  Solving time-harmonic wave propagation problems by iterative methods is a
difficult task, and over the last two decades, an important research effort has
gone into developing preconditioners for the simplest representative of such
wave propagation problems, the Helmholtz equation. A specific class of these
new preconditioners are considered here. They were developed by researchers
with various backgrounds using formulations and notations that are very
different, and all are among the most promising preconditioners for the
Helmholtz equation.
  The goal of the present manuscript is to show that this class of
preconditioners are based on a common mathematical principle, and they can all
be formulated in the context of domain decomposition methods called optimized
Schwarz methods. This common formulation allows us to explain in detail how and
why all these methods work. The domain decomposition formulation also allows us
to avoid technicalities in the implementation description we give of these
recent methods.
  The equivalence of these methods with optimized Schwarz methods translates at
the discrete level into equivalence with approximate block LU decomposition
preconditioners, and we give in each case the algebraic version, including a
detailed description of the approximations used. While we chose to use the
Helmholtz equation for which these methods were developed, our notation is
completely general and the algorithms we give are written for an arbitrary
second order elliptic operator. The algebraic versions are even more general,
assuming only a connectivity pattern in the discretization matrix.
