  This work deals with a regularization method enforcing solution sparsity of
linear ill-posed problems by appropriate discretization in the image space.
Namely, we formulate the so called least error method in an $\ell^1$ setting
and perform the convergence analysis by choosing the discretization level
according to an a priori rule, as well as two a posteriori rules, via the
discrepancy principle and the monotone error rule, respectively. Depending on
the setting, linear or sublinear convergence rates in the $\ell^1$-norm are
obtained under a source condition yielding sparsity of the solution. A part of
the study is devoted to analyzing the structure of the approximate solutions
and of the involved source elements.
