Discrepancy principle for the dynamical systems method

Abstract

Assume that Au=f is 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: u ̇ =−u+(B+ϵ(t)) −1A ∗f, u(0)=u 0, where B:=A ∗A , u ̇ := du/ dt , u 0 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 u 0, the Cauchy problem 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 Au= f. If f δ is given, such that ∥ f− f δ ∥⩽ δ, then u δ ( t) is defined as the solution to the Cauchy problem 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+ϵ(t)) −1A ∗f δ−f δ∥=δ, where it is assumed that ∥ f δ ∥> δ and f δ⊥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 ∥ f δ ∥> Cδ. For nonlinear monotone A a discrepancy principle is formulated and justified.