Quadratic fidelity ($\frac{1}{2}\Vert u-f{\Vert }_2^2$)

The squared-error term $\frac{1}{2}\Vert u-f{\Vert }_2^2$ is a strictly convex cost measuring agreement with data.

Let $w\in BV\cap L^{\infty }$ and $g\in BV\cap L^2$ on a bounded domain, $F_g(v)=\frac{1}{2}\Vert v-g{\Vert }_2^2$, and $w_{\pm t}=(1-\theta )w+\theta w\circ {\Phi }_{\pm t}$ for a smooth compactly supported local flow. The displayed formula is exact to an $o(t)$ remainder, and $g$ need not be bounded. Writing $J_t=\det D{\Phi }_t$, its cross-term identity isThe remainder is $o(t)$ because $1-J_{-t}=O(t)$ and weighted $L^1$ convergence follows fromFirst let $t\to 0$, then $K\to \infty$. The mass change of $w^2$ under opposite flows is $O(t^2)\Vert w{\Vert }_2^2$. Combined with the BV jump-product limit with one bounded factor, this evaluates the jump contribution without a bounded fidelity derivative.