A new discrepancy principle

The aim of this note is to prove a new discrepancy principle. The advantage of the new discrepancy principle compared with the known one consists of solving a minimization problem (see problem (2) below) approximately, rather than exactly, and in the proof of a stability result. To explain this in more detail, let us recall the usual discrepancy principle, which can be stated as follows. Consider an operator equation Au= f, (1) where A :H →H is a bounded linear operator on a Hilbert space H , and assume that the range R(A) is not closed, so that problem (1) is ill-posed. Assume that f =Ay where y is the minimal-norm solution to (1), and that noisy data fδ are given, such that ‖fδ −f ‖ δ. One wants to construct a stable approximation to y, given fδ . The variational regularization method for solving this problem consists of solving the minimization problem F(u) := ‖Au− fδ‖ + ε‖u‖2 = min. (2)