A stable method for the evaluation of Gaussian radial basis function solutions of interpolation and collocation problems

Radial basis functions (RBFs) are a powerful tool for approximating the solution of high-dimensional problems. They are often referred to as a meshfree method and can be spectrally accurate. The best accuracy can often be achieved when the so-called shape parameter of the basis functions is small, which in turn tends to make the interpolation matrix increasingly ill-conditioned. To overcome such instability in the numerical method, which arises for even the most stable problems, one needs to stabilize the method. In this paper we present a new stable method for evaluating Gaussian radial basis function interpolants based on the eigenfunction expansion for Gaussian RBFs. This work enhances the ideas proposed in Fasshauer and McCourt (2012), by exploiting the properties of the orthogonal eigenfunctions and their zeros. We develop our approach in one and two-dimensional spaces, with the extension to higher dimensions proceeding analogously. In the univariate setting the orthogonality of the eigenfunctions and our special collocation locations give rise to easily computable cardinal basis functions. The accuracy, robustness and computational efficiency of the method are tested by numerically solving several interpolation and boundary value problems in one and two dimensions. High accuracy, simple implementation and low complexity for high-dimensional problems are the advantages of our approach. On the down side, our method is currently limited to the use of tensor products of unevenly spaced one-dimensional data locations.