Research For Some Problems Of The Radial Basis Function Interpolation

In recent years, researchers proposed a new optimization method, the radial basis func-tion interpolation method, for the interpolation approximation problems. In the radial basis function interpolation method, we do not need a certain objective function and the information of derivative. We only need to choose a radial basis function and establish a comparative ac-curate simplified model by less function value points. Then we can find the approximation of the unknown function in the function space which is generated by the radial function. Hence, the radial basis function interpolation method is a new approach for solving the global opti-mization problems. For the black box function problems which are without certain objective functions and available derivative, we can solve the optimization problems of them by utilizing the radial basis function interpolation method.When the radial basis function is positive definite, a linear combination of the radial basis function can approximate any continuous function, these functions have been widely used in the field of scientific and engineering applications. Hence, the research of the radial basis function interpolation method has essential theoretical value and practical significance. In this paper, we mainly study the approach of choosing the shape parameter c, in radial basis function so that the interpolation deviation can be reduced as much as possible. What’s more, we propose two improved strategies on the radial basis function model methods for solving global optimization problems. The main contents are arranged as following:In Chapter 1, a briefly introduction about the research background and significance of the radial basis function is given. We summarize the research on radial basis functions and the main content of this thesis.In Chapter 2, we mainly introduce some basic knowledge of radial basis function (RBF), such as the definition and model of the radial basis function, the basic method of selecting the initial points in the SLHD method of the RBF algorithm, the approach of choosing the objective function value and the next iteration point in the algorithm and so on,In Chapter 3, the advantage of the radial basis function is illustrated by a numerical ex-ample. With the same interpolation error, the CPU time of this algorithm is less than that of the Newton interpolation algorithm. Furthermore, we investigate the approach of how to choose the shape parameter c in the MQ function and the Gaussian function of the radial basis functions by some numerical examples in order to have small interpolation error.In Chapter 4, we propose a new deformation function strategy to improve the optimization effect of the radial basis function interpolation method. Firstly, we demonstrate the feasibility of the strategy theoretically. Then we illustrate the advantages of SLHD method by some numerical examples. Furthermore, when there is little improvement by using new initial points in the SLHD method, we propose a restarting strategy of replacing the radial basis function which yields better numerical optimization performance.