  We study the prospects for three-dimensional mapping of the dark matter to
high redshift through the shearing of faint galaxies images at multiple
distances by gravitational lensing. Such maps could provide invaluable
information on the nature of the dark energy and dark matter. While in
principle well-posed, mapping by direct inversion introduces exceedingly large,
but usefully correlated noise into the reconstruction. By carefully propagating
the noise covariance, we show that lensing contains substantial information,
both direct and statistical, on the large-scale radial evolution of the density
field. This information can be efficiently distilled into low-order
signal-to-noise eigenmodes which may be used to compress the data by over an
order of magnitude. Such compression will be useful for the statistical
analysis of future large data sets. The reconstructed map also contains useful
information on the localization of individual massive dark matter halos, and
hence the dark energy from halo number counts, but its extraction depends
strongly on prior assumptions. We outline a procedure for maximum entropy and
point-source regularization of the maps that can identify alternate
reconstructions.
