We perform a local computation via the Partition of Unity (PU) method of rational Radial Basis Function (RBF) interpolants. We investigate the well-posedness of the problem and we provide error bounds. The resulting scheme, efficiently implemented by means of the Deflation Accelerated Conjugate Gradient (DACG), enables us to deal with huge data sets and, thanks to the use of Variably Scaled Kernels (VSKs), it turns out to be stable. For functions with steep gradients or discontinuities, which are truly common in applications, the results show that the new proposed method outperforms the classical and rescaled PU schemes.
