  We analyze the inverse problem of identifying the diffusivity coefficient of
a scalar elliptic equation as a function of the resolvent operator. We prove
that, within the class of measurable coefficients, bounded above and below by
positive constants, the resolvent determines the diffusivity in an unique
manner. Furthermore we prove that the inverse mapping from resolvent to the
coefficient is Lipschitz in suitable topologies. This result plays a key role
when applying greedy algorithms to the approximation of parameter-dependent
elliptic problems in an uniform and robust manner, independent of the given
source terms. In one space dimension the results can be improved using the
explicit expression of solutions, which allows to link distances between one
resolvent and a linear combination of finitely many others and the
corresponding distances on coefficients. These results are also extended to
multi-dimensional elliptic equations with variable density coefficients. We
also point out towards some possible extensions and open problems.
