pylops.signalprocessing.Fredholm1#

class pylops.signalprocessing.Fredholm1(*args, **kwargs)[source]#

Fredholm integral of first kind.

Implement a multi-dimensional Fredholm integral of first kind. Note that if the integral is two dimensional, this can be directly implemented using pylops.basicoperators.MatrixMult. A multi-dimensional Fredholm integral can be performed as a pylops.basicoperators.BlockDiag operator of a series of pylops.basicoperators.MatrixMult. However, here we take advantage of the structure of the kernel and perform it in a more efficient manner.

Parameters
Gnumpy.ndarray

Multi-dimensional convolution kernel of size \([n_{\text{slice}} \times n_x \times n_y]\)

nzint, optional

Additional dimension of model

saveGtbool, optional

Save G and G.H to speed up the computation of adjoint (True) or create G.H on-the-fly (False) Note that saveGt=True will double the amount of required memory

usematmulbool, optional

Use numpy.matmul (True) or for-loop with numpy.dot (False). As it is not possible to define which approach is more performant (this is highly dependent on the size of G and input arrays as well as the hardware used in the computation), we advise users to time both methods for their specific problem prior to making a choice.

dtypestr, optional

Type of elements in input array.

namestr, optional

New in version 2.0.0.

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

Notes

A multi-dimensional Fredholm integral of first kind can be expressed as

\[d(k, x, z) = \int{G(k, x, y) m(k, y, z) \,\mathrm{d}y} \quad \forall k=1,\ldots,n_{slice}\]

on the other hand its adjoin is expressed as

\[m(k, y, z) = \int{G^*(k, y, x) d(k, x, z) \,\mathrm{d}x} \quad \forall k=1,\ldots,n_{\text{slice}}\]

In discrete form, this operator can be seen as a block-diagonal matrix multiplication:

\[\begin{split}\begin{bmatrix} \mathbf{G}_{k=1} & \mathbf{0} & \ldots & \mathbf{0} \\ \mathbf{0} & \mathbf{G}_{k=2} & \ldots & \mathbf{0} \\ \vdots & \vdots & \ddots & \vdots \\ \mathbf{0} & \mathbf{0} & \ldots & \mathbf{G}_{k=N} \end{bmatrix} \begin{bmatrix} \mathbf{m}_{k=1} \\ \mathbf{m}_{k=2} \\ \vdots \\ \mathbf{m}_{k=N} \end{bmatrix}\end{split}\]
Attributes
shapetuple

Operator shape

explicitbool

Operator contains a matrix that can be solved explicitly (True) or not (False)

Methods

__init__(G[, nz, saveGt, usematmul, dtype, name])

Initialize this LinearOperator.

adjoint()

Hermitian 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()

Transpose this linear operator.