Solution
= Solution
An element $u^\dagger$ is a $J$-minimizing solution when
$$
Au^\dagger=f,
\qquad
J(u^\dagger)=\inf\{J(u):Au=f\}.
$$
It satisfies the <source condition in variational regularization> when there is a $w^\dagger\in Y^*$ such that
$$
\boxed{p^\dagger=A^*w^\dagger\in\partial J(u^\dagger)},
$$
where $\partial J(u^\dagger)$ is the <subdifferential> of the <convex function>[convex functional] $J$.