Convergent regularization of an inverse problem

A regularization $R_{\alpha }$ is convergent when its parameter can be chosen from the noise level and data so that the regularized reconstruction converges to the exact Moore–Penrose inverse of an operator solution as the noise level tends to zero.

For a bounded linear regularization operator $R_{\alpha }$ and noisy data with $\Vert f_{\delta }-f\Vert \le \delta$, the triangle inequality givesThe first term is noise amplification; the second is the approximation bias on exact data. Thus $\alpha (\delta )\to 0$ and $\delta \Vert R_{\alpha (\delta )}\Vert \to 0$ suffice for a convergent regularization of an inverse problem, provided $R_{\alpha }$ is consistent on exact data.