Discrepancy principle for DSM

Details Discrepancy principle for DSM Discrepancy principle for DSM

2006-03-27

arxiv.org/abs/math/0603632v1

Mathematics

Functional Analysis

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Discrepancy principle for DSM does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Discrepancy principle for DSM, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Discrepancy principle for DSM will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-413669