pylops.Gradient¶
- pylops.Gradient(dims, sampling=1, edge=False, kind='centered', dtype='float64')[source]¶
Gradient.
Apply gradient operator to a multi-dimensional array.
Note
At least 2 dimensions are required, use
pylops.FirstDerivative
for 1d arrays.- Parameters
- dims
tuple
Number of samples for each dimension.
- sampling
tuple
, optional Sampling steps for each direction.
- edge
bool
, optional Use reduced order derivative at edges (
True
) or ignore them (False
).- kind
str
, optional Derivative kind (
forward
,centered
, orbackward
).- dtype
str
, optional Type of elements in input array.
- dims
- Returns
- l2op
pylops.LinearOperator
Gradient linear operator
- l2op
Notes
The Gradient operator applies a first-order derivative to each dimension of a multi-dimensional array in forward mode.
For simplicity, given a three dimensional array, the Gradient in forward mode using a centered stencil can be expressed as:
\[\mathbf{g}_{i, j, k} = (f_{i+1, j, k} - f_{i-1, j, k}) / d_1 \mathbf{i_1} + (f_{i, j+1, k} - f_{i, j-1, k}) / d_2 \mathbf{i_2} + (f_{i, j, k+1} - f_{i, j, k-1}) / d_3 \mathbf{i_3}\]which is discretized as follows:
\[\begin{split}\mathbf{g} = \begin{bmatrix} \mathbf{df_1} \\ \mathbf{df_2} \\ \mathbf{df_3} \end{bmatrix}\end{split}\]In adjoint mode, the adjoints of the first derivatives along different axes are instead summed together.