Morozov's Discrepancy Principle under General Source Conditions

Abstract

In this paper we study linear ill-posed problems $Ax=y$ in a Hilbert space setting where instead of exact data $y$ noisy data $y^{\delta }$ are given satisfying $\Vert y-y^{\delta }\Vert \le \delta$ with known noise level $\delta$. Regularized approximations are obtained by a general regularization scheme where the regularization parameter is chosen from Morozov's discrepancy principle. Assuming the unknown solution belongs to some general source set $M$ we prove that the regularized approximation provides order optimal error bounds on the set $M$. Our results cover the special case of finitely smoothing operators $A$ and extend recent results for infinitely smoothing operators.