  We use \textit{ab initio} calculations to systematically study the phase
diagram of multiferroic Sr$_{1-x}$Ba$_x$MnO$_3$ ($0 \leq x \leq 1$) as a
function of chemical doping, epitaxial strain and external pressure. We find
that by replacing Sr with Ba in cubic SrMnO$_3$ and imposing epitaxial strain,
the material can be tuned to the vicinity of a first order transition between
two multiferroic phases, one antiferromagnetic with a smaller polarization and
one ferromagnetic with a larger polarization. A giant effective
magneto-electric coupling and cross-field control (electric field control of
magnetism or magnetic field control of polarization) can be achieved in the
vicinity of the transition. The dependence of the theoretically computed
transition point on the choice of exchange correlation functionals is
determined and is found to be non-negligible. We also show that the perovskite
structure of BaMnO$_3$ can be stabilized relative to its hexagonal polymorphs
at pressures larger than 20 GPa.
