Regularization of ill-posed problems

Remarks

Example: Tikhonov regularization $R_\alpha = (T^* T + \alpha I)^{-1} T^*$ with an a priori rule $\alpha = \alpha(\delta)$ such that $\alpha(\delta) \to 0$ and $\delta^2 / \alpha(\delta) \to 0$. Each $R_\alpha$ is bounded with $\norm{R_\alpha} \le 1/(2\sqrt{\alpha})$, which gives clause 1. The error splits as $\norm{R_\alpha y^\delta - T^\dagger y} \le \norm{R_\alpha y - T^\dagger y} + \delta/(2\sqrt{\alpha})$. The first term tends to $0$ as $\alpha \to 0$ by spectral calculus, and the second by the choice of $\alpha$, which gives clause 3. Non-example: $R_\alpha = T^\dagger$ for all $\alpha$ fails clause 1 whenever $\mathcal{R}(T)$ is not closed, because $T^\dagger$ is then unbounded. This happens for every injective compact operator with infinite-dimensional range, the typical model of an ill-posed problem. Other standard methods are truncated singular value decomposition, Landweber iteration with the iteration index as parameter, and the discrepancy principle as a parameter choice rule.