= Normal-operator source condition
{title2=$\operatorname{ran}(K^*K)\cap\partial J(u^\dagger)\ne\varnothing$}
This stronger <source condition in variational regularization> requires a <subgradient> $w=K^*Kv$ at $u^\dagger$. For each $\alpha>0$ it is equivalent to making $u^\dagger$ minimize $\|Ku-K\bar u\|^2/2+\alpha J(u)$ for artificial input $\bar u=u^\dagger+\alpha v$. The <subgradient optimality condition> proves both implications. The source vector may be zero: $K=I$ and $J\equiv0$ satisfy the condition but admit no nonzero source vector.
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