= Source-condition estimate for symmetric Bregman distance
{title2=$D_J^{\mathrm{sym}}(u_\alpha,u^\dagger)\leq\frac{\delta^2}{2\alpha}+\frac\alpha2\|w\|^2$}
Let $K:U\to V$ be a <bounded linear operator> from a real <Banach space> to a real <Hilbert space>, let $J$ be a proper lower-semicontinuous <convex function>, and let $\alpha>0$ and $\|f^\delta-Ku^\dagger\|\leq\delta$. For an existing minimizer of $\|Ku-f^\delta\|^2/2+\alpha J(u)$ and a <source condition in variational regularization> $p^\dagger=K^*w\in\partial J(u^\dagger)$, let $e=f^\delta-Ku^\dagger$ and $q=K(u_\alpha-u^\dagger)$. Its optimality <subgradient> is $p_\alpha=K^*(f^\delta-Ku_\alpha)/\alpha$. Then $\alpha D_J^{\mathrm{sym}}=\langle e-\alpha w,q\rangle-\|q\|^2\leq\|e-\alpha w\|^2/4$. The displayed estimate follows from $\|e-\alpha w\|^2\leq2\delta^2+2\alpha^2\|w\|^2$.
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