Discrepancy principle for DSM

Let $Ay=f$, $A$ is a linear operator in a Hilbert space $H$, $y\perp N(A):=\{u:Au=0\}$, $R(A):=\{h:h=Au,u\in D(A)\}$ is not closed, $\|f_\delta-f\|\leq\delta$. Given $f_\delta$, one wants to construct $u_\delta$ such that $\lim_{\delta\to 0}\|u_\delta-y\|=0$. A version of the DSM (dynamical systems method) for finding $u_\delta$ consists of solving the problem \bee \dotu_\delta(t)=-u_\delta(t)+T^{-1}_{a(t)} A^\ast f_\delta, \quad u(0)=u_0, \eqno{(\ast)}\eee where $T:=A^\ast A$, $T_a:=T+aI$, and $a=a(t)>0$, $a(t)\searrow 0$ as $t\to\infty$ is suitably chosen. It is proved that $u_\delta:=u_\delta(t_\delta)$ has the property $\lim_{\delta\to 0}\|u_\delta-y\|=0$. Here the stopping time $t_\delta$ is defined by the discrepancy principle: \bee \eqno{(\ast\ast)}\eee $c\in(1,2)$ is a constant. Equation $(\ast)$ defines $t_\delta$ uniquely and $\lim_{\delta\to 0}t_\delta=\infty$. Another version of the discrepancy principle is also proved in this paper.