The <normal equation for a linear inverse problem> is
$$
A^*Ax=A^*y.
$$
<Landweber iteration> is the stationary iteration
$$
\boxed{
x_{n+1}=x_n+\gamma A^*(y-Ax_n)
=(I-\gamma A^*A)x_n+\gamma A^*y}.
$$
A sufficient step-size condition is
$$
\boxed{0<\gamma<\frac2{\lVert A\rVert^2}}.
$$
If $y\in\mathcal D(A^\dagger)$ and $x_0\in(\ker A)^\perp$, then
$$
\boxed{x_n\longrightarrow A^\dagger y}.
$$
For an arbitrary initial iterate, its null-space component is unchanged and the limit is
$$
A^\dagger y+P_{\ker A}x_0.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 335 / 3 / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-335/3/ii/solution