SimPEG.regularization.PGIsmallness.f_m_deriv

PGIsmallness.f_m_deriv(m) → csr_matrix

Derivative of the regularization kernel function.

For Smallness regularization, the derivative of the regularization kernel function with respect to the model is given by:

\[\frac{\partial \mathbf{f_m}}{\partial \mathbf{m}} = \mathbf{I}\]

where \(\mathbf{I}\) is the identity matrix.

Parameters:

mnumpy.ndarray

The model.

Returns:

scipy.sparse.csr_matrix

The derivative of the regularization kernel function.

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

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

Thus, the derivative with respect to the model is:

\[\frac{\partial \mathbf{f_m}}{\partial \mathbf{m}} = \mathbf{I}\]

where \(\mathbf{I}\) is the identity matrix.