simpeg.regularization.SmoothnessSecondOrder.deriv2

SmoothnessSecondOrder.deriv2(m, 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 returns the second-derivative (Hessian) with respect to the model parameters:

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

or the second-derivative (Hessian) multiplied by a vector \((\mathbf{v})\):

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

Parameters:

m(n_param, ) numpy.ndarray

The model for which the Hessian is evaluated.

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

A vector.

Returns:

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

If the input argument v is None, the Hessian of the regularization function for the model provided is returned. If v is not None, the Hessian multiplied by the vector provided is returned.