scholarly journals Preconditioning of Radial Basis Function Interpolation Systems via Accelerated Iterated Approximate Moving Least Squares Approximation

2009 ◽  
pp. 57-75 ◽  
Author(s):  
Gregory E. Fasshauer ◽  
Jack G. Zhang
Keyword(s):  
Radial Basis Function ◽  
Least Squares ◽  
Basis Function ◽  
Moving Least Squares ◽  
Least Squares Approximation ◽  
Radial Basis Function Interpolation ◽  
Radial Basis ◽  
Interpolation Systems ◽  
Moving Least Squares Approximation

A Hybrid Method Using Response Surface and Pattern Search for Design Optimization

2005 ◽  
Author(s):  
T. Zhang ◽  
K. K. Choi ◽  
S. Rahman
Keyword(s):  
Radial Basis Function ◽  
Least Squares ◽  
Response Surface ◽  
Basis Function ◽  
Function Approximation ◽  
Least Squares Method ◽  
Order Approximation ◽  
Pattern Search ◽  
Moving Least Squares ◽  
Radial Basis

This paper presents a new method to construct response surface function and a new hybrid optimization method. For the response surface function, the radial basis function is used for a zeroth-order approximation, while new bases is proposed for the moving least squares method for a first-order approximation. For the new hybrid optimization method, the gradient-based algorithm and pattern search algorithm are integrated for robust and efficient optimization process. These methods are based on: (1) multi-point approximations of the objective and constraint functions; (2) a multi-quadric radial basis function for the zeroth-order function representation or radial basis function plus polynomial based moving least squares approximation for the first-order function approximation; and (3) a pattern search algorithm to impose a descent condition. Several numerical examples are presented to illustrate the accuracy and computational efficiency of the proposed method for both function approximation and design optimization. The examples for function approximation indicate that the multi-quadric radial basis function and the proposed radial basis function plus polynomial based moving least squares method can yield accurate estimates of arbitrary multivariate functions. Results also show that the hybrid method developed provides efficient and convergent solutions to both mathematical and structural optimization problems.

scholarly journals The Optimal Shape Parameter for the Least Squares Approximation Based on the Radial Basis Function

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1923
Author(s):  
Sanpeng Zheng ◽  
Renzhong Feng ◽  
Aitong Huang
Keyword(s):  
Radial Basis Function ◽  
Least Squares ◽  
Basis Function ◽  
Shape Parameter ◽  
Optimal Parameter ◽  
Optimal Shape ◽  
Least Squares Approximation ◽  
Scattered Data Approximation ◽  
Least Squares Problem ◽  
Radial Basis

The radial basis function (RBF) is a class of approximation functions commonly used in interpolation and least squares. The RBF is especially suitable for scattered data approximation and high dimensional function approximation. The smoothness and approximation accuracy of the RBF are affected by its shape parameter. There has been some research on the shape parameter, but the research on the optimal shape parameter of the least squares based on the RBF is scarce. This paper proposes a way for the measurement of the optimal shape parameter of the least squares approximation based on the RBF and an algorithm to solve the corresponding optimal parameter. The method consists of considering the shape parameter as an optimization variable of the least squares problem, such that the linear least squares problem becomes nonlinear. A dimensionality reduction is applied to the nonlinear least squares problem in order to simplify the objective function. To solve the optimization problem efficiently after the dimensional reduction, the derivative-free optimization is adopted. The numerical experiments indicate that the proposed method is efficient and reliable. Multiple kinds of RBFs are tested for their effects and compared. It is found through the experiments that the RBF least squares with the optimal shape parameter is much better than the polynomial least squares. The method is successfully applied to the fitting of real data.

Impulse Noise Removal Using Adaptive Radial Basis Function Interpolation

2016 ◽  
Vol 36 (3) ◽  
pp. 1192-1223 ◽  
Author(s):  
T. Veerakumar ◽  
Ravi Prasad K. Jagannath ◽  
Badri Narayan Subudhi ◽  
S. Esakkirajan
Keyword(s):  
Radial Basis Function ◽  
Basis Function ◽  
Impulse Noise ◽  
Noise Removal ◽  
Radial Basis Function Interpolation ◽  
Impulse Noise Removal ◽  
Radial Basis

scholarly journals Solution of the quantum fluid dynamical equations with radial basis function interpolation

2000 ◽  
Vol 61 (5) ◽  
pp. 5967-5976 ◽  
Author(s):  
Xu-Guang Hu ◽  
Tak-San Ho ◽  
Herschel Rabitz ◽  
Attila Askar
Keyword(s):  
Radial Basis Function ◽  
Basis Function ◽  
Dynamical Equations ◽  
Quantum Fluid ◽  
Radial Basis Function Interpolation ◽  
Radial Basis

scholarly journals Rapid Interactive and Intuitive Segmentation of 3D Medical Images Using Radial Basis Function Interpolation

2017 ◽  
Vol 3 (4) ◽  
pp. 56
Author(s):  
Tanja Kurzendorfer ◽  
Peter Fischer ◽  
Negar Mirshahzadeh ◽  
Thomas Pohl ◽  
Alexander Brost ◽  
...  
Keyword(s):  
Radial Basis Function ◽  
Basis Function ◽  
Medical Images ◽  
Radial Basis Function Interpolation ◽  
Radial Basis

scholarly journals A Frame-Based Conjugate Gradients Direct Search Method with Radial Basis Function Interpolation Model

2017 ◽  
Vol 2017 ◽  
pp. 1-9 ◽  
Author(s):  
Xiaowei Fang ◽  
Qin Ni
Keyword(s):  
Radial Basis Function ◽  
Basis Function ◽  
Direct Search ◽  
Search Method ◽  
Conjugate Gradients ◽  
Test Problems ◽  
Interpolation Model ◽  
Direct Search Method ◽  
Radial Basis Function Interpolation ◽  
Radial Basis

In this paper, we propose a new hybrid direct search method where a frame-based PRP conjugate gradients direct search algorithm is combined with radial basis function interpolation model. In addition, the rotational minimal positive basis is used to reduce the computation work at each iteration. Numerical results for solving the CUTEr test problems show that the proposed method is promising.

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