scholarly journals Solving the Linear Integral Equations Based on Radial Basis Function Interpolation

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Huaiqing Zhang ◽  
Yu Chen ◽  
Xin Nie
Keyword(s):  
Radial Basis Function ◽  
Integral Equations ◽  
Basis Function ◽  
Approximation Solution ◽  
Integration Schemes ◽  
Radial Basis ◽  
The Matrix ◽  
Linear Integral ◽  
Split Technique ◽  
Linear Integral Equations

The radial basis function (RBF) method, especially the multiquadric (MQ) function, was introduced in solving linear integral equations. The procedure of MQ method includes that the unknown function was firstly expressed in linear combination forms of RBFs, then the integral equation was transformed into collocation matrix of RBFs, and finally, solving the matrix equation and an approximation solution was obtained. Because of the superior interpolation performance of MQ, the method can acquire higher precision with fewer nodes and low computations which takes obvious advantages over thin plate splines (TPS) method. In implementation, two types of integration schemes as the Gauss quadrature formula and regional split technique were put forward. Numerical results showed that the MQ solution can achieve accuracy of1E-5. So, the MQ method is suitable and promising for integral equations.

scholarly journals Quantum speedup of training radial basis function networks

2019 ◽  
Vol 19 (7&8) ◽  
pp. 609-625
Author(s):  
Changpeng Shao
Keyword(s):  
Radial Basis Function ◽  
Basis Function ◽  
Quantum Computer ◽  
Quantum Algorithms ◽  
Rbf Network ◽  
Training Samples ◽  
Radial Basis ◽  
The Matrix ◽  
Hamiltonian Simulation ◽  
Classical Computer

Radial basis function (RBF) network is a simple but useful neural network model that contains wide applications in machine learning. The training of an RBF network reduces to solve a linear system, which is time consuming classically. Based on HHL algorithm, we propose two quantum algorithms to train RBF networks. To apply the HHL algorithm, we choose using the Hamiltonian simulation algorithm proposed in [P. Rebentrost, A. Steffens, I. Marvian and S. Lloyd, Phys. Rev. A 97, 012327, 2018]. However, to use this result, an oracle to query the entries of the matrix of the network should be constructed. We apply the amplitude estimation technique to build this oracle. The final results indicate that if the centers of the RBF network are the training samples, then the quantum computer achieves exponential speedup at the number and the dimension of training samples over the classical computer; if the centers are determined by the K-means algorithm, then the quantum computer achieves quadratic speedup at the number of samples and exponential speedup at the dimension of samples.

scholarly journals Application of Radial Basis Function Method for Solving Nonlinear Integral Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Huaiqing Zhang ◽  
Yu Chen ◽  
Chunxian Guo ◽  
Zhihong Fu
Keyword(s):  
Radial Basis Function ◽  
Integral Equations ◽  
Basis Function ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
Nonlinear Integral Equations ◽  
Mesh Free ◽  
Radial Basis ◽  
Background Mesh ◽  
Mesh Free Methods

The radial basis function (RBF) method, especially the multiquadric (MQ) function, was proposed for one- and two-dimensional nonlinear integral equations. The unknown function was firstly interpolated by MQ functions and then by forming the nonlinear algebraic equations by the collocation method. Finally, the coefficients of RBFs were determined by Newton’s iteration method and an approximate solution was obtained. In implementation, the Gauss quadrature formula was employed in one-dimensional and two-dimensional regular domain problems, while the quadrature background mesh technique originated in mesh-free methods was introduced for irregular situation. Due to the superior interpolation performance of MQ function, the method can acquire higher accuracy with fewer nodes, so it takes obvious advantage over the Gaussian RBF method which can be revealed from the numerical results.

scholarly journals The Quasi-Optimal Radial Basis Function Collocation Method: A Technical Note

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Juan Zhang ◽  
Mei Sun ◽  
Enran Hou ◽  
Zhaoxing Ma
Keyword(s):  
Radial Basis Function ◽  
Basis Function ◽  
Computational Cost ◽  
Technical Note ◽  
Function Parameter ◽  
Approximation Solution ◽  
Collocation Points ◽  
Radial Basis ◽  
Solution Accuracy ◽  
Parameter Values

The traditional radial basis function parameter controls the flatness of these functions and influences the precision and stability of approximation solution. The coupled radial basis function, which is based on the infinitely smooth radial basis functions and the conical spline, achieves an accurate and stable numerical solution, while the shape parameter values are almost independent. In this paper, we give a quasi-optimal conical spline which can improve the numerical results. Besides, we consider the collocation points in the Chebyshev-type which improves solution accuracy of the method with no additional computational cost.

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