from __future__ import division
import logging
import numpy as np
from scipy.linalg import solve, lstsq
from scipy.sparse.linalg import LinearOperator as spLinearOperator
from scipy.sparse.linalg import spsolve, lsqr
from scipy.linalg import eigvals
from scipy.sparse.linalg import eigs as sp_eigs
from scipy.sparse.linalg import eigsh as sp_eigsh
from scipy.sparse.linalg import lobpcg as sp_lobpcg
from scipy.sparse import csr_matrix
logging.basicConfig(format='%(levelname)s: %(message)s', level=logging.WARNING)
[docs]class LinearOperator(spLinearOperator):
"""Common interface for performing matrix-vector products.
This class is an overload of the
:py:class:`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 :ref:`addingoperator`.
Parameters
----------
Op : :obj:`scipy.sparse.linalg.LinearOperator` or :obj:`scipy.sparse.linalg._ProductLinearOperator` or :obj:`scipy.sparse.linalg._SumLinearOperator`
Operator
explicit : :obj:`bool`
Operator contains a matrix that can be solved explicitly
(``True``) or not (``False``)
"""
def __init__(self, Op=None, explicit=False):
self.explicit = explicit
if Op is not None:
self.Op = Op
self.shape = self.Op.shape
self.dtype = self.Op.dtype
def _matvec(self, x):
if callable(self.Op._matvec):
return self.Op._matvec(x)
def _rmatvec(self, x):
if callable(self.Op._rmatvec):
return self.Op._rmatvec(x)
def _matmat(self, X):
"""Matrix-matrix multiplication handler.
Modified version of scipy _matmat to avoid having trailing dimension
in col when provided to matvec
"""
return np.vstack([self.matvec(col.reshape(-1)) for col in X.T]).T
def __mul__(self, x):
y = super().__mul__(x)
if isinstance(y, spLinearOperator):
y = aslinearoperator(y)
return y
def __rmul__(self, x):
return aslinearoperator(super().__rmul__(x))
def __pow__(self, p):
return aslinearoperator(super().__pow__(p))
def __add__(self, x):
return aslinearoperator(super().__add__(x))
def __neg__(self):
return aslinearoperator(super().__neg__())
def __sub__(self, x):
return aslinearoperator(super().__sub__(x))
def _adjoint(self):
return aslinearoperator(super()._adjoint())
[docs] def div(self, y, niter=100):
r"""Solve the linear problem :math:`\mathbf{y}=\mathbf{A}\mathbf{x}`.
Overloading of operator ``/`` to improve expressivity of `Pylops`
when solving inverse problems.
Parameters
----------
y : :obj:`np.ndarray`
Data
niter : :obj:`int`, optional
Number of iterations (to be used only when ``explicit=False``)
Returns
-------
xest : :obj:`np.ndarray`
Estimated model
"""
xest = self.__truediv__(y, niter=niter)
return xest
def __truediv__(self, y, niter=100):
if self.explicit is True:
if isinstance(self.A, np.ndarray):
if self.A.shape[0] == self.A.shape[1]:
xest = solve(self.A, y)
else:
xest = lstsq(self.A, y)[0]
else:
xest = spsolve(self.A, y)
else:
xest = lsqr(self, y, iter_lim=niter)[0]
return xest
[docs] def todense(self):
r"""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 : :obj:`numpy.ndarray`
Dense matrix.
"""
# Wrap self into a LinearOperator. This is done for cases where self
# is a _SumLinearOperator or _ProductLinearOperator, so that it regains
# the dense method
Op = aslinearoperator(self)
identity = np.eye(self.shape[1], dtype=self.dtype)
matrix = Op.matmat(identity)
return matrix
[docs] def tosparse(self):
r"""Return sparse matrix.
The operator in converted into its sparse (CSR) matrix equivalent. In order
to do so, the operator is applied to series of unit vectors with length equal
to the number of coloumns in the original operator.
Returns
-------
matrix : :obj:`scipy.sparse.csr_matrix`
Sparse matrix.
