  This paper presents a multilevel FISTA algorithm, based on the use of the
Moreau envelope to build the correction brought by the coarse models, which is
easy to compute when the explicit form of the proximal operator of the
considered functions is known. This approach is supported by strong theoretical
guarantees: we prove both the rate of convergence and the convergence of the
iterates to a minimum in the convex case, an important result for ill-posed
problems. We evaluate our approach on image restoration problems and we show
that it outperforms classical FISTA for large-scale images.
