Take a minimizing sequence for the <variational regularization> functional
$$
F(u)=\frac12\|Au-f\|_Y^2+\alpha J(u).
$$
Its <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 $u\mapsto\|Au-f\|_Y^2$. Therefore
$$
F(u)\leq\liminf_nF(u_n)=\inf_XF,
$$
and $u$ is a minimizer. This is the <direct method in the calculus of variations>.

Solution

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

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Variational regularization balances data fidelity against a lower-semicontinuous penalty, for example by minimizing $\frac12\lVert Au-f\rVert^2+\alpha J(u)$.


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