# Source code for pymor.reductors.coercive

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

import numpy as np

from pymor.core.base import ImmutableObject
from pymor.operators.constructions import LincombOperator, induced_norm
from pymor.operators.numpy import NumpyMatrixOperator
from pymor.reductors.basic import StationaryRBReductor
from pymor.reductors.residual import ResidualReductor
from pymor.vectorarrays.numpy import NumpyVectorSpace

[docs]class CoerciveRBReductor(StationaryRBReductor):
"""Reduced Basis reductor for |StationaryModels| with coercive linear operator.

The only addition to :class:~pymor.reductors.basic.StationaryRBReductor is an error
estimator which evaluates the dual norm of the residual with respect to a given inner
product. For the reduction of the residual we use
:class:~pymor.reductors.residual.ResidualReductor for improved numerical stability
[BEOR14]_.

Parameters
----------
fom
The |Model| which is to be reduced.
RB
|VectorArray| containing the reduced basis on which to project.
product
Inner product for the orthonormalization of RB, the projection of the
|Operators| given by vector_ranged_operators and for the computation of
Riesz representatives of the residual. If None, the Euclidean product is used.
coercivity_estimator
None or a |Parameterfunctional| returning a lower bound for the coercivity
constant of the given problem. Note that the computed error estimate is only
guaranteed to be an upper bound for the error when an appropriate coercivity
estimate is specified.
"""

def __init__(self, fom, RB=None, product=None, coercivity_estimator=None,
check_orthonormality=None, check_tol=None):
super().__init__(fom, RB, product=product, check_orthonormality=check_orthonormality,
check_tol=check_tol)
self.coercivity_estimator = coercivity_estimator
self.residual_reductor = ResidualReductor(self.bases['RB'], self.fom.operator, self.fom.rhs,
product=product, riesz_representatives=True)

def assemble_estimator(self):
residual = self.residual_reductor.reduce()
estimator = CoerciveRBEstimator(residual, tuple(self.residual_reductor.residual_range_dims),
self.coercivity_estimator)
return estimator

def assemble_estimator_for_subbasis(self, dims):
return self._last_rom.estimator.restricted_to_subbasis(dims['RB'], m=self._last_rom)

[docs]class CoerciveRBEstimator(ImmutableObject):
"""Instantiated by :class:CoerciveRBReductor.

Not to be used directly.
"""

def __init__(self, residual, residual_range_dims, coercivity_estimator):
self.__auto_init(locals())

def estimate(self, U, mu, m):
est = self.residual.apply(U, mu=mu).l2_norm()
if self.coercivity_estimator:
est /= self.coercivity_estimator(mu)
return est

def restricted_to_subbasis(self, dim, m):
if self.residual_range_dims:
residual_range_dims = self.residual_range_dims[:dim + 1]
residual = self.residual.projected_to_subbasis(residual_range_dims[-1], dim)
return CoerciveRBEstimator(residual, residual_range_dims, self.coercivity_estimator)
else:
self.logger.warning('Cannot efficiently reduce to subbasis')
return CoerciveRBEstimator(self.residual.projected_to_subbasis(None, dim), None,
self.coercivity_estimator)

[docs]class SimpleCoerciveRBReductor(StationaryRBReductor):
"""Reductor for linear |StationaryModels| with affinely decomposed operator and rhs.

.. note::
The reductor :class:CoerciveRBReductor can be used for arbitrary coercive
|StationaryModels| and offers an improved error estimator
with better numerical stability.

The only addition is to :class:~pymor.reductors.basic.StationaryRBReductor is an error
estimator, which evaluates the norm of the residual with respect to a given inner product.

Parameters
----------
fom
The |Model| which is to be reduced.
RB
|VectorArray| containing the reduced basis on which to project.
product
Inner product for the orthonormalization of RB, the projection of the
|Operators| given by vector_ranged_operators and for the computation of
Riesz representatives of the residual. If None, the Euclidean product is used.
coercivity_estimator
None or a |Parameterfunctional| returning a lower bound for the coercivity
constant of the given problem. Note that the computed error estimate is only
guaranteed to be an upper bound for the error when an appropriate coercivity
estimate is specified.
"""

def __init__(self, fom, RB=None, product=None, coercivity_estimator=None,
check_orthonormality=None, check_tol=None):
assert fom.operator.linear and fom.rhs.linear
assert isinstance(fom.operator, LincombOperator)
assert all(not op.parametric for op in fom.operator.operators)
if fom.rhs.parametric:
assert isinstance(fom.rhs, LincombOperator)
assert all(not op.parametric for op in fom.rhs.operators)

super().__init__(fom, RB, product=product, check_orthonormality=check_orthonormality,
check_tol=check_tol)
self.coercivity_estimator = coercivity_estimator
self.extends = None

def assemble_estimator(self):
fom, RB, extends = self.fom, self.bases['RB'], self.extends
if extends:
old_RB_size = extends[0]
old_data = extends[1]
else:
old_RB_size = 0

