SimPEG.regularization.SmoothnessSecondOrder.f_m_deriv

SmoothnessSecondOrder.f_m_deriv(m) → csr_matrix

Derivative of the regularization kernel function.

For second-order smoothness regularization, the derivative of the regularization kernel function with respect to the model is given by:

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

where \(\mathbf{L_x}\) is the second-order derivative operator with respect to x.

Parameters:

mnumpy.ndarray

The model.

Returns:

scipy.sparse.csr_matrix

The derivative of the regularization kernel function.

Notes

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

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

where \(\mathbf{m}\) are the discrete model parameters (model), \(\mathbf{m}^{(ref)}\) is the reference model, \(\mathbf{L_x}\) is the second-order x-derivative operator, and \(\mathbf{W}\) is the weighting matrix. Similar for smoothness along y and z. See the SmoothnessSecondOrder class documentation for more detail.

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

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

such that

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

The derivative of the regularization kernel function with respect to the model is:

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