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

Take a minimizing sequence for the variational regularization functionalIts coercivity makes the sequence bounded. Since $X$ is a reflexive Banach space, a subsequence converges weakly to some $u\in X$. A convex norm-lower-semicontinuous functional has weak lower semicontinuity, so this applies to both $J$ and the convex continuous map $u\mapsto \Vert Au-f{\Vert }_Y^2$. Thereforeand $u$ is a minimizer. This is the direct method in the calculus of variations.

An element $u^{†}$ is a $J$-minimizing solution whenIt satisfies the source condition in variational regularization when there is a $w^{†}\in Y^{\ast }$ such thatwhere $\partial J(u^{†})$ is the subdifferential of the convex functional $J$.

Let ${\hat{u}}_{\alpha }$ minimize the nonsquared-residual objective. Comparison with $u^{†}$ givesThe source subgradient inequality givesConsequently,Thus forevery $0<\alpha <{\alpha }_0$ forces $A{\hat{u}}_{\alpha }=f$. The original comparison then gives $J({\hat{u}}_{\alpha })\le J(u^{†})$, so ${\hat{u}}_{\alpha }$ is itself $J$-minimizing. If $J$ is strictly convex, its restriction to the affine solution set has at most one minimizer, hence ${\hat{u}}_{\alpha }=u^{†}$. This is an exact penalty method.

For $p\in \partial J(v)$, the Bregman divergence isPut $r=A{\hat{u}}_{\alpha ,\delta }-f^{\delta }$ and $e=f^{\delta }-f$, so $\Vert e{\Vert }_Y\le \delta$. Comparison with $u^{†}$ givesUsing $p^{†}=A^{\ast }{w}^{†}$,For $0<\alpha <{\alpha }_0$ the last coefficient is negative, soThe estimate holds for any fixed admissible $\alpha$; it does not require $\alpha \to 0$ with the noise level.

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 $\boxed{C=(c+1)\Vert w^{†}{\Vert }_{Y^{\ast }}}$.