pylops.SecondDirectionalDerivative

pylops.SecondDirectionalDerivative(dims, v, sampling=1, edge=False, dtype='float64')

Second Directional derivative.

Apply a second directional derivative operator to a multi-dimensional array along either a single common direction or different directions for each point of the array.

Note

At least 2 dimensions are required, consider using pylops.SecondDerivative for 1d arrays.

Parameters:

dims : tuple

Number of samples for each dimension.

v : np.ndarray, optional

Single direction (array of size \(n_\text{dims}\)) or group of directions (array of size \([n_\text{dims} \times n_{d_0} \times ... \times n_{d_{n_\text{dims}}}]\))

sampling : tuple, optional

Sampling steps for each direction.

edge : bool, optional

Use reduced order derivative at edges (True) or ignore them (False).

dtype : str, optional

Type of elements in input array.

Returns:

ddop : pylops.LinearOperator

Second directional derivative linear operator

Notes

The SecondDirectionalDerivative applies a second-order derivative to a multi-dimensional array along the direction defined by the unitary vector \(\mathbf{v}\):

\[d^2f_\mathbf{v} = - D_\mathbf{v}^T [D_\mathbf{v} f]\]

where \(D_\mathbf{v}\) is the first-order directional derivative implemented by pylops.SecondDirectionalDerivative.

This operator is sometimes also referred to as directional Laplacian in the literature.

Examples using pylops.SecondDirectionalDerivative

Derivatives