Variable Tikhonov Parameter In the Tikhonov regularization problem, $\Vert Ax-b\Vert^{2}+\Vert\Gamma x\Vert^{2}$, with
$\Gamma=\alpha I$ .The solution from SVD is $x=VDU^\top$, where $A=U\Sigma V^\top$
and $D_{ii}=\dfrac{\sigma_i}{\sigma_i^2+\alpha^2}$, with
$\sigma_i$ given by the singular values from $\Sigma$.
The question? Is there a similar solution (in terms of SVD) when $\Gamma=\alpha_{i}I$, $i=1,...,$ number of rows.
 A: Tikhonov regularization in its most general form is the solution of the problem
$$\min_{\mathbf x}\|\mathbf A\mathbf x-\mathbf b\|^2+\alpha^2\|\mathbf L\mathbf x\|^2$$
or the problem
$$\min_{\mathbf x}\left\|\begin{pmatrix}\mathbf A\\ \alpha \mathbf L\end{pmatrix}\mathbf x-\begin{pmatrix}\mathbf b\\ \mathbf 0\end{pmatrix}\right\|$$
Lars Elden gave a method for converting this general Tikhonov problem into an equivalent "standard form" regularization problem. I assume in the sequel that $\mathbf L$ is invertible; for singular $\mathbf L$ (and in fact for rectangular $\mathbf L$ as well), refer to Elden's paper for the required transformations (which involves the use of the QR decomposition).
In particular, if we let $\mathbf y=\mathbf L\mathbf x$, one can see that an equivalent standard Tikhonov problem is
$$\min_{\mathbf x}\|\mathbf A\mathbf L^{-1}\mathbf y-\mathbf b\|^2+\alpha^2\|\mathbf y\|^2$$
from which you can use the usual formulae for Tikhonov regularization to solve for $\mathbf y$. From this, one obtains the solution to the general problem as $\mathbf x=\mathbf L^{-1}\mathbf y$.
A: Observe that 
$\min_x\left\|b - Ax \right\|_2 + \left\|\Gamma x\right\|_2=
\min_{x}
\left\|
\left[
\begin{array}{c}
             b \\
             0
\end{array} 
\right]
- \left[\begin{array}{c}
        A \\
        \Gamma \end{array}\right]x\right\|_2$
which is your original problem reformulated as a least squares problem. $x$ is a solution of the minimization problem if and only if it satisfies the normal equations
$\left(A^TA + \Gamma^T \Gamma\right) x = A^T b$. This is true for any $\Gamma$. For $\Gamma$ of full column rank we have a unique solution
$x = \left(A^TA + \Gamma^T \Gamma\right)^{-1} A^Tb$. Using this formula you can invoke the SVD of $A$ and study the effect of your regularization matrix $\Gamma^T \Gamma$ for various structures of $\Gamma$.
