# Source code for pymor.operators.constructions

# This file is part of the pyMOR project (http://www.pymor.org).

"""Module containing some constructions to obtain new operators from old ones."""

from functools import reduce
from numbers import Number

import numpy as np
import scipy.linalg as spla

from pymor.core.defaults import defaults
from pymor.core.exceptions import InversionError
from pymor.operators.interface import Operator
from pymor.parameters.base import ParametricObject
from pymor.parameters.functionals import ParameterFunctional, ConjugateParameterFunctional
from pymor.tools.deprecated import Deprecated
from pymor.vectorarrays.interface import VectorArray, VectorSpace
from pymor.vectorarrays.numpy import NumpyVectorSpace

[docs]class LincombOperator(Operator):
"""Linear combination of arbitrary |Operators|.

This |Operator| represents a (possibly |Parameter| dependent)
linear combination of a given list of |Operators|.

Parameters
----------
operators
List of |Operators| whose linear combination is formed.
coefficients
A list of linear coefficients. A linear coefficient can
either be a fixed number or a |ParameterFunctional|.
solver_options
The |solver_options| for the operator.
name
Name of the operator.
"""

def __init__(self, operators, coefficients, solver_options=None, name=None):
assert len(operators) > 0
assert len(operators) == len(coefficients)
assert all(isinstance(op, Operator) for op in operators)
assert all(isinstance(c, (ParameterFunctional, Number)) for c in coefficients)
assert all(op.source == operators[0].source for op in operators[1:])
assert all(op.range == operators[0].range for op in operators[1:])
operators = tuple(operators)
coefficients = tuple(coefficients)

self.__auto_init(locals())
self.source = operators[0].source
self.range = operators[0].range
self.linear = all(op.linear for op in operators)

@property
def H(self):
'inverse_adjoint': self.solver_options.get('inverse')} if self.solver_options else None
return self.with_(operators=[op.H for op in self.operators], solver_options=options,
coefficients=[ConjugateParameterFunctional(c) if isinstance(c, ParameterFunctional)
else np.conj(c)
for c in self.coefficients],

[docs]    def evaluate_coefficients(self, mu):
"""Compute the linear coefficients for given |parameter values|.

Parameters
----------
mu
|Parameter values| for which to compute the linear coefficients.

Returns
-------
List of linear coefficients.
"""
assert self.parameters.assert_compatible(mu)
return [c.evaluate(mu) if hasattr(c, 'evaluate') else c for c in self.coefficients]

[docs]    def apply(self, U, mu=None):
coeffs = self.evaluate_coefficients(mu)
if coeffs[0]:
R = self.operators[0].apply(U, mu=mu)
R.scal(coeffs[0])
else:
R = self.range.zeros(len(U))
for op, c in zip(self.operators[1:], coeffs[1:]):
if c:
R.axpy(c, op.apply(U, mu=mu))
return R

[docs]    def apply2(self, V, U, mu=None):
coeffs = self.evaluate_coefficients(mu)
coeffs_and_matrices = [(c, self.operators[i].apply2(V, U, mu=mu))
for i, c in enumerate(coeffs) if c]
if not coeffs_and_matrices:
return np.zeros((len(V), len(U)))
else:
coeffs, matrices = zip(*coeffs_and_matrices)
coeffs_dtype = reduce(np.promote_types, (type(c) for c in coeffs))
matrices_dtype = reduce(np.promote_types, (m.dtype for m in matrices))
common_dtype = np.promote_types(coeffs_dtype, matrices_dtype)
R = (coeffs[0] * matrices[0]).astype(common_dtype, copy=False)
for m, c in zip(matrices[1:], coeffs[1:]):
R += c * m
return R

[docs]    def pairwise_apply2(self, V, U, mu=None):
coeffs = self.evaluate_coefficients(mu)
coeffs_and_matrices = [(c, self.operators[i].pairwise_apply2(V, U, mu=mu))
for i, c in enumerate(coeffs) if c]
if not coeffs_and_matrices:
return np.zeros((len(V), len(U)))
else:
coeffs, matrices = zip(*coeffs_and_matrices)
coeffs_dtype = reduce(np.promote_types, (type(c) for c in coeffs))
matrices_dtype = reduce(np.promote_types, (m.dtype for m in matrices))
common_dtype = np.promote_types(coeffs_dtype, matrices_dtype)
R = (coeffs[0] * matrices[0]).astype(common_dtype, copy=False)
for m, c in zip(matrices[1:], coeffs[1:]):
R += c * m
return R

coeffs = self.evaluate_coefficients(mu)
if coeffs[0]:
R.scal(np.conj(coeffs[0]))
else:
R = self.source.zeros(len(V))
for op, c in zip(self.operators[1:], coeffs[1:]):
if c:
return R

