  In this paper we present a new approach to the inverse problem for
relativistic stars using the piecewise polytropic parametrization of the
equation of state. The algorithm is a piecewise polytropic meshing and
refinement method that reconstructs the neutron star equation of state from
experimental data of the mass and the tidal Love parameter. We use an initial
mesh of 65536 equations of state in a 4-volume of piecewise polytropic
parameters that contains most of the candidate equations of state used today.
The refinement process drives us to the reconstruction of the equation of state
with a certain precision. Using the reconstructed equation of state, we
calculate predictions for quasinormal modes and slow rotation parameters.
  In order to check the meshing and refinement method, we use as input data a
few (6) configurations of a given equation of state. We reconstruct the
equation of state in a quite good approximation, and then we compare the curves
of physical parameters from the original equation of state and the
reconstructed one. We obtain a relative difference for all the parameters
smaller than 7.5%.
  We also study the constraints that impose the GW170817 event on the piecewise
polytropic parameters $\{{\log_{10}p_1},{\Gamma_1},{\Gamma_2},{\Gamma_3}\}$. We
use the waveform model TaylorF2 for the low-spin scenario, and see that the
EOSs that lie outside the 90% credible region when
${\bar{\lambda}^{tid}_1}={\bar{\lambda}^{tid}_2}$ define a zone of polytropic
parameters that does not depend on ${\Gamma_3}$.
