pylops.signalprocessing.Fredholm1

class pylops.signalprocessing.Fredholm1(*args, **kwargs)

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.