  As galaxy surveys begin to measure the imprint of baryonic acoustic
oscillations (BAO) on large-scale structure at the sub-percent level,
reconstruction techniques that reduce the contamination from nonlinear
clustering become increasingly important. Inverting the nonlinear continuity
equation, we propose an Eulerian growth-shift reconstruction algorithm that
does not require the displacement of any objects, which is needed for the
standard Lagrangian BAO reconstruction algorithm. In real-space DM-only
simulations the algorithm yields 95% of the BAO signal-to-noise obtained from
standard reconstruction. The reconstructed power spectrum is obtained by adding
specific simple 3- and 4-point statistics to the pre-reconstruction power
spectrum, making it very transparent how additional BAO information from
higher-point statistics is included in the power spectrum through the
reconstruction process. Analytical models of the reconstructed density for the
two algorithms agree at second order. Based on similar modeling efforts, we
introduce four additional reconstruction algorithms and discuss their
performance.
