Discrepancy principle for the dynamical systems method

1 Discrepancy principle for the dynamical systems method Abstract 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)) −1 A * f, u(0) = u 0 , (2) where B := A * A, ˙ u := du dt , u 0 is arbitrary, and (t) > 0 is a continuously differentiable function, monotonically decaying to zero as t → ∞. A.G.Ramm has proved that, for any u 0 , problem (2) has a unique solution for all t > 0, there exists y := w(∞) := lim t→∞ u(t), Ay = f , and y is the unique minimal-norm solution to (1). If f δ is given, such that ||f − f δ || ≤ δ, then u δ (t) is defined as the solution to (2) with f replaced by f δ. The stopping time is defined as a number t δ such that lim δ→0 ||u δ (t δ) − y|| = 0, and lim δ→0 t δ = ∞. A discrepancy principle is proposed and proved in this paper. This principle yields t δ as the unique solution to the equation: ||A(B + (t)) −1 A * f δ − f δ || = δ, (3) where it is assumed that ||f δ || > δ and f δ ⊥ N (A *). For nonlinear monotone A a discrepancy principle is formulated and justified.