Model-space regularization

Model-space regularization implies adding equations to system  

$$\bold{L m \approx d}$$ (15)

to obtain a fully constrained (well-posed) inverse problem. The additional equations take the form  

$$\epsilon \bold{D m \approx 0} \;.$$ (16)

The full system of equations ()-() can be written in a short notation as  

$$\bold{G_m m} = \left[\begin{array} {c} \bold{L} \\ \epsilon... ... \bold{d} \\ \bold{0} \end{array}\right] = \hat{\bold{d}}\;,$$ (17)

where $\hat{\bold{d}}$ is the effective data vector:  

$$\hat{\bold{d}} = \left[\begin{array} {c} \bold{d} \\ \bold{0} \end{array}\right]\;,$$ (18)

and $\bold{G_m}$ is a column operator:  

$$\bold{G_m} = \left[\begin{array} {c} \bold{L} \\ \epsilon \bold{D} \end{array}\right]\;.$$ (19)

The estimation problem () is fully constrained. We can solve it by means of unconstrained least-squares optimization, minimizing the squared power $\hat{\bold{r}}^T \hat{\bold{r}}$ of the compound residual vector  

$$\hat{\bold{r}} = \hat{\bold{d}} - \bold{G_m m} = \left[\begi... ... \bold{d - L m}\\ - \epsilon \bold{D m} \end{array}\right]\;.$$ (20)

The formal solution of the regularized optimization problem has a known form, which coincides with formula (). One can carry out the optimization iteratively with the help of the conjugate-gradient method Hestenes and Steifel (1952) or its analogs Paige and Saunders (1982).

The next subsection introduces an alternative formulation of the optimization problem.