Source code for pymor.reductors.residual

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

import numpy as np

from pymor.algorithms.image import estimate_image_hierarchical
from pymor.algorithms.projection import project, project_to_subbasis
from pymor.core.base import BasicObject
from pymor.core.exceptions import ImageCollectionError
from pymor.operators.constructions import induced_norm
from pymor.operators.interface import Operator

[docs]class ResidualReductor(BasicObject):
"""Generic reduced basis residual reductor.

Given an operator and a right-hand side, the residual is given by::

residual.apply(U, mu) == operator.apply(U, mu) - rhs.as_range_array(mu)

When operator maps to functionals instead of vectors, we are interested in the Riesz
representative of the residual::

residual.apply(U, mu)
== product.apply_inverse(operator.apply(U, mu) - rhs.as_range_array(mu))

Given a basis RB of a subspace of the source space of operator, this reductor
uses :func:~pymor.algorithms.image.estimate_image_hierarchical to determine
a low-dimensional subspace containing the image of the subspace under
residual (resp. riesz_residual), computes an orthonormal basis
residual_range for this range space and then returns the Petrov-Galerkin projection ::

projected_residual
== project(residual, range_basis=residual_range, source_basis=RB)

of the residual operator. Given a reduced basis coefficient vector u, w.r.t.
RB, the (dual) norm of the residual can then be computed as ::

projected_residual.apply(u, mu).l2_norm()

Moreover, a reconstruct method is provided such that ::

residual_reductor.reconstruct(projected_residual.apply(u, mu))
== residual.apply(RB.lincomb(u), mu)

Parameters
----------
RB
|VectorArray| containing a basis of the reduced space onto which to project.
operator
See definition of residual.
rhs
See definition of residual. If None, zero right-hand side is assumed.
product
Inner product |Operator| w.r.t. which to orthonormalize and w.r.t. which to
compute the Riesz representatives in case operator maps to functionals.
riesz_representatives
If True compute the Riesz representative of the residual.
"""

def __init__(self, RB, operator, rhs=None, product=None, riesz_representatives=False):
assert RB in operator.source
assert rhs is None \
or (rhs.source.is_scalar and rhs.range == operator.range and rhs.linear)
assert product is None or product.source == product.range == operator.range

self.__auto_init(locals())
self.residual_range = operator.range.empty()
self.residual_range_dims = []

def reduce(self):
if self.residual_range is not False:
with self.logger.block('Estimating residual range ...'):
try:
self.residual_range, self.residual_range_dims = \
estimate_image_hierarchical([self.operator], [self.rhs],
self.RB,
(self.residual_range, self.residual_range_dims),
orthonormalize=True, product=self.product,
riesz_representatives=self.riesz_representatives)
except ImageCollectionError as e:
self.logger.warning(f'Cannot compute range of {e.op}. Evaluation will be slow.')
self.residual_range = False

if self.residual_range is False:
operator = project(self.operator, None, self.RB)
return NonProjectedResidualOperator(operator, self.rhs, self.riesz_representatives, self.product)

with self.logger.block('Projecting residual operator ...'):
if self.riesz_representatives:
operator = project(self.operator, self.residual_range, self.RB, product=None)  # the product cancels out.
rhs = project(self.rhs, self.residual_range, None, product=None)
else:
operator = project(self.operator, self.residual_range, self.RB, product=self.product)
rhs = project(self.rhs, self.residual_range, None, product=self.product)

return ResidualOperator(operator, rhs)

[docs]    def reconstruct(self, u):
"""Reconstruct high-dimensional residual vector from reduced vector u."""
if self.residual_range is False:
if self.product:
norm = induced_norm(self.product)
return u * (u.l2_norm() / norm(u))[0]
else:
return u
else:
return self.residual_range[:u.dim].lincomb(u.to_numpy())

[docs]class ResidualOperator(Operator):
"""Instantiated by :class:ResidualReductor."""

def __init__(self, operator, rhs, name=None):
self.__auto_init(locals())
self.source = operator.source
self.range = operator.range
self.linear = operator.linear
self.rhs_vector = rhs.as_range_array() if rhs and not rhs.parametric else None

[docs]    def apply(self, U, mu=None):
V = self.operator.apply(U, mu=mu)
if self.rhs:
F = self.rhs_vector or self.rhs.as_range_array(mu)
if len(V) > 1:
V -= F[[0]*len(V)]
else:
V -= F
return V

def projected_to_subbasis(self, dim_range=None, dim_source=None, name=None):
return ResidualOperator(project_to_subbasis(self.operator, dim_range, dim_source),
project_to_subbasis(self.rhs, dim_range, None),
name=name)

[docs]class NonProjectedResidualOperator(ResidualOperator):
"""Instantiated by :class:ResidualReductor.

