A Fast Solution for the Generalized Radial Basis Functions Interpolant

Abstract

In this paper, we propose a fast solution method of the generalized radial basis functions interpolant for global interpolation. The method can be used to efficiently interpolate large numbers of geometry constraints for implicit surface reconstruction. The basic idea of our approach is based on the far field expansion of the kernel and the preconditioned Krylov iteration using the domain decomposition method as a preconditioner. We present a fast evaluation method of the matrix-vector product for the linear system. To minimize the number of iterations for large numbers of constraints, the multi-level domain decomposition method is applied to improve overlap using a nested sequence of levels. The implemented solution algorithm generally achieves O(NlogN) complexity and O(N) storage. It is kernel independent both in the evaluation and solution processes without analytical expansions. It is very convenient to apply various types of RBF kernels in different applications without excessive modifications to the existing process. Numerical results show that the fast evaluation method has a good performance for the evaluation of the matrix-vector product and the preconditioned Krylov subspace iterative method has a good convergence rate with a small number of iterations. INDEX TERMS Radial basis functions, generalized radial basis functions, far field expansion, domain decomposition method, implicit surface reconstruction.