Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 326 / 1 / e / Solution

The exact solution is feasible becauseThe feasible set $E_{\delta }$ is convex and weakly closed. A minimizing sequence has bounded residual and bounded $J$; the coercivity assumption from part (a), applied to a fixed positive weighted objective, makes it bounded in $X$. Reflexivity gives a weakly convergent subsequence, and weak lower semicontinuity of the residual and $J$ keeps its limit feasible and minimizing.

Since ${\hat{u}}_{\delta }$ minimizes $J$ over $E_{\delta }$,The source condition in variational regularization and feasibility then yieldThus the claimed constant is $\boxed{C=(c+1)\Vert w^{†}{\Vert }_{Y^{\ast }}}$.