simpeg.regularization.BaseSparse.get_lp_weights#

BaseSparse.get_lp_weights(f_m)[source]#

Compute and return iteratively re-weighted least-squares (IRLS) weights.

For a regularization kernel function ${\mathbf{f}}_{\mathbf{m}}(\mathbf{m})$ evaluated at model $\mathbf{m}$, compute and return the IRLS weights. See Smallness.f_m() and SmoothnessFirstOrder.f_m() for examples of least-squares regularization kernels.

For SparseSmallness, f_m is a (n_cells, ) numpy.ndarray. For SparseSmoothness, f_m is a numpy.ndarray whose length corresponds to the number of faces along a particular orientation; e.g. for smoothness along x, the length is (n_faces_x, ).

Parameters:

f_mnumpy.ndarray

The regularization kernel function evaluated at the current model.

Notes

For a regularization kernel function ${\mathbf{f}}_{\mathbf{m}}$ evaluated at model $\mathbf{m}$, the IRLS weights are computed via:

$${\mathbf{w}}_{\mathbf{r}}=\mathit{\lambda }\oslash [{{\mathbf{f}}_{\mathbf{m}}}^2+{ϵ}^2{]}^{1-\mathbf{p}/2}$$

where $\oslash$ represents elementwise division, $ϵ$ is a small constant added for stability of the algorithm (set using the irls_threshold property), and $\mathbf{p}$ defines the norm at each cell.

$\mathit{\lambda }$ applies optional scaling to the IRLS weights (when the irls_scaled property is True). The scaling acts to preserve the balance between the data misfit and the components of the regularization based on the derivative of the l2-norm measure. And it assists the convergence by ensuring the model does not deviate aggressively from the global 2-norm solution during the first few IRLS iterations.

To apply elementwise scaling, let

$$f_{max}=\Vert {\mathbf{f}}_{\mathbf{m}}{\Vert }_{\mathrm{\infty }}$$

And define a vector array ${\stackrel{\mathbf{~}}{\mathbf{f}}}_{\mathbf{max}}$ such that:

$$\begin{array}{r}{\tilde{f}}_{i,max}=\{\begin{array}{l}{f}_{max}for{p}_i\ge 1\\ \frac{ϵ}{\sqrt{1-p_i}}for{p}_i<1\end{array}\end{array}$$

The elementwise scaling vector $\mathit{\lambda }$ is:

$$\text{Extra close brace or missing open brace}$$