Alternatives to conjugate direction optimization for sparse solutions to geophysical problems

$L_1$ Formulation

To satisfy the $L_1$ optimization criteria for a data fitting problem, with data ${\bf d}$ of length ${N_D}$, and model ${\bf m}$ of size ${N_M}$, modeled by $\vert{\bf Lm - d}\vert = r$, we set up the following to specify the terms in (15) and (16):

$${\bf A} = \begin{bmatrix}{\bf I}_{2N_D} & - {\bf L} \\ {\bf I}_{2N_D} & - {\bf L} \end{bmatrix}$$ (17)

$${\bf b} = \begin{bmatrix}{\bf d} \ {\bf -d} \end{bmatrix}$$ (18)

$${\bf c}^{T} = \begin{bmatrix}0 & {\bf 1}_{2N_M} \end{bmatrix}$$ (19)

The initial model can be stored in ${\bf x}$, which will be updated. The final value of auxiliary variables ${\bf x_s}$ represents the status of the model regularization constraints (which can also be assigned an initial value).

$${\bf x} = {\bf m}$$ (20)

$${\bf x_s} = {\bf m}_{regularization}$$ (21)

In this format, the matrices can be fed directly into the GLPK function call in C or Octave.

Alternatives to conjugate direction optimization for sparse solutions to geophysical problems