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

Piecewise polytropic meshing and refinement method for the  reconstruction of the neutron star equation of state using tidal  deformabilities and constraints in the piecewise polytropic parameters given  by the GW170817 event