 Solution

The source condition in variational regularization at an exact solution $u^{†}$ is the existence of a source element ${\mu }^{†}\in {\mathcal{V}}^{\ast }$ such thatFor a Hilbert data space one identifies ${\mathcal{V}}^{\ast }$ with $\mathcal{V}$. If $u$ satisfies $Au=f$ and $A^{\ast }\mu \in \partial J(u)$, then every other exact solution $v$ obeys the subgradient inequalityHence $u$ is a J-minimizing solution. This is source-condition certification of a penalty-minimizing solution.