pylops.LinearOperator

class pylops.LinearOperator(Op=None, explicit=False)[source]

Common interface for performing matrix-vector products.

This class is an overload of the scipy.sparse.linalg.LinearOperator class. It adds functionalities by overloading standard operators such as __truediv__ as well as creating convenience methods such as eigs, cond, and conj.

Note

End users of PyLops should not use this class directly but simply use operators that are already implemented. This class is meant for developers and it has to be used as the parent class of any new operator developed within PyLops. Find more details regarding implementation of new operators at Implementing new operators.

Parameters:
Op : scipy.sparse.linalg.LinearOperator or scipy.sparse.linalg._ProductLinearOperator or scipy.sparse.linalg._SumLinearOperator

Operator

explicit : bool

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

Methods

__init__(self[, Op, explicit]) Initialize this LinearOperator.
adjoint(self) Hermitian adjoint.
apply_columns(self, cols) Apply subset of columns of operator
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.
rmatmat(self, X) Adjoint matrix-matrix multiplication.
rmatvec(self, x) Adjoint matrix-vector multiplication.
todense(self) Return dense matrix.
transpose(self) Transpose this linear operator.
div(self, y, niter=100)[source]

Solve the linear problem \(\mathbf{y}=\mathbf{A}\mathbf{x}\).

Overloading of operator / to improve expressivity of Pylops when solving inverse problems.

Parameters:
y : np.ndarray

Data

niter : int, optional

Number of iterations (to be used only when explicit=False)
Returns:
xest : np.ndarray

Estimated model
todense(self)[source]

Return dense matrix.

The operator in converted into its dense matrix equivalent. In order to do so, the operator is applied to an identity matrix whose number of rows and columns is equivalent to the number of columns of the operator. Note that this operation may be costly for very large operators and it is only suggest it to use as a way to inspect the structure of the matricial equivalent of the operator.

Returns:
matrix : numpy.ndarray

Dense matrix.
eigs(self, neigs=None, symmetric=False, niter=None, **kwargs_eig)[source]

Most significant eigenvalues of linear operator.

Return an estimate of the most significant eigenvalues of the linear operator. If the operator has rectangular shape (shape[0]!=shape[1]), eigenvalues are first computed for the square operator \(\mathbf{A^H}\mathbf{A}\) and the square-root values are returned.

Parameters:
neigs : int

Number of eigenvalues to compute (if None, return all). Note that for explicit=False, only \(N-1\) eigenvalues can be computed where \(N\) is the size of the operator in the model space

symmetric : bool, optional

Operator is symmetric (True) or not (False). User should set this parameter to True only when it is guaranteed that the operator is real-symmetric or complex-hermitian matrices

niter : int, optional

Number of iterations for eigenvalue estimation

**kwargs_eig

Arbitrary keyword arguments for scipy.sparse.linalg.eigs or scipy.sparse.linalg.eigsh
Returns:
eigenvalues : numpy.ndarray

Operator eigenvalues.

Notes

Eigenvalues are estimated using scipy.sparse.linalg.eigs (explicit=True) or scipy.sparse.linalg.eigsh (explicit=False).

This is a port of ARPACK [1], a Fortran package which provides routines for quickly finding eigenvalues/eigenvectors of a matrix. As ARPACK requires only left-multiplication by the matrix in question, eigenvalues/eigenvectors can also be estimated for linear operators when the dense matrix is not available.

[1]http://www.caam.rice.edu/software/ARPACK/
cond(self, **kwargs_eig)[source]

Condition number of linear operator.

Return an estimate of the condition number of the linear operator as the ratio of the largest and lowest estimated eigenvalues.

Parameters:
**kwargs_eig

Arbitrary keyword arguments for scipy.sparse.linalg.eigs or scipy.sparse.linalg.eigsh
Returns:
eigenvalues : numpy.ndarray

Operator eigenvalues.

Notes

The condition number of a matrix (or linear operator) can be estimated as the ratio of the largest and lowest estimated eigenvalues:

\[k= \frac{\lambda_{max}}{\lambda_{min}}\]

The condition number provides an indication of the rate at which the solution of the inversion of the linear operator \(A\) will change with respect to a change in the data \(y\).

Thus, if the condition number is large, even a small error in \(y\) may cause a large error in \(x\). On the other hand, if the condition number is small then the error in \(x\) is not much bigger than the error in \(y\). A problem with a low condition number is said to be well-conditioned, while a problem with a high condition number is said to be ill-conditioned.

conj(self)[source]

Complex conjugate operator

Returns:
conjop : pylops.LinearOperator

Complex conjugate operator
apply_columns(self, cols)[source]

Apply subset of columns of operator

This method can be used to wrap a LinearOperator and mimic the action of a subset of columns of the operator on a reduced model in forward mode, and retrieve only the result of a subset of rows in adjoint mode.

Note that unless the operator has explicit=True, this is not optimal as the entire forward and adjoint passes of the original operator will have to be perfomed. It can however be useful for the implementation of solvers such as Orthogonal Matching Pursuit (OMP) that iteratively build a solution by evaluate only a subset of the columns of the operator.

Parameters:
cols : list

Columns to be selected
Returns:
colop : pylops.LinearOperator

Apply column operator