= Asymptotic regularization
Asymptotic regularization stops the <gradient flow> $x'=-A^*(Ax-f)$ at $t=1/\alpha$, starting from zero. With the <singular system of a compact operator> convention $Av_j=\sigma_j u_j$, its spectral filter gives
$$
R_\alpha f=\sum_j\frac{1-e^{-\sigma_j^2/\alpha}}{\sigma_j}\langle f,u_j\rangle v_j.
$$
The scalar coefficient solves $c_j'=-\sigma_j^2c_j+\sigma_j\langle f,u_j\rangle$ with zero initial value. Since $1-e^{-s}\leq\min(s,1)$ for $s\geq0$, the <operator norm> is at most $\alpha^{-1/2}$. On the domain of the <Moore–Penrose inverse of an operator>, <dominated convergence theorem> of the squared spectral coefficients proves $R_\alpha f\to A^\dagger f$. The <noise-bias decomposition for linear regularization> then gives noisy-data convergence when $\delta/\sqrt\alpha\to0$.
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