  The luminosity distance can be used to determine the properties of large
scale structure around the observer. To this purpose we develop a new inversion
method to map luminosity distance to a LTB metric based on the use of the exact
analytical solution for Einstein equations. The main advantages of this
approach are an improved numerical accuracy and stability, an exact analytical
setting of the initial conditions for the differential equations which need to
be solved and the validity for any sign of the functions determining the LTB
geometry. Given the fully analytical form of the differential equations, this
method also simplifies the calculation of the red-shift expansion around the
apparent horizon point where the numerical solution becomes unstable. We test
the method by inverting the supernovae Ia luminosity distance function
corresponding to the the best fit $\Lambda CDM$ model. We find that only a
limited range of initial conditions is compatible with observations, or a
transition from red to blue shift can occur at relatively low redshift. Despite
LTB solutions without a cosmological constant have been shown not to be
compatible with all different set of available observational data, those
studies normally fit data assuming a special functional ansatz for the
inhomogeneity profile, which often depend only on few parameters. Inversion
methods on the contrary are able to fully explore the freedom in fixing the
functions which determine a LTB solution. Another important possible
application is not about LTB solutions as cosmological models, but rather as
tools to study the effects on the observations made by a generic observer
located in an inhomogeneous region of the Universe where a fully non
perturbative treatment involving exact solutions of Einstein equations is
required.
