Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 326 / 2 / 1 / b

Suppose each $R_{\alpha }:\mathcal{V}\to \mathcal{U}$ is a bounded linear operator and $R_{\alpha }f\to K^{†}f$ for every $f\in \mathcal{D}(K^{†})$ as $\alpha \downarrow 0$. A standard sufficient a priori regularization parameter choice satisfiesFor every datum with $\Vert f^{\delta }-f\Vert \le \delta$, the noise-bias decomposition for linear regularization givesThus the parameter rule gives a convergent regularization of an inverse problem, uniformly over data in the prescribed noise ball for each fixed admissible exact datum. For Tikhonov regularization, $\Vert R_{\alpha }\Vert \le 1/(2\sqrt{\alpha })$, so $\alpha \to 0$ and $\delta /\sqrt{\alpha }\to 0$ suffice. The same sufficient noise scaling holds for spectral cutoff regularization.