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We propose a discrete approach for solving an inverse problem forSchr\"odinger's equation in two dimensions, where the unknown potential is tobe determined from boundary measurements of the Dirichlet to Neumann map. Forabsorptive potentials, and in the continuum, it is known that by using theLiouville identity we obtain an inverse conductivity problem. Its discreteanalogue is to find a resistor network that matches the measurements, and iswell understood. Here we show how to use a discrete Liouville identity totransform its solution to that of Schr\"odinger's problem. The discreteSchr\"odinger potential given by the discrete Liouville identity can be used toreconstruct the potential in the continuum in two ways. First, we can obtain adirect but coarse reconstruction by interpreting the values of the discreteSchr\"odinger potential as averages of the continuum Schr\"odinger potential ona special sensitivity grid. Second, the discrete Schr\"odinger potential may beused to reformulate the conventional nonlinear output least squaresoptimization formulation of the inverse Schr\"odinger problem. Instead ofminimizing the boundary measurement misfit, we minimize the misfit between thediscrete Schr\"odinger potentials. This results in a better behavedoptimization problem that converges in a single Gauss-Newton iteration, andgives good quality reconstructions of the potential, as illustrated by thenumerical results.

A discrete Liouville identity for numerical reconstruction of Schr\"odinger potentials