simpeg.regularization.SparseSmallness.f_m#

SparseSmallness.f_m(m)[source]#

Evaluate the regularization kernel function.

For smallness regularization, the regularization kernel function is given by:

$${\mathbf{f}}_{\mathbf{m}}(\mathbf{m})=\mathbf{m}-{\mathbf{m}}^{(ref)}$$

where $\mathbf{m}$ are the discrete model parameters and ${\mathbf{m}}^{(ref)}$ is a reference model. For a more detailed description, see the Notes section below.

Parameters:

mnumpy.ndarray

The model.

Returns:

numpy.ndarray

The regularization kernel function evaluated for the model provided.

Notes

The objective function for smallness regularization is given by:

$${\varphi }_m(\mathbf{m})=\Vert \mathbf{W}[\mathbf{m}-{\mathbf{m}}^{(ref)}]{\Vert }^2$$

where $\mathbf{m}$ are the discrete model parameters defined on the mesh (model), ${\mathbf{m}}^{(ref)}$ is the reference model, and $\mathbf{W}$ is the weighting matrix. See the Smallness class documentation for more detail.

We define the regularization kernel function ${\mathbf{f}}_{\mathbf{m}}$ as:

$${\mathbf{f}}_{\mathbf{m}}(\mathbf{m})=\mathbf{m}-{\mathbf{m}}^{(ref)}$$

such that

$${\varphi }_m(\mathbf{m})=\Vert \mathbf{W}{\mathbf{f}}_{\mathbf{m}}{\Vert }^2$$