Solving two inverse problems with same solution

Question

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.

Answer 1

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.