"""
Op = aslinearoperator(self)
(_, n) = self.shape
# stores non-zero data for the sparse matrix creation
entries = []
indices = []
# loop through columns of self
for i in range(n):
# make a unit vector for the ith column
unit_i = np.zeros(n)
unit_i[i] = 1
# multiply unit vector to self and find the non-zeros
res_i = Op * unit_i
rows_nz = np.where(res_i != 0)[0]
# append the non-zero values and indices to the lists
for j in rows_nz:
indices.append([i, j])
entries_i = res_i[rows_nz]
for e in entries_i:
entries.append(e)
# post process the entries / indices for scipy.sparse.csr_matrix
entries = np.array(entries)
indices = np.array(indices)
i, j = indices[:,0], indices[:,1]
# construct a sparse, CSR matrix from the entries / indices data.
matrix = csr_matrix((entries, (j, i)), shape=self.shape, dtype=self.dtype)
return matrix
[docs] def eigs(self, neigs=None, symmetric=False, niter=None,
uselobpcg=False, **kwargs_eig):
r"""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 :math:`\mathbf{A^H}\mathbf{A}`
and the square-root values are returned.
Parameters
----------
neigs : :obj:`int`
Number of eigenvalues to compute (if ``None``, return all). Note
that for ``explicit=False``, only :math:`N-1` eigenvalues can be
computed where :math:`N` is the size of the operator in the
model space
symmetric : :obj:`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 : :obj:`int`, optional
Number of iterations for eigenvalue estimation
uselobpcg : :obj:`bool`, optional
Use :func:`scipy.sparse.linalg.lobpcg`
**kwargs_eig
Arbitrary keyword arguments for :func:`scipy.sparse.linalg.eigs`,
:func:`scipy.sparse.linalg.eigsh`, or
:func:`scipy.sparse.linalg.lobpcg`
Returns
-------
eigenvalues : :obj:`numpy.ndarray`
Operator eigenvalues.
Raises
-------
ValueError
If ``uselobpcg=True`` for a non-symmetric square matrix with
complex type
Notes
-----
Depending on the size of the operator, whether it is explicit or not
and the number of eigenvalues requested, different algorithms are
used by this routine.
More precisely, when only a limited number of eigenvalues is requested
the :func:`scipy.sparse.linalg.eigsh` method is used in case of
``symmetric=True`` and the :func:`scipy.sparse.linalg.eigs` method
is used ``symmetric=False``. However, when the matrix is represented
explicitly within the linear operator (``explicit=True``) and all the
eigenvalues are requested the :func:`scipy.linalg.eigvals` is used
instead.
Finally, when only a limited number of eigenvalues is required,
it is also possible to explicitly choose to use the
``scipy.sparse.linalg.lobpcg`` method via the ``uselobpcg`` input
parameter flag.
Most of these algorithms are 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/
"""
if self.explicit and isinstance(self.A, np.ndarray):
if self.shape[0] == self.shape[1]:
if neigs is None or neigs == self.shape[1]:
eigenvalues = eigvals(self.A)
else:
if not symmetric and np.iscomplexobj(self) and uselobpcg:
raise ValueError('cannot use scipy.sparse.linalg.lobpcg '
'for non-symmetric square matrices of '
'complex type...')
if symmetric and uselobpcg:
X = np.random.rand(self.shape[0], neigs)
eigenvalues = \
sp_lobpcg(self.A, X=X, maxiter=niter,
**kwargs_eig)[0]
elif symmetric:
eigenvalues = sp_eigsh(self.A, k=neigs, maxiter=niter,
**kwargs_eig)[0]
else:
eigenvalues = sp_eigs(self.A, k=neigs, maxiter=niter,
**kwargs_eig)[0]
else:
if neigs is None or neigs == self.shape[1]:
eigenvalues = np.sqrt(eigvals(np.dot(np.conj(self.A.T),
self.A)))
else:
if uselobpcg:
X = np.random.rand(self.shape[1], neigs)
eigenvalues = np.sqrt(sp_lobpcg(
np.dot(np.conj(self.A.T), self.A),
X=X, maxiter=niter, **kwargs_eig)[0])
else:
eigenvalues = np.sqrt(sp_eigsh(
np.dot(np.conj(self.A.T), self.A),
k=neigs, maxiter=niter, **kwargs_eig)[0])
else:
if neigs is None or neigs >= self.shape[1]:
neigs = self.shape[1] - 2
if self.shape[0] == self.shape[1]:
if not symmetric and np.iscomplexobj(self) and uselobpcg:
raise ValueError('cannot use scipy.sparse.linalg.lobpcg for '
'non symmetric square matrices of '
'complex type...')
