simpeg.regularization.CrossReferenceRegularization.f_m_deriv#

CrossReferenceRegularization.f_m_deriv(m)[source]#

Derivative of the regularization kernel function.

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

$$\frac{\partial {\mathbf{f}}_{\mathbf{m}}}{\partial \mathbf{m}}=\mathbf{X}$$

where $\mathbf{X}$ is a linear operator that carries out the cross-product with a reference vector model.

Parameters:

mnumpy.ndarray

The vector model.

Returns:

scipy.sparse.csr_matrix

The derivative of the regularization kernel function.

Notes

The objective function for cross reference regularization is given by:

$${\varphi }_m(\mathbf{m})=\Vert \mathbf{W}\mathbf{X}\mathbf{m}{\Vert }^2$$

where $\mathbf{m}$ are the discrete vector model parameters defined on the mesh (model), $\mathbf{X}$ carries out the cross-product with a reference vector model, and $\mathbf{W}$ is the weighting matrix. See the CrossReferenceRegularization class documentation for more detail.

We define the regularization kernel function ${\mathbf{f}}_{\mathbf{m}}$ as:

$${\mathbf{f}}_{\mathbf{m}}(\mathbf{m})=\mathbf{X}\mathbf{m}$$

such that

$${\varphi }_m(\mathbf{m})=\Vert \mathbf{W}{\mathbf{f}}_{\mathbf{m}}{\Vert }^2$$

Thus, the derivative with respect to the model is:

$$\frac{\partial {\mathbf{f}}_{\mathbf{m}}}{\partial \mathbf{m}}=\mathbf{X}$$