  It has been a well-known problem in the $G$-framework that it is hard to
compute the sublinear expectation of the $G$-normal distribution
$\hat{\mathbb{E}}[\varphi(X)]$ when $\varphi$ is neither convex nor concave, if
not involving any PDE techniques to solve the corresponding $G$-heat equation.
Recently, we have established an efficient iterative method able to compute the
sublinear expectation of \emph{arbitrary} functions of the $G$-normal
distribution, which directly applies the \emph{Nonlinear Central Limit Theorem}
in the $G$-framework to a sequence of variance-uncertain random variables
following the \emph{Semi-$G$-normal Distribution}, a newly defined concept with
a nice \emph{Integral Representation}, behaving like a ladder in both theory
and intuition, helping us climb from the ground of classical normal
distribution to approach the peak of $G$-normal distribution through the
\emph{iteratively maximizing} steps. The series of iteration functions actually
produce the whole \emph{solution surface} of the $G$-heat equation on a given
time grid.
