pylops.Gradient

class pylops.Gradient(dims, sampling=1.0, edge=False, kind='centered', dtype='float64', name='G')

Gradient.

Apply gradient operator to a multi-dimensional array.

Note

At least 2 dimensions are required, use pylops.FirstDerivative for 1d arrays.

Parameters:

dimstuple

Number of samples for each dimension.

samplingtuple or float, optional

Sampling steps for each direction. If a single float is provided, it is used for all directions.

edgebool, optional

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

kindstr, optional

Derivative kind (forward, centered, or backward).

dtypestr, optional

Type of elements in input array.

namestr, optional

Added in version 2.0.0.

Name of operator (to be used by pylops.utils.describe.describe)

Attributes:

dimstuple

Shape of the array after the adjoint, but before flattening.

For example, x_reshaped = (Op.H * y.ravel()).reshape(Op.dims).

dimsdtuple

Shape of the array after the forward, but before flattening.

For example, y_reshaped = (Op * x.ravel()).reshape(Op.dimsd).

shapetuple

Operator shape.

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.

Methods

__init__(dims[, sampling, edge, kind, ...])
adjoint()
apply_columns(cols)	Apply subset of columns of operator
cond([uselobpcg])	Condition number of linear operator.
conj()	Complex conjugate operator
div(y[, niter, densesolver])	Solve the linear problem \(\mathbf{y}=\mathbf{A}\mathbf{x}\).
dot(x)	Matrix-matrix or matrix-vector multiplication.
eigs([neigs, symmetric, niter, uselobpcg])	Most significant eigenvalues of linear operator.
matmat(X)	Matrix-matrix multiplication.
matvec(x)	Matrix-vector multiplication.
reset_count()	Reset counters
rmatmat(X)	Matrix-matrix multiplication.
rmatvec(x)	Adjoint matrix-vector multiplication.
todense([backend])	Return dense matrix.
toimag([forw, adj])	Imag operator
toreal([forw, adj])	Real operator
tosparse()	Return sparse matrix.
trace([neval, method, backend])	Trace of linear operator.
transpose()

Examples using pylops.Gradient

Derivatives

Derivatives