Non-parametric strong lens inversion of SDSS J1004+4112
  In this article we study the well-known strong lensing system SDSS
J1004+4112. Not only does it host a large-separation lensed quasar with
measured time-delay information, but several other lensed galaxies have been
identified as well. A previously developed strong lens inversion procedure that
is designed to handle a wide variety of constraints, is applied to this lensing
system and compared to results reported in other works. Without the inclusion
of a tentative central image of one of the galaxies as a constraint, we find
that the model recovered by the other constraints indeed predicts an image at
that location. An inversion which includes the central image provides tighter
constraints on the shape of the central part of the mass map. The resulting
model also predicts a central image of a second galaxy where indeed an object
is visible in the available ACS images. We find masses of 2.5x10^13 M_O and
6.1x10^13 M_O within a radius of 60 kpc and 110 kpc respectively, confirming
the results from other authors. The resulting mass map is compatible with an
elliptical generalization of a projected NFW profile, with r_s = 58_{-13}^{+21}
arcsec and c_vir = 3.91 +/- 0.74. The orientation of the elliptical NFW profile
follows closely the orientation of the central cluster galaxy and the overall
distribution of cluster members.
