Landweber iteration with a Tikhonov initial value ($x_0=(A^{\ast }A+\alpha I{)}^{-1}{A}^{\ast }y,x_{n+1}=x_n+{\alpha }^{-1}{A}^{\ast }(y-A{x}_n)$)

Initialize Landweber iteration by Tikhonov regularization, then use step $1/\alpha$. Writing $B=A^{\ast }A$ and $R=I-B/\alpha$, the iterate is $x_n=R^n(B+\alpha I{)}^{-1}{A}^{\ast }y+{\alpha }^{-1}{\sum }_{j=0}^{n-1}{R}^j{A}^{\ast }y$. For nonzero $A$, the sufficient convergence condition is $\alpha >\Vert A{\Vert }^2/2$. This explicit method differs from Iterated Tikhonov regularization, whose update is $(B+\alpha I)x_{n+1}=\alpha x_n+A^{\ast }y$.

In a singular system of a compact operator, Landweber iteration with a Tikhonov initial value has coefficients $g_{\alpha ,n}({\sigma }_i)(y,u_i)$. Summing the finite geometric series gives the displayed filter. For fixed $n$, $g_{\alpha ,n}(\sigma )=(n+1)\sigma /\alpha +O({\sigma }^3)$ at zero, so each finite iterate is bounded. Under $\alpha >\Vert A{\Vert }^2/2$ the filter approaches $1/\sigma$ on every positive singular mode; noisy data require early stopping of Landweber iteration.