[docs]    def assemble(self, mu=None):
from pymor.algorithms.lincomb import assemble_lincomb
operators = tuple(op.assemble(mu) for op in self.operators)
coefficients = self.evaluate_coefficients(mu)
op = assemble_lincomb(operators, coefficients, solver_options=self.solver_options,
name=self.name + '_assembled')
if op:
return op
else:
if self.parametric or operators != self.operators:
return LincombOperator(operators, coefficients, solver_options=self.solver_options,
name=self.name + '_assembled')
else:  # this can only happen when both operators and self.operators are tuples!
return self  # avoid infinite recursion

[docs]    def jacobian(self, U, mu=None):
from pymor.algorithms.lincomb import assemble_lincomb
if self.linear:
return self.assemble(mu)
jacobians = [op.jacobian(U, mu) for op in self.operators]
coefficients = self.evaluate_coefficients(mu)
options = self.solver_options.get('jacobian') if self.solver_options else None
jac = assemble_lincomb(jacobians, coefficients, solver_options=options,
name=self.name + '_jacobian')
if jac is None:
return LincombOperator(jacobians, coefficients, solver_options=options,
name=self.name + '_jacobian')
else:
return jac

[docs]    def d_mu(self, parameter, index=0):
for op in self.operators:
if parameter in op.parameters:
raise NotImplementedError
derivative_coefficients = []
for coef in self.coefficients:
if isinstance(coef, ParametricObject):
derivative_coefficients.append(coef.d_mu(parameter, index))
else:
derivative_coefficients.append(0.)
return self.with_(coefficients=derivative_coefficients, name=self.name + '_d_mu')

[docs]    def apply_inverse(self, V, mu=None, initial_guess=None, least_squares=False):
if len(self.operators) == 1:
if self.coefficients[0] == 0.:
if least_squares:
return self.source.zeros(len(V))
else:
raise InversionError
else:
U = self.operators[0].apply_inverse(V, mu=mu, initial_guess=initial_guess, least_squares=least_squares)
U *= (1. / self.coefficients[0])
return U
else:
return super().apply_inverse(V, mu=mu, initial_guess=initial_guess, least_squares=least_squares)

[docs]    def apply_inverse_adjoint(self, U, mu=None, initial_guess=None, least_squares=False):
if len(self.operators) == 1:
if self.coefficients[0] == 0.:
if least_squares:
return self.range.zeros(len(U))
else:
raise InversionError
else:
initial_guess=initial_guess, least_squares=least_squares)
V *= (1. / self.coefficients[0])
return V
else:

def _as_array(self, source, mu):
coefficients = np.array(self.evaluate_coefficients(mu))
arrays = [op.as_source_array(mu) if source else op.as_range_array(mu) for op in self.operators]
R = arrays[0]
R.scal(coefficients[0])
for c, v in zip(coefficients[1:], arrays[1:]):
R.axpy(c, v)
return R

[docs]    def as_range_array(self, mu=None):
return self._as_array(False, mu)

[docs]    def as_source_array(self, mu=None):
return self._as_array(True, mu)

[docs]class ConcatenationOperator(Operator):
"""|Operator| representing the concatenation of two |Operators|.

Parameters
----------
operators
Tuple  of |Operators| to concatenate. operators[-1]
is the first applied operator, operators[0] is the last
applied operator.
solver_options
The |solver_options| for the operator.
name
Name of the operator.
"""

def __init__(self, operators, solver_options=None, name=None):
assert all(isinstance(op, Operator) for op in operators)
assert all(operators[i].source == operators[i+1].range for i in range(len(operators)-1))
operators = tuple(operators)

self.__auto_init(locals())
self.source = operators[-1].source
self.range = operators[0].range
self.linear = all(op.linear for op in operators)

@property
def H(self):
'inverse_adjoint': self.solver_options.get('inverse')} if self.solver_options else None
return type(self)(tuple(op.H for op in self.operators[::-1]), solver_options=options,

[docs]    def apply(self, U, mu=None):
assert self.parameters.assert_compatible(mu)
for op in self.operators[::-1]:
U = op.apply(U, mu=mu)
return U

assert self.parameters.assert_compatible(mu)
for op in self.operators:
return V

[docs]    def jacobian(self, U, mu=None):
assert len(U) == 1
Us = [U]
for op in self.operators[:0:-1]:
Us.append(op.apply(Us[-1], mu=mu))
options = self.solver_options.get('jacobian') if self.solver_options else None
return ConcatenationOperator(tuple(op.jacobian(U, mu=mu) for op, U in zip(self.operators, Us[::-1])),
solver_options=options, name=self.name + '_jacobian')

[docs]    def restricted(self, dofs):
restricted_ops = []
for op in self.operators:
rop, dofs = op.restricted(dofs)
restricted_ops.append(rop)
return ConcatenationOperator(restricted_ops), dofs

[docs]    def __matmul__(self, other):
if not isinstance(other, Operator):
return NotImplemented

if self.name != 'ConcatenationOperator':
if isinstance(other, ConcatenationOperator) and other.name == 'ConcatenationOperator':
operators = (self,) + other.operators
else:
operators = (self, other)
elif isinstance(other, ConcatenationOperator) and other.name == 'ConcatenationOperator':
operators = self.operators + other.operators
else:
operators = self.operators + (other,)

return ConcatenationOperator(operators, solver_options=self.solver_options)

def __rmatmul__(self, other):
if not isinstance(other, Operator):
return NotImplemented

# note that 'other' can never be a ConcatenationOperator
if self.name != 'ConcatenationOperator':
operators = (other, self)
else:
operators = (other,) + self.operators

return ConcatenationOperator(operators, solver_options=other.solver_options)

[docs]@Deprecated(ConcatenationOperator)
def Concatenation(*args, **kwargs):
return ConcatenationOperator(*args, **kwargs)

[docs]class ProjectedOperator(Operator):
"""Generic |Operator| representing the projection of an |Operator| to a subspace.

