= Solution
A <regularization of an inverse problem> is a family of continuous maps $R_\alpha:\mathcal V\to\mathcal U$, $\alpha>0$, such that
$$
R_\alpha f\longrightarrow A^\dagger f\qquad(\alpha\downarrow0)
$$
for every $f$ in the domain of the <Moore–Penrose inverse of an operator>. A linear regularization has each $R_\alpha$ bounded and linear. For example, <Tikhonov regularization> gives
$$
R_\alpha=(A^*A+\alpha I)^{-1}A^*.
$$
A nonlinear example is <quartic-norm variational regularization>:
$$
R_\alpha f=\arg\min_u\left\{\|Au-f\|^2+\alpha\|u\|^4\right\}.
$$
Its strictly convex norm penalty selects the same <minimum-norm least-squares solution> as $\alpha\downarrow0$, but the map from data to minimizer is generally nonlinear.
A <convergent regularization of an inverse problem> additionally has a parameter rule $\alpha(\delta,f_\delta)>0$ such that, for every admissible exact $f$,
$$
\boxed{\sup_{\|f_\delta-f\|\leq\delta}
\|R_{\alpha(\delta,f_\delta)}f_\delta-A^\dagger f\|\longrightarrow0
\quad(\delta\downarrow0).}
$$
An <a priori regularization parameter choice> depends only on $\delta$ and fixed information about the problem.
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