SimPEG.regularization.SparseSmoothness.f_m

SparseSmoothness.f_m(m)

Evaluate the regularization kernel function.

For first-order smoothness regularization in the x-direction, the regularization kernel function is given by:

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

where \(\mathbf{G_x}\) is the partial cell gradient operator along the x-direction (i.e. x-derivative), \(\mathbf{m}\) are the discrete model parameters defined on the mesh and \(\mathbf{m}^{(ref)}\) is the reference model (optional). Similarly for smoothness along y and z.

Parameters:

mnumpy.ndarray

The model.

Returns:

numpy.ndarray

The regularization kernel function.

Notes

The objective function for first-order smoothness regularization along the x-direction is given by:

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

where \(\mathbf{m}\) are the discrete model parameters (model), \(\mathbf{m}^{(ref)}\) is the reference model, \(\mathbf{G_x}\) is the partial cell gradient operator along the x-direction (i.e. x-derivative), and \(\mathbf{W}\) is the weighting matrix. Similar for smoothness along y and z. See the SmoothnessFirstOrder class documentation for more detail.

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

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

such that

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