This operator is implemented as the concatenation of the linear combination with
source_basis, application of the original operator and projection onto
range_basis. As such, this operator can be used to obtain a reduced basis
projection of any given |Operator|. However, no offline/online decomposition is
performed, so this operator is mainly useful for testing before implementing
offline/online decomposition for a specific application.

This operator is instantiated in :func:pymor.algorithms.projection.project
as a default implementation for parametric or nonlinear operators.

Parameters
----------
operator
The |Operator| to project.
range_basis
See :func:pymor.algorithms.projection.project.
source_basis
See :func:pymor.algorithms.projection.project.
product
See :func:pymor.algorithms.projection.project.
solver_options
The |solver_options| for the projected operator.
"""

linear = False

def __init__(self, operator, range_basis, source_basis, product=None, solver_options=None):
assert isinstance(operator, Operator)
assert source_basis is None or source_basis in operator.source
assert range_basis is None or range_basis in operator.range
assert (product is None
or (isinstance(product, Operator)
and range_basis is not None
and operator.range == product.source
and product.range == product.source))
if source_basis is not None:
source_basis = source_basis.copy()
if range_basis is not None:
range_basis = range_basis.copy()
self.__auto_init(locals())
self.source = NumpyVectorSpace(len(source_basis)) if source_basis is not None else operator.source
self.range = NumpyVectorSpace(len(range_basis)) if range_basis is not None else operator.range
self.linear = operator.linear

@property
def H(self):
if self.product:
return super().H
else:
'inverse_adjoint': self.solver_options.get('inverse')} if self.solver_options else None
return ProjectedOperator(self.operator.H, self.source_basis, self.range_basis, solver_options=options)

[docs]    def apply(self, U, mu=None):
assert self.parameters.assert_compatible(mu)
if self.source_basis is None:
if self.range_basis is None:
return self.operator.apply(U, mu=mu)
elif self.product is None:
return self.range.make_array(self.operator.apply2(self.range_basis, U, mu=mu).T)
else:
V = self.operator.apply(U, mu=mu)
return self.range.make_array(self.product.apply2(V, self.range_basis))
else:
UU = self.source_basis.lincomb(U.to_numpy())
if self.range_basis is None:
return self.operator.apply(UU, mu=mu)
elif self.product is None:
return self.range.make_array(self.operator.apply2(self.range_basis, UU, mu=mu).T)
else:
V = self.operator.apply(UU, mu=mu)
return self.range.make_array(self.product.apply2(V, self.range_basis))

[docs]    def jacobian(self, U, mu=None):
assert len(U) == 1
assert self.parameters.assert_compatible(mu)
if self.linear:
return self.assemble(mu)
if self.source_basis is None:
J = self.operator.jacobian(U, mu=mu)
else:
J = self.operator.jacobian(self.source_basis.lincomb(U.to_numpy()), mu=mu)
from pymor.algorithms.projection import project
pop = project(J, range_basis=self.range_basis, source_basis=self.source_basis,
product=self.product)
if self.solver_options:
options = self.solver_options.get('jacobian')
if options:
pop = pop.with_(solver_options=options)
return pop

[docs]    def assemble(self, mu=None):
op = self.operator.assemble(mu=mu)
if op == self.operator:  # avoid infinite recursion in apply_inverse default impl
return self
from pymor.algorithms.projection import project
pop = project(op, range_basis=self.range_basis, source_basis=self.source_basis,
product=self.product)
if self.solver_options:
pop = pop.with_(solver_options=self.solver_options)
return pop

assert V in self.range
if self.range_basis is not None:
V = self.range_basis.lincomb(V.to_numpy())
if self.source_basis is not None:
U = self.source.make_array(U.inner(self.source_basis))
return U

[docs]class LowRankOperator(Operator):
"""Non-parametric low-rank operator.

Represents an operator of the form :math:L C R^H or
:math:L C^{-1} R^H where :math:L and :math:R are
|VectorArrays| of column vectors and :math:C a 2D |NumPy array|.

