Tikhonov stability bound ($\Vert (A^{\ast }A+\alpha I{)}^{-1}{A}^{\ast }\Vert \le (2\sqrt{\alpha }{)}^{-1}$)

The reconstruction operator in Tikhonov regularization has scalar gain $\sigma /({\sigma }^2+\alpha )$. Its maximum over $\sigma \ge 0$ is $1/(2\sqrt{\alpha })$, so a data perturbation of norm at most $\delta$ changes the solution by at most $\delta /(2\sqrt{\alpha })$. A convergent noisy-data parameter choice satisfies $\alpha \to 0$ and $\delta /\sqrt{\alpha }\to 0$. Fixed-parameter stability is distinct from stability of the unregularized inverse.