Quadratic functional estimation in inverse problems

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.