On the discrepancy principle for the dynamical systems method

Abstract

Assume that $$ Au=f,\quad (1) $$ is a solvable linear equation in a Hilbert space, $||A||<\infty$, 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: $$ \dot u= -u +(B+\ep(t))^{-1}A^*f, \quad u(0)=u_0, \quad (2) $$ where $B:=A^*A$, $\dot u:=\frac {du}{dt}$, $u_0$ is arbitrary, and $\ep(t)>0$ is a continuously differentiable function, monotonically decaying to zero as $t\to \infty$. A.G.Ramm has proved that, for any $u_0$, problem (2) has a unique solution for all $t>0$, there exists $y:=w(\infty):=\lim_{t\to \infty}u(t)$, $Ay=f$, and $y$ is the unique minimal-norm solution to (1). If $f_\d$ is given, such that $||f-f_\d||\leq \d$, then $u_\d(t)$ is defined as the solution to (2) with $f$ replaced by $f_\d$. The stopping time is defined as a number $t_\d$such that $\lim_{\d \to 0}||u_\d (t_\d)-y||=0$, and $\lim_{\d \to 0}t_\d=\infty$. A discrepancy principle is proposed and proved in this paper. This principle yields $t_\d$ as the unique solution to the equation: $$ ||A(B+\ep(t))^{-1}A^*f_\d -f_\d||=\d, \quad (3) $$ where it is assumed that $||f_\d||>\d$ and $f_\d\perp N(A^*)$. For nonlinear monotone $A$ a discrepancy principle is formulated and justified.