simpeg.regularization.JointTotalVariation.deriv2

JointTotalVariation.deriv2(model, v=None)

Hessian of the regularization function evaluated for the model provided.

Where \(\phi (\mathbf{m})\) is the discrete regularization function (objective function), this method evalutate and returns the second derivative (Hessian) with respect to the model parameters. For a model \(\mathbf{m}\) consisting of multiple physical properties \(\mathbf{m_1}, \; \mathbf{m_2}, \; ...\) such that:

\[\begin{split}\mathbf{m} = \begin{bmatrix} \mathbf{m_1} \\ \mathbf{m_2} \\ \vdots \end{bmatrix}\end{split}\]

The Hessian has the form:

\[\begin{split}\frac{\partial^2 \phi}{\partial \mathbf{m}^2} = \begin{bmatrix} \dfrac{\partial^2 \phi}{\partial \mathbf{m_1}^2} & \dfrac{\partial^2 \phi}{\partial \mathbf{m_1} \partial \mathbf{m_2}} & \cdots \\ \dfrac{\partial^2 \phi}{\partial \mathbf{m_2} \partial \mathbf{m_1}} & \dfrac{\partial^2 \phi}{\partial \mathbf{m_2}^2} & \; \\ \vdots & \; & \ddots \end{bmatrix}\end{split}\]

When a vector \((\mathbf{v})\) is supplied, the method returns the Hessian times the vector:

\[\frac{\partial^2 \phi}{\partial \mathbf{m}^2} \, \mathbf{v}\]

Parameters:

model(n_param, ) numpy.ndarray

The model; a vector array containing all physical properties.

vnumpy.ndarray, optional

An array to multiply the Hessian by.

Returns:

numpy.ndarray or scipy.sparse.csr_matrix

Hessian of the regularization function evaluated for the model provided. The Hessian of joint total variation with respect to the model times a vector or the full Hessian if v is None.