SimPEG.regularization.Smallness.f_m#
- Smallness.f_m(m) ndarray[source]#
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:
- m
numpy.ndarray The model.
- m
- Returns:
numpy.ndarrayThe regularization kernel function evaluated for the model provided.
Notes
The objective function for smallness regularization is given by:
\[\phi_m (\mathbf{m}) = \Big \| \mathbf{W} \big [ \mathbf{m} - \mathbf{m}^{(ref)} \big ] \Big \|^2\]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
Smallnessclass 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
\[\phi_m (\mathbf{m}) = \Big \| \mathbf{W} \, \mathbf{f_m} \Big \|^2\]