= Iterated Tikhonov regularization
{c}
With fixed $\alpha>0$, iterated Tikhonov regularization repeatedly fits the data while penalizing displacement from the preceding estimate:
$$
x_{k+1}=\operatorname*{argmin}_x\{\|Ax-y\|^2+\alpha\|x-x_k\|^2\}.
$$
For a <bounded linear operator> between <Hilbert spaces>, each step is uniquely defined and obeys
$$
(A^*A+\alpha I)x_{k+1}=A^*y+\alpha x_k.
$$
Equivalently the correction solves $(A^*A+\alpha I)(x_{k+1}-x_k)=A^*(y-Ax_k)$. This is an implicit residual correction; <Landweber iteration> instead multiplies the residual gradient by a scalar step.
Starting from zero, a <singular value> $\sigma>0$ has inverse coefficient $[1-(\alpha/(\alpha+\sigma^2))^k]/\sigma$. For each fixed mode it approaches $1/\sigma$, while finite iteration limits noise amplification. Holding the penalty parameter fixed therefore does not remove the need for a stopping rule on noisy data. With $\tau=1/\alpha$, this is the <iterated Tikhonov spectral filter>.
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