Discrepancy principle for the dynamical systems method

Assume thatAu=fis a solvable linear equation in a Hilbert space, ∥A∥<∞, and R(A) is not closed, so this problem is ill-posed. Here R(A) is the range of the linear operator A. A dynamical systems method for solving this problem, consists of solving the following Cauchy problem:u̇=−u+(B+ϵ(t))−1A∗f,u(0)=u0,where B:=A∗A, u̇:=du/dt, u0 is arbitrary, and ϵ(t)>0 is a continuously differentiable function, monotonically decaying to zero as t→∞. Ramm has proved [Commun Nonlin Sci Numer Simul 9(4) (2004) 383] that, for any u0, the Cauchy problem has a unique solution for all t>0, there exists y:=w(∞):=limt→∞u(t), Ay=f, and y is the unique minimal-norm solution to Au=f. If fδ is given, such that ∥f−fδ∥⩽δ, then uδ(t) is defined as the solution to the Cauchy problem with f replaced by fδ. The stopping time is defined as a number tδ such that limδ→0∥uδ(tδ)−y∥=0 and limδ→0tδ=∞. A discrepancy principle is proposed and proved in this paper. This principle yields tδ as the unique solution to the equation:∥A(B+ϵ(t))−1A∗fδ−fδ∥=δ,where it is assumed that ∥fδ∥>δ and fδ⊥N(A∗). The last assumption is removed, and if it does not hold, then the right-hand side of the above equation is replaced by Cδ, where C=const>1, and one assumes that ∥fδ∥>Cδ. For nonlinear monotone A a discrepancy principle is formulated and justified.