Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 326 / 2 / 1 / Solution

A regularization of an inverse problem is a family of continuous maps $R_{\alpha }:V\to U$, $\alpha >0$, such that $R_{\alpha }f\to K^{†}f$ for each $f\in \mathcal{D}(K^{†})$ as $\alpha \downarrow 0$. 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,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 $\alpha (\delta )\to 0$ and $\delta \Vert R_{\alpha (\delta )}\Vert \to 0$ are sufficient.

For example, Tikhonov regularization with identity penalty has $R_{\alpha }=(K^{\ast }K+\alpha I{)}^{-1}{K}^{\ast }$ and $\Vert R_{\alpha }\Vert \le 1/(2\sqrt{\alpha })$. 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.