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. No discretization for f is needed. 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. In the case where the operator has some invariance properties the storage needs are dramatically reduced.