Parameters
----------
left
|VectorArray| representing :math:L.
core
|NumPy array| representing :math:C.
right
|VectorArray| representing :math:R.
inverted
Whether :math:C is inverted.
solver_options
The |solver_options| for the operator.
name
Name of the operator.
"""

linear = True

def __init__(self, left, core, right, inverted=False, solver_options=None, name=None):
assert isinstance(left, VectorArray)
assert isinstance(right, VectorArray)
assert len(left) == len(right)
assert (isinstance(core, np.ndarray)
and core.ndim == 2
and core.shape[0] == core.shape[1] == len(left))

self.__auto_init(locals())
self.source = right.space
self.range = left.space

@property
def H(self):
options = {
} if self.solver_options else None
return type(self)(self.right,
self.core.T.conj(),
self.left,
inverted=self.inverted,
solver_options=options,

[docs]    def apply(self, U, mu=None):
assert U in self.source
V = self.right.inner(U)
if self.inverted:
V = spla.solve(self.core, V)
else:
V = self.core @ V
return self.left.lincomb(V.T)

assert V in self.range
U = self.left.inner(V)
if self.inverted:
U = spla.solve(self.core.T.conj(), U)
else:
U = self.core.T.conj() @ U
return self.right.lincomb(U.T)

[docs]class LowRankUpdatedOperator(LincombOperator):
r"""|Operator| plus :class:LowRankOperator.

Represents a linear combination of an |Operator| and
:class:LowRankOperator. Uses the Sherman-Morrison-Woodbury formula
in apply_inverse and apply_inverse_adjoint:

.. math::
\left(\alpha A + \beta L C R^H \right)^{-1}
& = \alpha^{-1} A^{-1}
- \alpha^{-1} \beta A^{-1} L C
\left(\alpha C + \beta C R^H A^{-1} L C \right)^{-1}
C R^H A^{-1}, \\
\left(\alpha A + \beta L C^{-1} R^H \right)^{-1}
& = \alpha^{-1} A^{-1}
- \alpha^{-1} \beta A^{-1} L
\left(\alpha C + \beta R^H A^{-1} L \right)^{-1}
R^H A^{-1}.

Parameters
----------
operator
|Operator|.
lr_operator
:class:LowRankOperator.
coeff
A linear coefficient for operator. Can either be a fixed
number or a |ParameterFunctional|.
lr_coeff
A linear coefficient for lr_operator. Can either be a fixed
number or a |ParameterFunctional|.
solver_options
The |solver_options| for the operator.
name
Name of the operator.
"""

def __init__(self, operator, lr_operator, coeff, lr_coeff,
solver_options=None, name=None):
assert isinstance(lr_operator, LowRankOperator)
super().__init__([operator, lr_operator], [coeff, lr_coeff],
solver_options=solver_options, name=name)
self.__auto_init(locals())

[docs]    def apply_inverse(self, V, mu=None, initial_guess=None, least_squares=False):
if least_squares:
return super().apply_inverse(V, mu=mu, initial_guess=initial_guess, least_squares=True)
A, LR = self.operators
L, C, R = LR.left, LR.core, LR.right
if not LR.inverted:
L = L.lincomb(C.T)
R = R.lincomb(C.conj())
alpha, beta = self.evaluate_coefficients(mu)
AinvV = A.apply_inverse(V)
AinvL = A.apply_inverse(L)
mat = alpha * C + beta * R.inner(AinvL)
RhAinvV = R.inner(AinvV)
U = AinvV
U.axpy(-beta, AinvL.lincomb(spla.solve(mat, RhAinvV).T))
U.scal(1 / alpha)
return U

[docs]    def apply_inverse_adjoint(self, U, mu=None, initial_guess=None, least_squares=False):
if least_squares:
A, LR = self.operators
L, C, R = LR.left, LR.core, LR.right
if not LR.inverted:
L = L.lincomb(C.T)
R = R.lincomb(C.conj())
alpha, beta = (c.conjugate() for c in self.evaluate_coefficients(mu))
mat = alpha * C.T.conj() + beta * L.inner(AinvhR)
LhAinvhU = L.inner(AinvhU)
V = AinvhU
V.axpy(-beta, AinvhR.lincomb(spla.solve(mat, LhAinvhU).T))
V.scal(1 / alpha)
return V

[docs]class ComponentProjectionOperator(Operator):
"""|Operator| representing the projection of a |VectorArray| onto some of its components.

Parameters
----------
components
List or 1D |NumPy array| of the indices of the vector
:meth:~pymor.vectorarrays.interface.VectorArray.components that are
to be extracted by the operator.
source
Source |VectorSpace| of the operator.
name
Name of the operator.
"""

linear = True

def __init__(self, components, source, name=None):
assert all(0 <= c < source.dim for c in components)
components = np.array(components, dtype=np.int32)

self.__auto_init(locals())
self.range = NumpyVectorSpace(len(components))

[docs]    def apply(self, U, mu=None):
assert U in self.source
return self.range.make_array(U.dofs(self.components))

[docs]    def restricted(self, dofs):
assert all(0 <= c < self.range.dim for c in dofs)
source_dofs = self.components[dofs]
return IdentityOperator(NumpyVectorSpace(len(source_dofs))), source_dofs

[docs]class IdentityOperator(Operator):
"""The identity |Operator|.

