  Accurate estimates of the horizontal electric field on the Sun's visible
surface are important not only for estimating the Poynting flux of magnetic
energy into the corona but also for driving time-dependent magnetohydrodynamic
models of the corona. In this paper, a method is developed for estimating the
horizontal electric field from a sequence of radial-component magnetic field
maps. This problem of inverting Faraday's law has no unique solution.
Unfortunately, the simplest solution (a divergence-free electric field) is not
realistically localized in regions of non-zero magnetic field, as would be
expected from Ohm's law. Our new method generates instead a localized solution,
using a basis pursuit algorithm to find a sparse solution for the electric
field. The method is shown to perform well on test cases where the input
magnetic maps are flux balanced, in both Cartesian and spherical geometries.
However, we show that if the input maps have a significant imbalance of flux -
usually arising from data assimilation - then it is not possible to find a
localized, realistic, electric field solution. This is the main obstacle to
driving coronal models from time sequences of solar surface magnetic maps.
