Q:

Solving two inverse problems with same solution

I've got two inverse problems,
$$A_1 ~ x = b_1 \qquad A_2 ~ x = b_2$$
So far I've been solving them independently using Tikhonov Regularization and getting two estimates for $x$. However in my case $x$ represents the same solution in both equations. Is it possible to do a 'simultaneous' solve? Ideally I would be finding the answer for
$$\min \left( \lVert A_1 x - b_1 \rVert^2 + \lVert A_2 x - b_2 \rVert^2 + \lVert\Gamma x\lVert^2 \right)$$
Where $\Gamma = \alpha ~ I$ and $I$ is the identity matrix as in Tikhonov Regularization (aka ridge regression). I suppose I could just take the average of both solutions, wondering if there is a more statistically powerful way of approaching this however.

A:

You can write your problem as 
$\min \| Fm - g \|_{2}^{2}$
where
$F=\left[
\begin{array}{c}
A_{1} \\
A_{2} \\
\alpha I \\
\end{array}
\right]
$
and
$g=\left[
\begin{array}{c}
b_{1} \\
b_{2} \\
0 \\
\end{array}
\right].
$
You can use whatever linear least squares solver you want to solve this problem.  

