SimPEG.regularization.CrossReferenceRegularization.f_m

CrossReferenceRegularization.f_m(m)

Evaluate the regularization kernel function.

For cross reference regularization, the regularization kernel function is given by:

\[\mathbf{f_m}(\mathbf{m}) = \mathbf{X m}\]

where \(\mathbf{m}\) are the discrete model parameters and \(\mathbf{X}\) carries out the cross-product with a reference vector model. For a more detailed description, see the Notes section below.

Parameters:

mnumpy.ndarray

The vector model.

Returns:

numpy.ndarray

The regularization kernel function evaluated for the model provided.

Notes

The objective function for cross reference regularization is given by:

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

where \(\mathbf{m}\) are the discrete vector model parameters defined on the mesh (model), \(\mathbf{X}\) carries out the cross-product with a reference vector model, and \(\mathbf{W}\) is the weighting matrix. See the CrossReferenceRegularization class documentation for more detail.

We define the regularization kernel function \(\mathbf{f_m}\) as:

\[\mathbf{f_m}(\mathbf{m}) = \mathbf{X m}\]

such that

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