SimPEG.regularization.PGIsmallness.f_m

PGIsmallness.f_m(m) → ndarray

Evaluate the regularization kernel function.

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

\[\mathbf{f_m}(\mathbf{m}) = \mathbf{m} - \mathbf{m}^{(ref)}\]

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:

\[\phi_m (\mathbf{m}) = \frac{1}{2} \Big \| \mathbf{W} \big [ \mathbf{m} - \mathbf{m}^{(ref)} \big ] \Big \|^2\]

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 \(\mathbf{f_m}\) as:

\[\mathbf{f_m}(\mathbf{m}) = \mathbf{m} - \mathbf{m}^{(ref)}\]

such that

\[\phi_m (\mathbf{m}) = \frac{1}{2} \Big \| \mathbf{W} \, \mathbf{f_m} \Big \|^2\]