Discrepancy principle for DSM II

Let Ay = f, A is a linear operator in a Hilbert space H, y ⊥ N(A) ≔ {u : Au = 0}, R(A) ≔ {h : h = Au, u ∈ D(A)} is not closed, ∥fδ − f∥ ⩽ δ. Given fδ, one wants to construct uδ such that limδ→0∥uδ − y∥ = 0. Two versions of discrepancy principles for the DSM (dynamical systems method) for finding the stopping time and calculating the stable solution uδ to the original equation Ay = f are formulated and mathematically justified.