On the discrepancy principle for the dynamical systems method

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Assume that Au=f, (1) is a solvable linear equation in a Hilbert space, ||A||<∞, and R(A) is not closed, so problem (1) is ill-posed. Here R(A) is the range of the linear operator A. A DSM (dynamical systems method) for solving (1), consists of solving the following Cauchy problem: u= -u +(B+(t))⁻¹A*f, u(0)=u₀, (2) where B:=A*A, u:=du/dt, u₀ is arbitrary, and (t)>0 is a continuously differentiable function, monotonically decaying to zero as t→ ∞. A.G.Ramm has proved that, for any u₀, problem (2) has a unique solution for all t>0, there exists y:=w(∞):=ₜ→ ∞u(t), Ay=f, and y is the unique minimal-norm solution to (1). If f is given, such that ||f-f||≤, then u(t) is defined as the solution to (2) with f replaced by f. The stopping time is defined as a number t that → ₀||u (t)-y||=0, and → ₀t=∞. A discrepancy principle is proposed and proved in this paper. This principle yields t as the unique solution to the equation: ||A(B+(t))⁻¹A*f -f||=, (3) where it is assumed that ||f||> and f⊥ N(A*). For nonlinear monotone A a discrepancy principle is formulated and justified.