In other words::

op.apply(U) == U

Parameters
----------
space
The |VectorSpace| the operator acts on.
name
Name of the operator.
"""

linear = True

def __init__(self, space, name=None):
self.__auto_init(locals())
self.source = self.range = space

@property
def H(self):
return self

[docs]    def apply(self, U, mu=None):
assert U in self.source
return U.copy()

assert V in self.range
return V.copy()

[docs]    def apply_inverse(self, V, mu=None, initial_guess=None, least_squares=False):
assert V in self.range
assert initial_guess is None or initial_guess in self.source and len(initial_guess) == len(V)
return V.copy()

[docs]    def apply_inverse_adjoint(self, U, mu=None, initial_guess=None, least_squares=False):
assert U in self.source
assert initial_guess is None or initial_guess in self.range and len(initial_guess) == len(U)
return U.copy()

[docs]    def assemble(self, mu=None):
return self

[docs]    def restricted(self, dofs):
assert all(0 <= c < self.range.dim for c in dofs)
return IdentityOperator(NumpyVectorSpace(len(dofs))), dofs

[docs]class ConstantOperator(Operator):
"""A constant |Operator| always returning the same vector.

Parameters
----------
value
A |VectorArray| of length 1 containing the vector which is
returned by the operator.
source
Source |VectorSpace| of the operator.
name
Name of the operator.
"""

linear = False

def __init__(self, value, source, name=None):
assert isinstance(value, VectorArray)
assert len(value) == 1
value = value.copy()

self.__auto_init(locals())
self.range = value.space

[docs]    def apply(self, U, mu=None):
assert U in self.source
return self.value[[0] * len(U)].copy()

[docs]    def jacobian(self, U, mu=None):
assert U in self.source
assert len(U) == 1
return ZeroOperator(self.range, self.source, name=self.name + '_jacobian')

[docs]    def restricted(self, dofs):
assert all(0 <= c < self.range.dim for c in dofs)
restricted_value = NumpyVectorSpace.make_array(self.value.dofs(dofs))
return ConstantOperator(restricted_value, NumpyVectorSpace(len(dofs))), dofs

[docs]    def apply_inverse(self, V, mu=None, initial_guess=None, least_squares=False):
if not least_squares:
raise InversionError('ConstantOperator is not invertible.')
return self.source.zeros(len(V))

[docs]@Deprecated(ComponentProjectionOperator)
def ComponentProjection(*args, **kwargs):
return ComponentProjectionOperator(*args, **kwargs)

[docs]class ZeroOperator(Operator):
"""The |Operator| which maps every vector to zero.

Parameters
----------
range
Range |VectorSpace| of the operator.
source
Source |VectorSpace| of the operator.
name
Name of the operator.
"""

linear = True

def __init__(self, range, source, name=None):
assert isinstance(range, VectorSpace)
assert isinstance(source, VectorSpace)
self.__auto_init(locals())

@property
def H(self):
return type(self)(self.source, self.range, name=self.name + '_adjoint')

[docs]    def apply(self, U, mu=None):
assert U in self.source
return self.range.zeros(len(U))

assert V in self.range
return self.source.zeros(len(V))

[docs]    def apply_inverse(self, V, mu=None, initial_guess=None, least_squares=False):
assert V in self.range
assert initial_guess is None or initial_guess in self.source and len(initial_guess) == len(V)
if not least_squares:
raise InversionError
return self.source.zeros(len(V))

[docs]    def apply_inverse_adjoint(self, U, mu=None, initial_guess=None, least_squares=False):
assert U in self.source
assert initial_guess is None or initial_guess in self.range and len(initial_guess) == len(U)
if not least_squares:
raise InversionError
return self.range.zeros(len(U))

[docs]    def restricted(self, dofs):
assert all(0 <= c < self.range.dim for c in dofs)
return ZeroOperator(NumpyVectorSpace(len(dofs)), NumpyVectorSpace(0)), np.array([], dtype=np.int32)

[docs]class VectorArrayOperator(Operator):
"""Wraps a |VectorArray| as an |Operator|.

If adjoint is False, the operator maps from NumpyVectorSpace(len(array))
to array.space by forming linear combinations of the vectors in the array
with given coefficient arrays.

If adjoint == True, the operator maps from array.space to
NumpyVectorSpace(len(array)) by forming the inner products of the argument
with the vectors in the given array.

