Discrepancy principle for DSM

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, f δ − f ≤ δ. Given f δ , one wants to construct u δ such that lim δ→0 u δ − y = 0. A version of the DSM (dynamical systems method) for finding u δ consists of solving the problem ˙ u δ (t) = −u δ (t) + T −1 a(t) A * f δ , u(0) = u 0 , (*) where T := A * A, T a := T + aI, and a = a(t) > 0, a(t) ց 0 as t → ∞ is suitably chosen. It is proved that u δ := u δ (t δ) has the property lim δ→0 u δ − y = 0. Here the stopping time t δ is defined by the discrepancy principle: t 0 e −(t−s) a(s)Q −1 a(s) f δ ds = cδ, (* *) c ∈ (1, 2) is a constant. Equation (*) defines t δ uniquely and lim δ→0 t δ = ∞. Another version of the discrepancy principle is proved in this paper: let a δ be the solution to the equation aQ −1 a f δ = cδ, and t δ be the solution to the equation a(t) = a δ. If lim t→∞ ˙ a(t) a 2 (t) = 0, then lim δ→0 u δ − y = 0, where u δ := u δ (t δ) and u δ (t) solves (*).