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It's a well-known result that the 2d tomography problem is weakly ill-posed (singular values decay as $O(1/\sqrt{n})$ even with full data and strongly ill-posed (the singular values decay exponentiall …

You want to minimize $\min \| Ax -y \|_{2}^{2} + x^{T}B^{T}Bx=\| Ax -y \|_{2}^{2} + \| Bx \|_{2}^{2}$ Recall that $\| u \|_{2}^{2} + \| v \|_{2}^{2}= \left\| \left[ \begin{array}{c} u \\ v \end{ar …

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 …

There's a large literature on "trans-dimensional" Markov Chain Monte Carlo methods in geophysics where the geophysical model consists of a number of layers and the inversion process adjusts the number …

Changing the relative weight of the two terms is equivalent to changing your prior. The fundamental issue here is that you've got a problem with far more parameters to estimate than data points. Y …

There has been a lot of research in this area over the last 20 years. It's appropriate to start by using search engines such as Google Scholar and Web of Science to look for survey and review article …

The short answer is that in general this problem is intractable (NP-Hard) with the $\| x \|_{0}$ regularization. Are you willing to consider minimizing $ f(x) + \lambda \| x \|_{1}$ instead? Th …