simpeg.regularization.SparseSmallness.f_m_deriv#

SparseSmallness.f_m_deriv(m)[source]#

Derivative of the regularization kernel function.

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

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

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:

$${\varphi }_m(\mathbf{m})=\Vert \mathbf{W}[\mathbf{m}-{\mathbf{m}}^{(ref)}]{\Vert }^2$$

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 ${\mathbf{f}}_{\mathbf{m}}$ as:

$${\mathbf{f}}_{\mathbf{m}}(\mathbf{m})=\mathbf{m}-{\mathbf{m}}^{(ref)}$$

such that

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

Thus, the derivative with respect to the model is:

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

where $\mathbf{I}$ is the identity matrix.