simpeg.regularization.CrossGradient.deriv2#

CrossGradient.deriv2(model, v=None)[source]#

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 two physical properties 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:

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

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.

vNone, (n_param, ) numpy.ndarray (optional)

A numpy array to model the Hessian by.

Returns:

(n_param, n_param) scipy.sparse.csr_matrix | (n_param, ) numpy.ndarray

If the input argument v is None, the Hessian for the models provided is returned. If v is not None, the Hessian multiplied by the vector provided is returned.