Lavrentiev Regularization for Linear Ill-Posed Problems under General Source Conditions

Abstract

In this paper we study the problem of identifying the solution $x^{†}$ of linear ill-posed problems $Ax=y$ with non-negative and self-adjoint operators $A$ on a Hilbert space $X$ where instead of exact data $y$ noisy data $y^{\delta }\in X$ are given satisfying $\Vert y-y^{\delta }\Vert \le \delta$ with known noise level $\delta$. Regularized approximations $x_{\alpha }^{\delta }$ are obtained by the method of Lavrentiev regularization, that is, $x_{\alpha }^{\delta }$ is the solution of the singularly perturbed operator equation $Ax+\alpha x=y^{\delta }$, and the regularization parameter $\alpha$ is chosen either a priori or a posteriori by the rule of Raus. Assuming the unknown solution belongs to some general source set $M$, we prove that the regularized approximations provide 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. In addition, we generalize our results to the method of iterated Lavrentiev regularization of order $m$ and discuss a special ill-posed problem arising in inverse heat conduction.