simpeg.regularization.PGIsmallness.f_m

PGIsmallness.f_m(m)

Evaluate the regularization kernel function.

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

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

where \(\mathbf{m}\) are the discrete model parameters, \(\mathbf{m}^\text{ref}\) is a reference model, and \(\mu\) is the mapping function. 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:

\[\phi_m (\mathbf{m}) = \left\lVert \mathbf{W} \left[ \mu(\mathbf{m}) - \mu(\mathbf{m}^\text{ref}) \right] \right\rVert^2\]

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

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

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

such that

\[\phi_m(\mathbf{m}) = \left\lVert \mathbf{W} \, \mathbf{f_m} \right\rVert^2\]