 Normal-operator source condition

This stronger source condition in variational regularization requires a subgradient $w=K^{\ast }Kv$ at $u^{†}$. For each $\alpha >0$ it is equivalent to making $u^{†}$ minimize $\Vert Ku-K\bar{u}{\Vert }^2/2+\alpha J(u)$ for artificial input $\bar{u}=u^{†}+\alpha v$. The subgradient optimality condition proves both implications. The source vector may be zero: $K=I$ and $J\equiv 0$ satisfy the condition but admit no nonzero source vector.