  In many statistical linear inverse problems, one needs to recover classes of
similar curves from their noisy images under an operator that does not have a
bounded inverse. Problems of this kind appear in many areas of application.
Routinely, in such problems clustering is carried out at the pre-processing
step and then the inverse problem is solved for each of the cluster averages
separately. As a result, the errors of the procedures are usually examined for
the estimation step only. The objective of this paper is to examine, both
theoretically and via simulations, the effect of clustering on the accuracy of
the solutions of general ill-posed linear inverse problems. In particular, we
assume that one observes $X_m = A f_m + \delta \epsilon_m$, $m=1, \cdots, M$,
where functions $f_m$ can be grouped into $K$ classes and one needs to recover
a vector function ${\bf f}= (f_1,\cdots, f_M)^T$. We construct an estimators
for ${\bf f}$ as a solution of a penalized optimization problem and derive an
oracle inequality for its precision. By deriving upper and minimax lower bounds
for the error, we confirm that the estimator is minimax optimal or nearly
minimax optimal up to a logarithmic factor of the number of observations. One
of the advantages of our estimation procedure is that we do not assume that the
number of clusters is known in advance. We conclude that clustering at the
pre-processing step is beneficial when the problem is moderately ill-posed. It
should be applied with extreme care when the problem is severely ill-posed.
