class pylops.signalprocessing.Fredholm1(G, nz=1, saveGt=True, usematmul=True, dtype='float64')[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.

G : numpy.ndarray

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

nz : numpy.ndarray, optional

Additional dimension of model

saveGt : bool, 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

usematmul : bool, optional

Use numpy.matmul (True) or for-loop with (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 compution), we advise users to time both methods for their specific problem prior to making a choice.

dtype : str, optional

Type of elements in input array.


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

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

on the other hand its adjoin is expressed as

\[m(sl, y, z) = \int{G^*(sl, y, x) d(sl, x, z) dx} \quad \forall sl=1,n_{slice}\]

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

\[\begin{split}\begin{bmatrix} \mathbf{G}_{sl1} & \mathbf{0} & ... & \mathbf{0} \\ \mathbf{0} & \mathbf{G}_{sl2} & ... & \mathbf{0} \\ ... & ... & ... & ... \\ \mathbf{0} & \mathbf{0} & ... & \mathbf{G}_{slN} \end{bmatrix} \begin{bmatrix} \mathbf{m}_{sl1} \\ \mathbf{m}_{sl2} \\ ... \\ \mathbf{m}_{slN} \end{bmatrix}\end{split}\]
shape : tuple

Operator shape

explicit : bool

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


__init__(self, G[, nz, saveGt, usematmul, dtype]) Initialize this LinearOperator.
adjoint(self) Hermitian adjoint.
cond(self, \*\*kwargs_eig) Condition number of linear operator.
conj(self) Complex conjugate operator
div(self, y[, niter]) Solve the linear problem \(\mathbf{y}=\mathbf{A}\mathbf{x}\).
dot(self, x) Matrix-matrix or matrix-vector multiplication.
eigs(self[, neigs, symmetric, niter]) Most significant eigenvalues of linear operator.
matmat(self, X) Matrix-matrix multiplication.
matvec(self, x) Matrix-vector multiplication.
rmatvec(self, x) Adjoint matrix-vector multiplication.
transpose(self) Transpose this linear operator.