= Solution
A <regularization of an inverse problem> is a family of continuous maps $R_\alpha:V\to U$, $\alpha>0$, such that $R_\alpha f\to K^\dagger f$ for each $f\in\mathcal D(K^\dagger)$ as $\alpha\downarrow0$. A <linear regularization> requires each $R_\alpha$ to be a <bounded linear operator>.
A <regularization parameter choice> assigns $\alpha=\alpha(\delta,f^\delta)>0$ from the noise bound and possibly the measured data. A <convergent regularization of an inverse problem> satisfies, for every fixed admissible exact datum,
$$
\boxed{\sup_{\|g-f\|\leq\delta}\|R_{\alpha(\delta,g)}g-K^\dagger f\|\longrightarrow0\quad(\delta\downarrow0).}
$$
In the usual definition the chosen parameter also tends to zero uniformly over this noise ball. For a linear regularization with exact-data consistency, the conditions $\alpha(\delta)\to0$ and $\delta\|R_{\alpha(\delta)}\|\to0$ are sufficient.
For example, <Tikhonov regularization> with identity penalty has $R_\alpha=(K^*K+\alpha I)^{-1}K^*$ and $\|R_\alpha\|\leq1/(2\sqrt\alpha)$. Its spectral filter converges to the <Moore–Penrose inverse of an operator> on its domain. Taking $\alpha(\delta)=\delta$ for small positive noise levels gives both required limits.
Back to article page
