Recent Developments in Inverse Problems - Abstract

Hämarik, Uno

On comparison of accuracy of approximate solutions of ill-posed problems

Let $A colon Xto Y$ be a linear bounded operator between Hilbert spaces. We are interested in finding the minimum norm solution $x_*$ of the ill-posed problem beginequationlabeleq1 Ax = y, qquad y in mathcalR(A) not =overline mathcalR(A). endequation Instead of exact data $y$, noisy data $y^delta$, $ y-y^delta leq delta$ with noise level $delta$ are given. The following simple result may be useful for comparison of accuracy of different approximate solutions of problem $Ax=y$. Theorem. If $x, x' in X$, $x'=x A^*z, zin Y$ and $w=x A^*z/2$, then it holds the implication[D(z):=frac(y^delta-Aw,z) z > delta Longrightarrow x'-x_* < \ x-x_*\ .\] Moreover \begin{multline*} \frac{1}{2}(\ x'-x_*\ ^2-\ x-x_*\ ^2) =(Aw-y,z)\\ \leq \frac{1}{2}\ A^*z\ ^2 (Ax-y^{\delta},z) \delta \ z\ =\ z\ (\delta -D(z)). \end{multline*} For given $x$ we may find $z \in Y$ which minimizes the function in the last estimate $F(z):=\ A^*z\ ^2/2 (Ax-y^{\delta},z) \delta \ z\ $. This $z$ solves also the equation $AA^*z \delta z/\ z\ =y^{\delta}-Ax $. Note that $x_\alpha:=x A^*z_\alpha$ with $z_\alpha=(\alpha I 1/2AA^*)^{-1}(y^{\delta}-Ax)$ is more accurate approximate solution than $x$, if $D(z_\alpha)=\alpha \ z_\alpha \ >delta$. We consider also other possibilities for construction of more accurate approximate solutions than given $x$.