  We propose a discrete approach for solving an inverse problem for
Schr\"odinger's equation in two dimensions, where the unknown potential is to
be determined from boundary measurements of the Dirichlet to Neumann map. For
absorptive potentials, and in the continuum, it is known that by using the
Liouville identity we obtain an inverse conductivity problem. Its discrete
analogue is to find a resistor network that matches the measurements, and is
well understood. Here we show how to use a discrete Liouville identity to
transform its solution to that of Schr\"odinger's problem. The discrete
Schr\"odinger potential given by the discrete Liouville identity can be used to
reconstruct the potential in the continuum in two ways. First, we can obtain a
direct but coarse reconstruction by interpreting the values of the discrete
Schr\"odinger potential as averages of the continuum Schr\"odinger potential on
a special sensitivity grid. Second, the discrete Schr\"odinger potential may be
used to reformulate the conventional nonlinear output least squares
optimization formulation of the inverse Schr\"odinger problem. Instead of
minimizing the boundary measurement misfit, we minimize the misfit between the
discrete Schr\"odinger potentials. This results in a better behaved
optimization problem that converges in a single Gauss-Newton iteration, and
gives good quality reconstructions of the potential, as illustrated by the
numerical results.
