For the solution of an ill-posed equation (1) \(Af= g\), defined by an operator \(A\) between Hilbert spaces, methods are considered for constructing an approximate inverse which maps the data \(g\) to a regularized solution of (1) and which is independent of the data. In essence, instead of \(f\), the approximation \(\langle f, e_\gamma\rangle\) with suitable mollifiers \(e_\gamma\) is computed. This can be done for linear and certain nonlinear problems. In the case of linear operators a mollified version of the minimum-norm solution is obtained [see also the author and P. Maass, Inverse Probl. 6, No. 3, 427-440 (1990; Zbl 0713.65040)]; for nonlinear operators the method improves on an approach of R. Snieder [Inverse Probl. 7, No. 3, 409-433 (1991; Zbl 0737.35155)].
It is shown that this class of regularization operators contains, as special cases, the classical methods of Tikhonov-Phillips, iteration methods, as well as discretization methods. For operators with some invariance properties, group representations can be applied to reduce the storage needs considerably. Some numerical experiments for a nonlinear problem are given.