  In \cite{hien}, M. Hien introduced rapid decay homology group
$\Homo^{rd}_{*}(U, (\nabla, E))$ associated to an irregular connection
$(\nabla, E)$ on a smooth complex affine variety $U$, and showed that it is the
dual group of the algebraic de Rham cohomology group
$\Homo^*_{dR}(U,(\nabla^{\vee}, E^{\vee}))$. On the other hand, F. Pham has
already introduced his version of rapid decay homology when $(\nabla, E)$ is
the so-called elementary irregular connection (\cite{Sab}) in \cite{Pham}. In
this report, we will state a comparison theorem of these homology groups and
give an outline of its proof. This can be regarded as a homological counterpart
of the result \cite{Sab} of C. Sabbah. As an application, we construct a basis
of some rapid decay homologies associated to a hyperplane arrangement and
hypersphere arrangement of Schl\"ofli type.
