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.
Solving two inverse problems with same solution
linear algebra;numerical analysis;inverse problem
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.  