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 \([n_{\text{slice}} \times n_x \times n_y]\)

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

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

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:

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

Operator shape

explicit : bool

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


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