Smoothing Algorithms and High-order Singularities in Gravitational   Lensing
  We propose a new smoothing method for obtaining surface densities from discrete particle positions from numerical simulations. This is an essential step for many applications in gravitational lensing. This method is based on the ``scatter'' interpretation of the discrete density field in the Smoothed Particle Hydrodynamics. We use Monte Carlo simulations of uniform density fields and one isothermal ellipsoid to empirically derive the noise properties, and best smoothing parameters (such as the number of nearest neighbors used). A cluster from high-resolution simulations is then used to assess the reality of high-order singularities such as swallowtails and butterflies in caustics, which are important for the interpretation of substructures in gravitational lenses. We also compare our method with the Delaunay tesselation field estimator using the galaxy studied by Bradac et al. (2004), and find good agreements. We show that higher order singularities are not only connected with bound subhaloes but also with the satellite streams. However, the presence of high-order singularities are sensitive to not only the fluctuation amplitude of the surface density, but also the detailed form of the underlying smooth lensing potential (such as ellipticity and external shear). 
Guo-Liang Li, S. Mao, Y.P. Jing, X. Kang, M. Bartelmann