Discrepancy principle for DSM

Abstract

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, $\|f_δ-f\|\leqδ$. Given $f_δ$, one wants to construct $u_δ$ such that $\lim_{δ\to 0}\|u_δ-y\|=0$. A version of the DSM (dynamical systems method) for finding $u_δ$ consists of solving the problem \bee \dotu_δ(t)=-u_δ(t)+T^{-1}_{a(t)} A^\ast f_δ, \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 $u_δ:=u_δ(t_δ)$ has the property $\lim_{δ\to 0}\|u_δ-y\|=0$. Here the stopping time $t_δ$ is defined by the discrepancy principle: \bee \eqno{(\ast\ast)}\eee $c\in(1,2)$ is a constant. Equation $(\ast)$ defines $t_δ$ uniquely and $\lim_{δ\to 0}t_δ=\infty$. Another version of the discrepancy principle is also proved in this paper.