  The aim of this paper is the derivation of an a-posteriori error estimate for
a numerical method based on an exponential scheme in time and spectral Galerkin
methods in space. We obtain analytically a rigorous bound on the mean square
error conditioned to the calculated data, which is numerically computable and
uses the given numerical approximation. Thus one can check a-posteriori the
error for a given numerical computation without relying on an asymptotic
result.
  All estimates are only based on the numerical data and the structure of the
equation, but they do not use any a-priori information of the solution, which
makes the approach applicable to equations where global existence of solutions
is not known. For simplicity of presentation, we develop the method here in a
relatively simple situation of a stable one-dimensional Allen-Cahn equation
with additive forcing.
