  In this paper we develop a procedure to deal with a family of
parameter-dependent ill-posed problems, for which the exact solution in general
does not exist. The original problems are relaxed by considering corresponding
approximate ones, whose optimal solutions are well dfined, where the optimality
is determined by the minimal norm requirement. The procedure is based upon
greedy algorithms that preserve, at least asymptotically, Kolmogorov
approximation rates. In order to provide a-priori estimates for the algorithm,
a Tychonff-type regularization is applied, which adds an additional parameter
to the model. The theory is developed in an abstract theoretical framework that
allows its application to different kinds of problems. We present a specific
example that considers a family of ill-posed elliptic problems. The required
general assumptions in this case translate to rather natural uniform lower and
upper bounds on coefficients of the considered operators.