if symmetric and uselobpcg:
X = np.random.rand(self.shape[0], neigs)
eigenvalues = \
sp_lobpcg(self, X=X, maxiter=niter, **kwargs_eig)[0]
elif symmetric:
eigenvalues = sp_eigsh(self, k=neigs, maxiter=niter,
**kwargs_eig)[0]
else:
eigenvalues = sp_eigs(self, k=neigs, maxiter=niter,
**kwargs_eig)[0]
else:
if uselobpcg:
X = np.random.rand(self.shape[1], neigs)
eigenvalues = np.sqrt(sp_lobpcg(self.H * self, X=X,
maxiter=niter,
**kwargs_eig)[0])
else:
eigenvalues = np.sqrt(sp_eigs(self.H * self, k=neigs,
maxiter=niter,
**kwargs_eig)[0])
return -np.sort(-eigenvalues)
[docs] def cond(self, uselobpcg=False, **kwargs_eig):
r"""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
----------
uselobpcg : :obj:`bool`, optional
Use :func:`scipy.sparse.linalg.lobpcg` to compute eigenvalues
**kwargs_eig
Arbitrary keyword arguments for :func:`scipy.sparse.linalg.eigs`,
:func:`scipy.sparse.linalg.eigsh`, or
:func:`scipy.sparse.linalg.lobpcg`
Returns
-------
eigenvalues : :obj:`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:
.. math::
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 :math:`A` will
change with respect to a change in the data :math:`y`.
Thus, if the condition number is large, even a small error in :math:`y`
may cause a large error in :math:`x`. On the other hand, if the
condition number is small then the error in :math:`x` is not much
bigger than the error in :math:`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*.
"""
if not uselobpcg:
cond = np.asscalar(self.eigs(neigs=1, which='LM', **kwargs_eig))/ \
np.asscalar(self.eigs(neigs=1, which='SM', **kwargs_eig))
else:
print('here')
cond = np.asscalar(self.eigs(neigs=1, uselobpcg=True, largest=True,
**kwargs_eig)) / \
np.asscalar(self.eigs(neigs=1, uselobpcg=True, largest=False,
**kwargs_eig))
return cond
[docs] def conj(self):
"""Complex conjugate operator
Returns
-------
conjop : :obj:`pylops.LinearOperator`
Complex conjugate operator
"""
conjop = _ConjLinearOperator(self)
return conjop
[docs] def apply_columns(self, cols):
"""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 : :obj:`list`
Columns to be selected
Returns
-------
colop : :obj:`pylops.LinearOperator`
Apply column operator
"""
colop = _ColumnLinearOperator(self, cols)
return colop
class _ConjLinearOperator(LinearOperator):
"""Complex conjugate linear operator
"""
def __init__(self, Op):
if not isinstance(Op, spLinearOperator):
raise TypeError('Op must be a LinearOperator')
super(_ConjLinearOperator, self).__init__(Op, Op.shape)
self.Op = Op
def _matvec(self, x):
return (self.Op._matvec(x.conj())).conj()
def _rmatvec(self, x):
return (self.Op._rmatvec(x.conj())).conj()
def _adjoint(self):
return _ConjLinearOperator(self.Op.H)
class _ColumnLinearOperator(LinearOperator):
"""Column selector linear operator
Produces the forward and adjoint passes with a subset of columns of the
original operator
"""
def __init__(self, Op, cols):
if not isinstance(Op, spLinearOperator):
raise TypeError('Op must be a LinearOperator')
super(_ColumnLinearOperator, self).__init__(Op, Op.explicit)
self.Op = Op
self.cols = cols
self.shape = (Op.shape[0], len(cols))
if self.explicit:
self.Opcol = Op.A[:, cols]
def _matvec(self, x):
if self.explicit:
y = self.Opcol @ x
else:
y = np.zeros(self.Op.shape[1], dtype=self.dtype)
y[self.cols] = x
y = self.Op._matvec(y)
return y
def _rmatvec(self, x):
if self.explicit:
y = self.Opcol.T @ x
else:
y = self.Op._rmatvec(x)
y = y[self.cols]
return y
def aslinearoperator(Op):
"""Return Op as a LinearOperator.
Converts any operator into a LinearOperator. This can be used when `Op`
is a private operator to ensure that the return operator has all properties
and methods of the parent class.
Parameters
----------
Op : :obj:`pylops.LinearOperator` or any other Operator
Operator of any type
Returns
-------
Op : :obj:`pylops.LinearOperator`
Operator of type :obj:`pylops.LinearOperator`
"""
if isinstance(Op, LinearOperator):
return Op
else:
return LinearOperator(Op)