  We propose and test the first Reduced Radial Basis Function Method (R$^2$BFM)
for solving parametric partial differential equations on irregular domains. The
two major ingredients are a stable Radial Basis Function (RBF) solver that has
an optimized set of centers chosen through a reduced-basis-type greedy
algorithm, and a collocation-based model reduction approach that systematically
generates a reduced-order approximation whose dimension is orders of magnitude
smaller than the total number of RBF centers. The resulting algorithm is
efficient and accurate as demonstrated through two- and three-dimensional test
problems.
