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.

On the forward modelling of three-dimensional magnetotelluric data using a radial-basis-function-based mesh-free method

2019 ◽  
Vol 219 (1) ◽  
pp. 394-416 ◽  
Author(s):  
Jianbo Long ◽  
Colin G Farquharson
Keyword(s):  
Finite Difference ◽  
Radial Basis Function ◽  
Basis Function ◽  
Weak Form ◽  
Forward Modelling ◽  
Magnetotelluric Data ◽  
Mesh Free ◽  
Radial Basis ◽  
Mesh Free Method ◽  
Mesh Free Methods

SUMMARY The investigation of using a novel radial-basis-function-based mesh-free method for forward modelling magnetotelluric data is presented. The mesh-free method, which can be termed as radial-basis-function-based finite difference (RBF-FD), uses only a cloud of unconnected points to obtain the numerical solution throughout the computational domain. Unlike mesh-based numerical methods (e.g. grid-based finite difference, finite volume and finite element), the mesh-free method has the unique feature that the discretization of the conductivity model can be decoupled from the discretization used for numerical computation, thus avoiding traditional expensive mesh generation and allowing complicated geometries of the model be easily represented. To accelerate the computation, unstructured point discretization with local refinements is employed. Maxwell’s equations in the frequency domain are re-formulated using $\mathbf {A}$-ψ potentials in conjunction with the Coulomb gauge condition, and are solved numerically with a direct solver to obtain magnetotelluric responses. A major obstacle in applying common mesh-free methods in modelling geophysical electromagnetic data is that they are incapable of reproducing discontinuous fields such as the discontinuous electric field over conductivity jumps, causing spurious solutions. The occurrence of spurious, or non-physical, solutions when applying standard mesh-free methods is removed here by proposing a novel mixed scheme of the RBF-FD and a Galerkin-type weak-form treatment in discretizing the equations. The RBF-FD is applied to the points in uniform conductivity regions, whereas the weak-form treatment is introduced to points located on the interfaces separating different homogeneous conductivity regions. The effectiveness of the proposed mesh-free method is validated with two numerical examples of modelling the magnetotelluric responses over 3-D conductivity models.

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.

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