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, $\Vert f_{\delta }-f\Vert \le \delta$. Given $f_{\delta }$, one wants to construct $u_{\delta }$ such that ${\lim }_{\delta \to 0}\Vert u_{\delta }-y\Vert =0$. A version of the DSM (dynamical systems method) for finding $u_{\delta }$ 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 $u_{\delta }:=u_{\delta }(t_{\delta })$ has the property ${\lim }_{\delta \to 0}\Vert u_{\delta }-y\Vert =0$. 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.