 Iterated Tikhonov regularization

With fixed $\alpha >0$, iterated Tikhonov regularization repeatedly fits the data while penalizing displacement from the preceding estimate:For a bounded linear operator between Hilbert spaces, each step is uniquely defined and obeysEquivalently the correction solves $(A^{\ast }A+\alpha I)(x_{k+1}-x_k)=A^{\ast }(y-A{x}_k)$. 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.