  The inverse stellar structure problem determines the equation of state of the
matter in stars from a knowledge of their macroscopic observables (e.g. their
masses and radii). This problem was solved in a previous paper by constructing
a spectral representation of the equation of state whose stellar models match a
prescribed set of macroscopic observables. This paper improves and extends that
work in two significant ways: i) The method is made more robust by accounting
for an unexpected feature of the enthalpy based representations of the
equations of state used in this work. After making the appropriate
modifications, accurate initial guesses for the spectral parameters are no
longer needed so Monte-Carlo techniques can now be used to ensure the best fit
to the observables. ii) The method is extended here to use masses and tidal
deformabilities (which will be measured by gravitational wave observations of
neutron-star mergers) as the macroscopic observables instead of masses and
radii. The accuracy and reliability of this extended and more robust spectral
method is evaluated in this paper using mock data for observables from stars
based on 34 different theoretical models of the high density neutron-star
equation of state. In qualitative agreement with earlier work, these tests
suggest the high density part of the neutron-star equation of state could be
determined at the few-percent accuracy level using high quality measurements of
the masses and radii (or masses and tidal deformabilities) of just two or three
neutron stars.
