Let $(\sigma_j,u_j,v_j)$ be a singular system for $A$. The <Picard criterion> for $f\in\operatorname{dom}(A^\dagger)$ is
$$
\sum_j\frac{|\langle f,v_j\rangle|^2}{\sigma_j^2}<\infty
$$
after discarding the component in $(\operatorname{ran}A)^\perp$. For $0<\tau<\|A\|^{-2}$, the partial series acts diagonally:
$$
Q_Nf
=\sum_j
\frac{1-(1-\tau\sigma_j^2)^{N+1}}{\sigma_j}
\langle f,v_j\rangle u_j.
$$
For each $j$ the multiplier in the numerator tends to one and lies in $[0,1]$. The Picard summability condition therefore supplies an $\ell^2$ dominating sequence, so the <dominated convergence theorem> gives
$$
\boxed{Q_Nf\longrightarrow
\sum_j\frac{\langle f,v_j\rangle}{\sigma_j}u_j
=A^\dagger f}.
$$
This is the series form of <Landweber iteration>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 326 / 3 / e / Solution

{c}
{wiki}

Landweber iteration applies $u_{n+1}=u_n+\tau A^*(f-Au_n)$. Starting at zero,
$$
u_N=\tau\sum_{n=0}^{N-1}(I-\tau A^*A)^nA^*f,
$$
and early stopping regularizes the inverse problem.


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