simpeg.regularization.AmplitudeSmoothnessFirstOrder.update_weights#

AmplitudeSmoothnessFirstOrder.update_weights(m)[source]#

Update the IRLS weights for sparse smoothness regularization.

Parameters:

mnumpy.ndarray

The model used to update the IRLS weights.

Notes

Let us consider the IRLS weights for sparse smoothness along the x-direction. When the class property gradient_type`=’components’`, IRLS weights are computed using the regularization kernel function and we define:

$${\mathbf{f}}_{\mathbf{m}}={\mathbf{G}}_{\mathbf{x}}[\mathbf{m}-{\mathbf{m}}^{(ref)}]$$

where $\mathbf{m}$ is the model provided, ${\mathbf{G}}_{\mathbf{x}}$ is the partial cell gradient operator along x (i.e. x-derivative), and ${\mathbf{m}}^{(ref)}$ is a reference model (optional, activated using reference_model_in_smooth). See SmoothnessFirstOrder.f_m() for a more comprehensive definition of the regularization kernel function.

However, when the class property gradient_type`=’total’`, IRLS weights are computed using the magnitude of the total gradient and we define:

$${\mathbf{f}}_{\mathbf{m}}={\mathbf{A}}_{\mathbf{cx}}\sum _{j=x,y,z}|{\mathbf{A}}_{\mathbf{j}}{\mathbf{G}}_{\mathbf{j}}[\mathbf{m}-{\mathbf{m}}^{(ref)}]|$$

where ${\mathbf{A}}_{\mathbf{j}}$ for $j=x,y,z$ averages the partial gradients from their respective faces to cell centers, and ${\mathbf{A}}_{\mathbf{cx}}$ averages the sum of the absolute values back to the appropriate faces.

Once ${\mathbf{f}}_{\mathbf{m}}$ is obtained, 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 for each element (set using the norm property).

$\mathit{\lambda }$ 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 the 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 scaling vector $\mathit{\lambda }$ is:

$$\mathit{\lambda }=[\frac{f_{max}}{{\stackrel{\mathbf{~}}{\mathbf{f}}}_{\mathbf{max}}}]\odot [{{\stackrel{\mathbf{~}}{\mathbf{f}}}_{\mathbf{max}}}^2+{ϵ}^2{]}^{1-\mathbf{p}/2}$$