# compute data for estimator
space = fom.operator.source

# compute the Riesz representative of (U, .)_L2 with respect to product
def riesz_representative(U):
if self.products['RB'] is None:
return U.copy()
else:
return self.products['RB'].apply_inverse(U)

def append_vector(U, R, RR):
RR.append(riesz_representative(U), remove_from_other=True)
R.append(U, remove_from_other=True)

# compute all components of the residual
if extends:
R_R, RR_R = old_data['R_R'], old_data['RR_R']
elif not fom.rhs.parametric:
R_R = space.empty(reserve=1)
RR_R = space.empty(reserve=1)
append_vector(fom.rhs.as_range_array(), R_R, RR_R)
else:
R_R = space.empty(reserve=len(fom.rhs.operators))
RR_R = space.empty(reserve=len(fom.rhs.operators))
for op in fom.rhs.operators:
append_vector(op.as_range_array(), R_R, RR_R)

if len(RB) == 0:
R_Os = [space.empty()]
RR_Os = [space.empty()]
elif not fom.operator.parametric:
R_Os = [space.empty(reserve=len(RB))]
RR_Os = [space.empty(reserve=len(RB))]
for i in range(len(RB)):
append_vector(-fom.operator.apply(RB[i]), R_Os[0], RR_Os[0])
else:
R_Os = [space.empty(reserve=len(RB)) for _ in range(len(fom.operator.operators))]
RR_Os = [space.empty(reserve=len(RB)) for _ in range(len(fom.operator.operators))]
if old_RB_size > 0:
for op, R_O, RR_O, old_R_O, old_RR_O in zip(fom.operator.operators, R_Os, RR_Os,
old_data['R_Os'], old_data['RR_Os']):
R_O.append(old_R_O)
RR_O.append(old_RR_O)
for op, R_O, RR_O in zip(fom.operator.operators, R_Os, RR_Os):
for i in range(old_RB_size, len(RB)):
append_vector(-op.apply(RB[i]), R_O, RR_O)

# compute Gram matrix of the residuals
R_RR = RR_R.dot(R_R)
R_RO = np.hstack([RR_R.dot(R_O) for R_O in R_Os])
R_OO = np.vstack([np.hstack([RR_O.dot(R_O) for R_O in R_Os]) for RR_O in RR_Os])

estimator_matrix = np.empty((len(R_RR) + len(R_OO),) * 2)
estimator_matrix[:len(R_RR), :len(R_RR)] = R_RR
estimator_matrix[len(R_RR):, len(R_RR):] = R_OO
estimator_matrix[:len(R_RR), len(R_RR):] = R_RO
estimator_matrix[len(R_RR):, :len(R_RR)] = R_RO.T

estimator_matrix = NumpyMatrixOperator(estimator_matrix)

estimator = SimpleCoerciveRBEstimator(estimator_matrix, self.coercivity_estimator)
self.extends = (len(RB), dict(R_R=R_R, RR_R=RR_R, R_Os=R_Os, RR_Os=RR_Os))

return estimator

def assemble_estimator_for_subbasis(self, dims):
return self._last_rom.estimator.restricted_to_subbasis(dims['RB'], m=self._last_rom)

[docs]class SimpleCoerciveRBEstimator(ImmutableObject):
"""Instantiated by :class:SimpleCoerciveRBReductor.

Not to be used directly.
"""

def __init__(self, estimator_matrix, coercivity_estimator):
self.__auto_init(locals())
self.norm = induced_norm(estimator_matrix)

def estimate(self, U, mu, m):
if len(U) > 1:
raise NotImplementedError
if not m.rhs.parametric:
CR = np.ones(1)
else:
CR = np.array(m.rhs.evaluate_coefficients(mu))

if not m.operator.parametric:
CO = np.ones(1)
else:
CO = np.array(m.operator.evaluate_coefficients(mu))

C = np.hstack((CR, np.dot(CO[..., np.newaxis], U.to_numpy()).ravel()))

est = self.norm(NumpyVectorSpace.make_array(C))
if self.coercivity_estimator:
est /= self.coercivity_estimator(mu)

return est

def restricted_to_subbasis(self, dim, m):
cr = 1 if not m.rhs.parametric else len(m.rhs.operators)
co = 1 if not m.operator.parametric else len(m.operator.operators)
old_dim = m.operator.source.dim

indices = np.concatenate((np.arange(cr),
((np.arange(co)*old_dim)[..., np.newaxis] + np.arange(dim)).ravel() + cr))
matrix = self.estimator_matrix.matrix[indices, :][:, indices]

return SimpleCoerciveRBEstimator(NumpyMatrixOperator(matrix), self.coercivity_estimator)