Parameters
----------
array
The |VectorArray| which is to be treated as an operator.
See description above.
space_id
Id of the source (range) |VectorSpace| in case adjoint is
False (True).
name
The name of the operator.
"""

linear = True

def __init__(self, array, adjoint=False, space_id=None, name=None):
array = array.copy()

self.__auto_init(locals())
self.source = array.space
self.range = NumpyVectorSpace(len(array), space_id)
else:
self.source = NumpyVectorSpace(len(array), space_id)
self.range = array.space

@property
def H(self):

[docs]    def apply(self, U, mu=None):
assert U in self.source
return self.array.lincomb(U.to_numpy())
else:
return self.range.make_array(self.array.inner(U).T)

[docs]    def apply_inverse(self, V, mu=None, initial_guess=None, least_squares=False):
if not least_squares and len(self.array) != self.array.dim:
raise InversionError

from pymor.algorithms.gram_schmidt import gram_schmidt
from numpy.linalg import lstsq

Q, R = gram_schmidt(self.array, return_R=True, reiterate=False)
v = lstsq(R.T.conj(), V.to_numpy().T)[0]
U = Q.lincomb(v.T)
else:
v = Q.inner(V)
u = lstsq(R, v)[0]
U = self.source.make_array(u.T)

return U

assert V in self.range
return self.source.make_array(self.array.inner(V).T)
else:
return self.array.lincomb(V.to_numpy())

[docs]    def apply_inverse_adjoint(self, U, mu=None, initial_guess=None, least_squares=False):

[docs]    def as_range_array(self, mu=None):
return self.array.copy()
else:
return super().as_range_array(mu)

[docs]    def as_source_array(self, mu=None):
return self.array.copy()
else:
return super().as_source_array(mu)

[docs]    def restricted(self, dofs):
assert all(0 <= c < self.range.dim for c in dofs)
restricted_value = NumpyVectorSpace.make_array(self.array.dofs(dofs))
return VectorArrayOperator(restricted_value, False), np.arange(self.source.dim, dtype=np.int32)
else:
raise NotImplementedError

[docs]class VectorOperator(VectorArrayOperator):
"""Wrap a vector as a vector-like |Operator|.

Given a vector v of dimension d, this class represents
the operator ::

op: R^1 ----> R^d
x  |---> xâ‹…v

In particular::

VectorOperator(vector).as_range_array() == vector

Parameters
----------
vector
|VectorArray| of length 1 containing the vector v.
name
Name of the operator.
"""

linear = True
source = NumpyVectorSpace(1)

def __init__(self, vector, name=None):
assert isinstance(vector, VectorArray)
assert len(vector) == 1
self.vector = self.array  # do not init with vector arg, as vector gets copied in VectorArrayOperator.__init__

[docs]class VectorFunctional(VectorArrayOperator):
"""Wrap a vector as a linear |Functional|.

Given a vector v of dimension d, this class represents
the functional ::

f: R^d ----> R^1
u  |---> (u, v)

where ( , ) denotes the inner product given by product.

In particular, if product is None ::

VectorFunctional(vector).as_source_array() == vector.

If product is not none, we obtain ::

VectorFunctional(vector).as_source_array() == product.apply(vector).

Parameters
----------
vector
|VectorArray| of length 1 containing the vector v.
product
|Operator| representing the scalar product to use.
name
Name of the operator.
"""

linear = True
range = NumpyVectorSpace(1)

def __init__(self, vector, product=None, name=None):
assert isinstance(vector, VectorArray)
assert len(vector) == 1
assert product is None or isinstance(product, Operator) and vector in product.source
if product is None:
else:
self.vector = self.array  # do not init with vector arg, as vector gets copied in VectorArrayOperator.__init__
self.product = product

[docs]class ProxyOperator(Operator):
"""Forwards all interface calls to given |Operator|.

Mainly useful as base class for other |Operator| implementations.

Parameters
----------
operator
The |Operator| to wrap.
name
Name of the wrapping operator.
"""

def __init__(self, operator, name=None):
assert isinstance(operator, Operator)
self.__auto_init(locals())
self.source = operator.source
self.range = operator.range
self.linear = operator.linear

@property
def H(self):

[docs]    def apply(self, U, mu=None):
return self.operator.apply(U, mu=mu)

[docs]    def apply_inverse(self, V, mu=None, initial_guess=None, least_squares=False):
return self.operator.apply_inverse(V, mu=mu, initial_guess=initial_guess, least_squares=least_squares)

[docs]    def apply_inverse_adjoint(self, U, mu=None, initial_guess=None, least_squares=False):

[docs]    def jacobian(self, U, mu=None):
return self.operator.jacobian(U, mu=mu)

[docs]    def restricted(self, dofs):
op, source_dofs = self.operator.restricted(dofs)
return self.with_(operator=op), source_dofs

[docs]class FixedParameterOperator(ProxyOperator):
"""Makes an |Operator| |Parameter|-independent by setting fixed |parameter values|.

