  The Partition of Unity (PU) method, performed with local Radial Basis
Function (RBF) approximants, has been proved to be an effective tool for
solving large scattered data interpolation problems. However, in order to
achieve a good accuracy, the question about how many points we have to consider
on each local subdomain, i.e. how large can be the local data sets, needs to be
answered. Moreover, it is well-known that also the shape parameter affects the
accuracy of the local RBF approximants and, as a consequence, of the PU
interpolant. Thus here, both the shape parameter used to fit the local problems
and the size of the associated linear systems are supposed to vary among the
subdomains. They are selected by minimizing an a priori error estimate. As
evident from extensive numerical experiments and applications provided in the
paper, the proposed method turns out to be extremely accurate also when data
with non-homogeneous density are considered.
