 Source-condition certification of a penalty-minimizing solution

If $Au=f$ and $A^{\ast }\mu \in \partial J(u)$, then $u$ is a J-minimizing solution. For every other exact solution $v$, the subgradient inequality gives $J(v)\ge J(u)+⟨\mu ,Av-Au⟩=J(u)$. This is a certificate of global optimality supplied by the range of the adjoint operator.