and ease of implementation. We also present a theoretical framework to show the conservativeness and convergence of the proposed method. As the numerical experiments show, it not only offers a high order accuracy but also has a good property of long-time tracking capability.

Multiquadric trigonometric spline quasi-interpolation for numerical differentiation of noisy data: a stochastic perspective

Based on multiquadric trigonometric spline quasi-interpolation, the paper proposes a scheme for numerical differentiation of noisy data, which is a well-known ill-posed problem in practical applications. In addition, in the perspective of kernel regression, the paper studies its large sample properties including optimal bandwidth selection, convergence rate, almost sure convergence, and uniformly asymptotic normality. Simulations are provided at the end of the paper to demonstrate features of the scheme. Both theoretical results and simulations show that the scheme is simple, easy to compute, and efficient for numerical differentiation of noisy data.

Multi-quadric quasi-interpolation method coupled with FDM for the Degasperis–Procesi equation

In this paper, the Degasperis–Procesi (DP) equation which contains nonlinear high order derivatives and discontinuous solutions is recast to its equivalent form: u−P. Based on the equivalent formulation of DP equation, a new numerical method is presented. The multi-quadric (MQ) quasi-interpolation method coupled with finite difference method is applied to approximate the spatial derivatives and the third order TVD method is used to approximate the time derivative. Several examples are presented to demonstrate the effectiveness of the proposed method.