Parameters
----------
operator
The |Operator| to wrap.
mu
The fixed |parameter values| that will be fed to the
:meth:~pymor.operators.interface.Operator.apply method
(and related methods) of operator.
"""

def __init__(self, operator, mu=None, name=None):
super().__init__(operator, name)
assert operator.parameters.assert_compatible(mu)
self.mu = mu
if mu:
self.parameters_internal = mu.parameters

[docs]    def apply(self, U, mu=None):
return self.operator.apply(U, mu=self.mu)

[docs]    def apply_inverse(self, V, mu=None, initial_guess=None, least_squares=False):
return self.operator.apply_inverse(V, mu=self.mu, initial_guess=initial_guess, least_squares=least_squares)

[docs]    def apply_inverse_adjoint(self, U, mu=None, initial_guess=None, least_squares=False):
initial_guess=initial_guess, least_squares=least_squares)

[docs]    def jacobian(self, U, mu=None):
return self.operator.jacobian(U, mu=self.mu)

[docs]class LinearOperator(ProxyOperator):
"""Mark the wrapped |Operator| to be linear."""

def __init__(self, operator, name=None):
super().__init__(operator, name)
self.linear = True

[docs]class AffineOperator(ProxyOperator):
"""Decompose an affine |Operator| into affine_shift and linear_part. """

def __init__(self, operator, name=None):
if operator.parametric:
raise NotImplementedError
super().__init__(operator, name)
self.affine_shift = ConstantOperator(operator.apply(operator.source.zeros()), source=operator.source)
self.linear_part = LinearOperator(operator - self.affine_shift, name=operator.name + '_linear_part')

[docs]    def jacobian(self, U, mu=None):
return self.linear_part.jacobian(U, mu)

[docs]class InverseOperator(Operator):
"""Represents the inverse of a given |Operator|.

Parameters
----------
operator
The |Operator| of which the inverse is formed.
name
If not None, name of the operator.
"""

def __init__(self, operator, name=None):
assert isinstance(operator, Operator)
name or operator.name + '_inverse'

self.__auto_init(locals())
self.source = operator.range
self.range = operator.source
self.linear = operator.linear

@property
def H(self):

[docs]    def apply(self, U, mu=None):
assert U in self.source
return self.operator.apply_inverse(U, mu=mu)

assert V in self.range

[docs]    def apply_inverse(self, V, mu=None, initial_guess=None, least_squares=False):
assert V in self.range
assert initial_guess is None or initial_guess in self.source and len(initial_guess) == len(V)
return self.operator.apply(V, mu=mu)

[docs]    def apply_inverse_adjoint(self, U, mu=None, initial_guess=None, least_squares=False):
assert U in self.source
assert initial_guess is None or initial_guess in self.range and len(initial_guess) == len(U)

"""Represents the inverse adjoint of a given |Operator|.

Parameters
----------
operator
The |Operator| of which the inverse adjoint is formed.
name
If not None, name of the operator.
"""

linear = True

def __init__(self, operator, name=None):
assert isinstance(operator, Operator)
assert operator.linear
name = name or operator.name + '_inverse_adjoint'

self.__auto_init(locals())
self.source = operator.source
self.range = operator.range

@property
def H(self):
return InverseOperator(self.operator)

[docs]    def apply(self, U, mu=None):
assert U in self.source

assert V in self.range
return self.operator.apply_inverse(V, mu=mu)

[docs]    def apply_inverse(self, V, mu=None, initial_guess=None, least_squares=False):
assert V in self.range

[docs]    def apply_inverse_adjoint(self, U, mu=None, initial_guess=None, least_squares=False):
assert U in self.source
return self.operator.apply(U, mu=mu)

"""Represents the adjoint of a given linear |Operator|.

For a linear |Operator| op the adjoint op^* of op is given by::

(op^*(v), u)_s = (v, op(u))_r,

where ( , )_s and ( , )_r denote the inner products on the source
and range space of op. If two products are given by P_s and P_r, then::

op^*(v) = P_s^(-1) o op.H o P_r,

Thus, if ( , )_s and ( , )_r are the Euclidean inner products,
op^*v is simply given by application of the
|Operator|.

Parameters
----------
operator
The |Operator| of which the adjoint is formed.
source_product
If not None, inner product |Operator| for the source |VectorSpace|
w.r.t. which to take the adjoint.
range_product
If not None, inner product |Operator| for the range |VectorSpace|
w.r.t. which to take the adjoint.
name
If not None, name of the operator.
with_apply_inverse
If True, provide own :meth:~pymor.operators.interface.Operator.apply_inverse
and :meth:~pymor.operators.interface.Operator.apply_inverse_adjoint
implementations by calling these methods on the given operator.
(Is set to False in the default implementation of
and :meth:~pymor.operators.interface.Operator.apply_inverse_adjoint.)
solver_options
When with_apply_inverse is False, the |solver_options| to use for
the apply_inverse default implementation.
"""

linear = True

def __init__(self, operator, source_product=None, range_product=None, name=None,
with_apply_inverse=True, solver_options=None):
assert isinstance(operator, Operator)
assert operator.linear
assert not with_apply_inverse or solver_options is None

self.__auto_init(locals())
self.source = operator.range
self.range = operator.source

@property
def H(self):
if not self.source_product and not self.range_product:
return self.operator
else:
'inverse_adjoint': self.solver_options.get('inverse')} if self.solver_options else None
range_product=self.source_product, solver_options=options)

[docs]    def apply(self, U, mu=None):
assert U in self.source
if self.range_product:
U = self.range_product.apply(U)
if self.source_product:
V = self.source_product.apply_inverse(V)
return V

assert V in self.range
if self.source_product:
V = self.source_product.apply_inverse(V)
U = self.operator.apply(V, mu=mu)
if self.range_product:
U = self.range_product.apply(U)
return U

