  We consider in this paper a Gaussian sequence model of observations $Y_i$,
$i\geq 1$ having mean (or signal) $\theta_i$ and variance $\sigma_i$ which is
growing polynomially like $i^\gamma$, $\gamma >0$. This model describes a large
panel of inverse problems. We estimate the quadratic functional of the unknown
signal $\sum_{i\geq 1}\theta_i^2$ when the signal belongs to ellipsoids of both
finite smoothness functions (polynomial weights $i^\alpha$, $\alpha>0$) and
infinite smoothness (exponential weights $e^{\beta i^r}$, $\beta >0$, $0<r \leq
2$). We propose a Pinsker type projection estimator in each case and study its
quadratic risk. When the signal is sufficiently smoother than the difficulty of
the inverse problem ($\alpha>\gamma+1/4$ or in the case of exponential
weights), we obtain the parametric rate and the efficiency constant associated
to it. Moreover, we give upper bounds of the second order term in the risk and
conjecture that they are asymptotically sharp minimax. When the signal is
finitely smooth with $\alpha \leq \gamma +1/4$, we compute non parametric upper
bounds of the risk of and we presume also that the constant is asymptotically
sharp.
