Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 326 / 2 / 1 / c / Solution

Starting from zero, the Landweber iteration isFor $K\ne 0$, take the step size $0<\tau <2/\Vert K{\Vert }^2$. Iterating the linear update givesand its Landweber spectral filter in a singular system of a compact operator isThe strict upper step size bound makes $|1-\tau {\sigma }_j^2|<1$ for every positive singular value. A common more restrictive choice is $0<\tau \le \Vert K{\Vert }^{-2}$, giving nonnegative damping factors. Each finite iterate is bounded and linear; early stopping of Landweber iteration controls the amplification of noise as smaller singular values are progressively inverted. If $K=0$, every iterate is zero for any positive step size.