= Solution
The <source condition in variational regularization> at an exact solution $u^\dagger$ is the existence of a source element $\mu^\dagger\in\mathcal V^*$ such that
$$
\boxed{A^*\mu^\dagger\in\partial J(u^\dagger).}
$$
For a Hilbert data space one identifies $\mathcal V^*$ with $\mathcal V$. If $u$ satisfies $Au=f$ and $A^*\mu\in\partial J(u)$, then every other exact solution $v$ obeys the <subgradient inequality>
$$
J(v)\geq J(u)+\langle A^*\mu,v-u\rangle
=J(u)+\langle\mu,Av-Au\rangle=J(u).
$$
Hence $u$ is a <J-minimizing solution>. This is <source-condition certification of a penalty-minimizing solution>.
Back to article page
