On the discrepancy principle for the dynamical systems method

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