SimPEG.regularization.JointTotalVariation.deriv2

JointTotalVariation.deriv2(model, v=None)

Hessian of the regularization function evaluated for the model provided.

Where $\varphi (\mathbf{m})$ 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 ${\mathbf{m}}_{\mathbf{1}},{\mathbf{m}}_{\mathbf{2}},...$ such that:

$$\begin{array}{r}\mathbf{m}=\left[\begin{array}{c}{\mathbf{m}}_{\mathbf{1}}\\ {\mathbf{m}}_{\mathbf{2}}\\ ⋮\end{array}\right]\end{array}$$

The Hessian has the form:

$$\begin{array}{r}\frac{{\partial }^2\varphi }{\partial {\mathbf{m}}^2}=\left[\begin{array}{ccc}{\displaystyle \frac{\partial {\varphi }^2}{\partial {{\mathbf{m}}_{\mathbf{1}}}^2}}& {\displaystyle \frac{\partial {\varphi }^2}{\partial {\mathbf{m}}_{\mathbf{1}}\partial {\mathbf{m}}_{\mathbf{2}}}}& \cdots \\ {\displaystyle \frac{\partial {\varphi }^2}{\partial {\mathbf{m}}_{\mathbf{2}}\partial {\mathbf{m}}_{\mathbf{1}}}}& {\displaystyle \frac{\partial {\varphi }^2}{\partial {{\mathbf{m}}_{\mathbf{2}}}^2}}& \\ ⋮& & \ddots \end{array}\right]\end{array}$$

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

$$\frac{{\partial }^2\varphi }{\partial {\mathbf{m}}^2}\mathbf{v}$$

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.