# SimPEG.regularization.Smallness.f_m#

Smallness.f_m(m) [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:
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

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