  Many applications and algorithms in the field of gravitational lensing make
use of meshes with a finite number of nodes to analyze and manipulate data.
Specific examples in lensing are astronomical CCD images in general, the
reconstruction of density distributions from lensing data, lens-source plane
mapping or the characterization and interpolation of a point-spread-function.
We present a numerical framework to interpolate and differentiate in the
mesh-free domain, defined by nodes with coordinates that follow no regular
pattern. The framework is based on radial basis functions (RBFs) to smoothly
represent data around the nodes. We demonstrate the performance of Gaussian
RBF-based, mesh-free interpolation and differentiation, which reaches the
sub-percent level in both cases. We use our newly developed framework to
translate ideas of free-form mass reconstruction from lensing onto the
mesh-free domain. By reconstructing a simulated mock lens we find that strong
lensing only reconstructions achieve < 10% accuracy in the areas where these
constraints are available but provide poorer results when departing from these
regions. Weak-lensing only reconstructions give < 10% percent accuracy outside
the strong lensing regime, but cannot resolve the inner core structure of the
lens. Once both regimes are combined, accurate reconstructions can be achieved
over the full field of view. The reconstruction of a simulated lens, using
constraints that mimics real observations, yields accurate results in terms of
surface-mass density, NFW parameter, Einstein radius and magnification map
recovery, encouraging the application of this method to real data.
