Recent Developments in Inverse Problems - Abstract

Kangro, Urve

On self-regularization of ill-posed problems in Banach spaces by the least squares and the least error methods

with B. Kaltenbacher, U. Hämarik, E. Resmerita We consider an ill-posed problem $Au=f$ with linear operator $A in mathcalL(E,F)$ acting between Banach spaces $E$, $F$. The exact right-hand side $f$ is unknown; instead noisy data $f^delta$ satisfying $ f^delta-f le delta$ with known noise level $delta$ are given. For finding approximation $u_n$ to the solution $u_*$ of the problem we discuss the least squares method, where one minimizes $ Au_n-f^delta $ over a $n$-dimensional subspace $E_n subset E$, and the least error method, where $ u_n$ is the minimal norm solution of the projected equation. For the case of exact data we give conditions under which there is a unique solution $u_n$ and $ u_n-u_* to 0$ as $n to infty$. If the data are noisy, then one can choose the dimension $n=n(delta)$ as the regularization parameter depending on the noise level $delta$ in such way that $ u_n(delta)-u_* to 0$ as $delta to 0$. For the choice of dimension $n=n(delta)$ we consider a priori rule and a posteriori choice by the discrepancy principle and by the monotone error rule.