= Solution
A <regularization of an inverse problem> consists of bounded operators $R_\alpha:Y\to X$ and a parameter rule $\alpha=\alpha(\delta,y^\delta)$ such that, whenever $\|y^\delta-y\|\leq\delta$ and $y$ lies in the domain of $A^\dagger$,
$$
\alpha(\delta,y^\delta)\to0,
\qquad
R_{\alpha(\delta,y^\delta)}y^\delta\to A^\dagger y
$$
as $\delta\to0$. It is needed because a compact operator on an infinite-dimensional space has singular values tending to zero, so direct inversion divides noisy data by arbitrarily small numbers and is generally discontinuous.
<Tikhonov regularization> defines
$$
x_\alpha^\delta
=\underset{x\in X}{\operatorname{argmin}}
\left(\|Ax-y^\delta\|^2+\alpha\|x\|^2\right),
$$
and its normal equation gives
$$
\boxed{x_\alpha^\delta
=(A^*A+\alpha I)^{-1}A^*y^\delta.}
$$
For every fixed $\alpha>0$, $A^*A+\alpha I$ is bounded below by $\alpha I$, and the data-to-solution operator is bounded. Small changes in $y^\delta$ therefore produce small changes in $x_\alpha^\delta$.
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