pylops.SecondDirectionalDerivative

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

Second Directional derivative.

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

Note

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

Parameters

dimstuple

Number of samples for each dimension.

vnumpy.ndarray, optional

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

samplingtuple, optional

Sampling steps for each direction.

edgebool, optional

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

dtypestr, optional

Type of elements in input array.

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.

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.

Methods

__init__(dims, v[, sampling, edge, dtype, name])
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.SecondDirectionalDerivative

Derivatives

Derivatives