[docs]    def apply_inverse(self, V, mu=None, initial_guess=None, least_squares=False):
if not self.with_apply_inverse:
return super().apply_inverse(V, mu=mu, initial_guess=initial_guess, least_squares=least_squares)

assert V in self.range
if self.source_product:
V = self.source_product(V)
initial_guess=initial_guess if not self.range_product else None,
least_squares=least_squares)
if self.range_product:
U = self.range_product.apply_inverse(U)
return U

[docs]    def apply_inverse_adjoint(self, U, mu=None, initial_guess=None, least_squares=False):
if not self.with_apply_inverse:

assert U in self.source
if self.range_product:
U = self.range_product.apply_inverse(U)
V = self.operator.apply_inverse(U, mu=mu,
initial_guess=initial_guess if not self.source_product else None,
least_squares=least_squares)
if self.source_product:
V = self.source_product.apply(V)
return V

[docs]class SelectionOperator(Operator):
"""An |Operator| selected from a list of |Operators|.

operators[i] is used if parameter_functional(mu) is less or
equal than boundaries[i] and greater than boundaries[i-1]::

-infty ------- boundaries[i] ---------- boundaries[i+1] ------- infty
|                        |
--- operators[i] ---|---- operators[i+1] ----|---- operators[i+2]
|                        |

Parameters
----------
operators
List of |Operators| from which one |Operator| is
selected based on the given |parameter values|.
parameter_functional
The |ParameterFunctional| used for the selection of one |Operator|.
boundaries
The interval boundaries as defined above.
name
Name of the operator.

"""
def __init__(self, operators, parameter_functional, boundaries, name=None):
assert len(operators) > 0
assert len(boundaries) == len(operators) - 1
# check that boundaries are ascending:
for i in range(len(boundaries)-1):
assert boundaries[i] <= boundaries[i+1]
assert all(isinstance(op, Operator) for op in operators)
assert all(op.source == operators[0].source for op in operators[1:])
assert all(op.range == operators[0].range for op in operators[1:])
operators = tuple(operators)
boundaries = tuple(boundaries)

self.__auto_init(locals())
self.source = operators[0].source
self.range = operators[0].range
self.linear = all(op.linear for op in operators)

@property
def H(self):
return self.with_(operators=[op.H for op in self.operators],

def _get_operator_number(self, mu):
value = self.parameter_functional.evaluate(mu)
for i in range(len(self.boundaries)):
if self.boundaries[i] >= value:
return i
return len(self.boundaries)

[docs]    def assemble(self, mu=None):
assert self.parameters.assert_compatible(mu)
op = self.operators[self._get_operator_number(mu)]
return op.assemble(mu)

[docs]    def apply(self, U, mu=None):
assert self.parameters.assert_compatible(mu)
operator_number = self._get_operator_number(mu)
return self.operators[operator_number].apply(U, mu=mu)

assert self.parameters.assert_compatible(mu)
op = self.operators[self._get_operator_number(mu)]

[docs]    def as_range_array(self, mu=None):
assert self.parameters.assert_compatible(mu)
operator_number = self._get_operator_number(mu)
return self.operators[operator_number].as_range_array(mu=mu)

[docs]    def as_source_array(self, mu=None):
assert self.parameters.assert_compatible(mu)
operator_number = self._get_operator_number(mu)
return self.operators[operator_number].as_source_array(mu=mu)

[docs]@defaults('raise_negative', 'tol')
def induced_norm(product, raise_negative=True, tol=1e-10, name=None):
"""Obtain induced norm of an inner product.

The norm of the vectors in a |VectorArray| U is calculated by
calling ::

product.pairwise_apply2(U, U, mu=mu).

In addition, negative norm squares of absolute value smaller
than tol are clipped to 0.
If raise_negative is True, a :exc:ValueError exception
is raised if there are negative norm squares of absolute value
larger than tol.

Parameters
----------
product
The inner product |Operator| for which the norm is to be
calculated.
raise_negative
If True, raise an exception if calculated norm is negative.
tol
See above.
name
optional, if None product's name is used

Returns
-------
norm
A function norm(U, mu=None) taking a |VectorArray| U
as input together with the |parameter values| mu which
are passed to the product.
"""
return InducedNorm(product, raise_negative, tol, name)

[docs]class InducedNorm(ParametricObject):
"""Instantiated by :func:induced_norm. Do not use directly."""

def __init__(self, product, raise_negative, tol, name):
name = name or product.name
self.__auto_init(locals())

[docs]    def __call__(self, U, mu=None):
norm_squared = self.product.pairwise_apply2(U, U, mu=mu).real
if self.tol > 0:
norm_squared = np.where(np.logical_and(0 > norm_squared, norm_squared > - self.tol),
0, norm_squared)
if self.raise_negative and np.any(norm_squared < 0):
raise ValueError(f'norm is negative (square = {norm_squared})')
return np.sqrt(norm_squared)
`