  We study and develop multilevel methods for the numerical approximation of a
log-concave probability $\pi$ on $\mathbb{R}^d$, based on (over-damped)
Langevin diffusion. In the continuity of \cite{art:egeapanloup2021multilevel}
concentrated on the uniformly log-concave setting, we here study the procedure
in the absence of the uniformity assumption. More precisely, we first adapt an
idea of \cite{art:DalalyanRiouKaragulyan} by adding a penalization term to the
potential to recover the uniformly convex setting. Such approach leads to an
\textit{$\varepsilon$-complexity} of the order $\varepsilon^{-5} \pi(|.|^2)^{3}
d$ (up to logarithmic terms). Then, in the spirit of \cite{art:gadat2020cost},
we propose to explore the robustness of the method in a weakly convex
parametric setting where the lowest eigenvalue of the Hessian of the potential
$U$ is controlled by the function $U(x)^{-r}$ for $r \in (0,1)$. In this
intermediary framework between the strongly convex setting ($r=0$) and the
``Laplace case'' ($r=1$), we show that with the help of the control of
exponential moments of the Euler scheme, we can adapt some fundamental
properties for the efficiency of the method. In the ``best'' setting where $U$
is ${\mathcal{C}}^3$ and $U(x)^{-r}$ control the largest eigenvalue of the
Hessian, we obtain an $\varepsilon$-complexity of the order
$c_{\rho,\delta}\varepsilon^{-2-\rho} d^{1+\frac{\rho}{2}+(4-\rho+\delta) r}$
for any $\rho>0$ (but with a constant $c_{\rho,\delta}$ which increases when
$\rho$ and $\delta$ go to $0$).
