Approximate inverse for linear and some nonlinear problems

In this paper we present a method for solving problems such as Af=g by constructing an approximate inverse which maps the data g to a regularized solution of this equation of the first kind. The solution operator can be precomputed independently of the data. This works for linear problems and for nonlinear problems with a special structure. The regularization is achieved by computing mollified versions of the (minimum-norm) solution. It is shown that this class of regularization operators contains as special cases the classical methods such as Tikhonov-Phillips, iteration methods and also discretization methods.