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In this paper we present a new approach to the inverse problem forrelativistic stars using quasinormal modes and the piecewise polytropicparametrization of the equation of state. The algorithm is a piecewisepolytropic meshing and refinement method that reconstructs the neutron starequation of state from experimental data of the mass and the $wI$-quasinormalmodes. We present an algorithm able to numerically calculate axial quasinormalmodes of neutron stars in an efficient way. We use an initial mesh of $27440$equations of state in a $4$-volume of piecewise polytropic parameters thatcontains most of the candidate equations of state used today. The refinementprocess drives us to the reconstruction of the equation of state with a certainprecision. Using the reconstructed equation of state, we calculate predictionsfor tidal deformability and slow rotation parameters (moment of inertia andquadrupole moment, for example).  In order to check the method with an explicit example, we use as input data afew (five) 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 of the parameterssmaller than $2.5\%$ except for the tidal deformability, for which we obtain arelative difference smaller than $6.5\%$. We also study constraints fromGW170817 event for the reconstructed equation of state.

Reconstruction of the neutron star equation of state from $w$-quasinormal modes spectra with a piecewise polytropic meshing and refinement method