MQ-RBF with an optimal value of shape parameter determined by a size-independence technique

Abstract

This paper proposes a new method to find the optimal value of shape parameter in the multiquadric radial basis function (MQ-RBF), which is formulated by a size-independence technique (SIT). The size n for a high-dense MQ-RBF is reduced to m < n, with n simultaneously denoting the number of radial bases, the number of unknown coefficients and the dimension of linear system. Then a lower m-dimensional linear system is solved several times to determine the optimal value of shape parameter, using the golden section search algorithm to minimize the maximal error to fit the given data. Inserting the optimal value obtained from a lower m dimensional linear system into a high-dense MQ-RBF, one can construct a highly accurate interpolant. To solve elliptic type equations we adapt the MQ-RBF by SIT. For searching a good shape parameter in the high-dense MQ-RBF the SIT can improve the accuracy and save CPU time.