  We report multipronged progress on the stochastic averaging approach to
numerical analytic continuation of quantum Monte Carlo data. With the sampled
spectrum parametrized with delta-functions in continuous frequency space, a
calculation of the configurational entropy lends support to a simple
goodness-of-fit criterion for the optimal sampling temperature. To further
investigate entropic effects, we compare spectra sampled in continuous
frequency with results of amplitudes sampled on a fixed frequency grid. We
demonstrate equivalences between sampling and optimizing spectral functions
with the maximum-entropy approach with different forms of the entropy. These
insights revise prevailing notions of the maximum-entropy method and its
relationship to stochastic analytic continuation. We further explore various
adjustable (optimized) constraints that allow sharp spectral features to be
resolved, in particular at the lower frequency edge. The constraints, e.g., the
location of the edge or the spectral weight of a quasi-particle peak, are
optimized using a statistical criterion. We show that this method can correctly
reproduce both narrow and broad quasi-particle peaks. We next introduce a
parametrization for more intricate spectral functions with sharp edges, e.g.,
power-law singularities. Tests with synthetic data as well as with real
simulation data for the spin-1/2 Heisenberg chain demonstrate that constrained
sampling methods can reproduce spectral functions with sharp edge features at
unprecedented fidelity. We present new results for S=1/2 Heisenberg 2-leg and
3-leg ladders to illustrate the ability of the methods to resolve spectral
features arising from both elementary and composite excitations. Finally, we
also propose how the methods developed here could be used as "pre processors"
for analytic continuation by machine learning.
