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

Lipschitz dependence of the coefficients on the resolvent and greedy approximation for scalar elliptic problems