Two results on ill-posed problems

Abstract

Let $A=A^*$ be a linear operator in a Hilbert space $H$. Assume that equation $Au=f \quad (1)$ is solvable, not necessarily uniquely, and $y$ is its minimal-norm solution. Assume that problem (1) is ill-posed. Let $f_\d$, $||f-f_d||\leq \d$, be noisy data, which are given, while $f$ is not known. Variational regularization of problem (1) leads to an equation $A^*Au+\a u=A^*f_\d$. Operation count for solving this equation is much higher, than for solving the equation $(A+ia)u=f_\d \quad (2)$. The first result is the theorem which says that if $a=a(\d)$, $\lim_{\d \to 0}a(\d)=0$ and $\lim_{\d \to 0}\frac \d {a(\d)}=0$, then the unique solution $u_\d$ to equation (2), with $a=a(\d),$ has the property $\lim_{\d \to 0}||u_\d-y||=0$. The second result is an iterative method for stable calculation of the values of unbounded operator on elements given with an error.