The author develops a general theory of mollification for approximate solution of ill-posed linear problems in Banach space. For a given family of subspaces on which the problem is well-posed the idea is to construct a corresponding family of mollification operators which map the problem into a well-posed problem on the subspace. This is accomplished by minimizing an appropriate functional.
Error estimates and optimal or “quasi-optimal” parameter choice strategies are established and the method is applied to problems of numerical differentiation, parabolic equations reversed in time, Cauchy problems for the Laplace equation, and other problems. In addition, new Hölder type estimates are established for the backward heat equation and for certain non-characteristic Cauchy problems for parabolic equations.