Not to be used directly.
"""

def __init__(self, operator, rhs, riesz_representatives, product):
super().__init__(operator, rhs)
self.__auto_init(locals())

[docs]    def apply(self, U, mu=None):
R = super().apply(U, mu=mu)
if self.product:
if self.riesz_representatives:
R_riesz = self.product.apply_inverse(R)
# divide by norm, except when norm is zero:
inversel2 = 1./R_riesz.l2_norm()
inversel2 = np.nan_to_num(inversel2)
R_riesz.scal(np.sqrt(R_riesz.pairwise_dot(R)) * inversel2)
return R_riesz
else:
# divide by norm, except when norm is zero:
inversel2 = 1./R.l2_norm()
inversel2 = np.nan_to_num(inversel2)
R.scal(np.sqrt(self.product.pairwise_apply2(R, R)) * inversel2)
return R
else:
return R

def projected_to_subbasis(self, dim_range=None, dim_source=None, name=None):
return self.with_(operator=project_to_subbasis(self.operator, None, dim_source))

[docs]class ImplicitEulerResidualReductor(BasicObject):
"""Reduced basis residual reductor with mass operator for implicit Euler timestepping.

Given an operator, mass and a functional, the concatenation of residual operator
with the Riesz isomorphism is given by::

riesz_residual.apply(U, U_old, mu)
== product.apply_inverse(operator.apply(U, mu) + 1/dt*mass.apply(U, mu) - 1/dt*mass.apply(U_old, mu)
- rhs.as_vector(mu))

This reductor determines a low-dimensional subspace of the image of a reduced
basis space under riesz_residual using :func:~pymor.algorithms.image.estimate_image_hierarchical,
computes an orthonormal basis residual_range of this range space and then
returns the Petrov-Galerkin projection ::

projected_riesz_residual
== riesz_residual.projected(range_basis=residual_range, source_basis=RB)

of the riesz_residual operator. Given reduced basis coefficient vectors u and u_old,
the dual norm of the residual can then be computed as ::

projected_riesz_residual.apply(u, u_old, mu).l2_norm()

Moreover, a reconstruct method is provided such that ::

residual_reductor.reconstruct(projected_riesz_residual.apply(u, u_old, mu))
== riesz_residual.apply(RB.lincomb(u), RB.lincomb(u_old), mu)

Parameters
----------
operator
See definition of riesz_residual.
mass
The mass operator. See definition of riesz_residual.
dt
The time step size. See definition of riesz_residual.
rhs
See definition of riesz_residual. If None, zero right-hand side is assumed.
RB
|VectorArray| containing a basis of the reduced space onto which to project.
product
Inner product |Operator| w.r.t. which to compute the Riesz representatives.
"""
def __init__(self, RB, operator, mass, dt, rhs=None, product=None):
assert RB in operator.source
assert rhs is None \
or rhs.source.is_scalar and rhs.range == operator.range and rhs.linear
assert product is None or product.source == product.range == operator.range

self.__auto_init(locals())
self.residual_range = operator.range.empty()
self.residual_range_dims = []

def reduce(self):
if self.residual_range is not False:
with self.logger.block('Estimating residual range ...'):
try:
self.residual_range, self.residual_range_dims = \
estimate_image_hierarchical([self.operator, self.mass], [self.rhs],
self.RB,
(self.residual_range, self.residual_range_dims),
orthonormalize=True, product=self.product,
riesz_representatives=True)
except ImageCollectionError as e:
self.logger.warning(f'Cannot compute range of {e.op}. Evaluation will be slow.')
self.residual_range = False

if self.residual_range is False:
operator = project(self.operator, None, self.RB)
mass = project(self.mass, None, self.RB)
return NonProjectedImplicitEulerResidualOperator(operator, mass, self.rhs, self.dt, self.product)

with self.logger.block('Projecting residual operator ...'):
operator = project(self.operator, self.residual_range, self.RB, product=None)  # the product always cancels out.
mass = project(self.mass, self.residual_range, self.RB, product=None)
rhs = project(self.rhs, self.residual_range, None, product=None)

return ImplicitEulerResidualOperator(operator, mass, rhs, self.dt)

[docs]    def reconstruct(self, u):
"""Reconstruct high-dimensional residual vector from reduced vector u."""
if self.residual_range is False:
if self.product:
norm = induced_norm(self.product)
return u * (u.l2_norm() / norm(u))[0]
else:
return u
else:
return self.residual_range[:u.dim].lincomb(u.to_numpy())

[docs]class ImplicitEulerResidualOperator(Operator):
"""Instantiated by :class:ImplicitEulerResidualReductor."""

def __init__(self, operator, mass, rhs, dt, name=None):
self.__auto_init(locals())
self.source = operator.source
self.range = operator.range
self.linear = operator.linear
self.rhs_vector = rhs.as_range_array() if rhs and not rhs.parametric else None

[docs]    def apply(self, U, U_old, mu=None):
V = self.operator.apply(U, mu=mu)
V.axpy(1./self.dt, self.mass.apply(U, mu=mu))
V.axpy(-1./self.dt, self.mass.apply(U_old, mu=mu))
if self.rhs:
F = self.rhs_vector or self.rhs.as_range_array(mu)
if len(V) > 1:
V -= F[[0]*len(V)]
else:
V -= F
return V

def projected_to_subbasis(self, dim_range=None, dim_source=None, name=None):
return ImplicitEulerResidualOperator(project_to_subbasis(self.operator, dim_range, dim_source),
project_to_subbasis(self.mass, dim_range, dim_source),
project_to_subbasis(self.rhs, dim_range, None),
self.dt,
name=name)

[docs]class NonProjectedImplicitEulerResidualOperator(ImplicitEulerResidualOperator):
"""Instantiated by :class:ImplicitEulerResidualReductor.

Not to be used directly.
"""

def __init__(self, operator, mass, rhs, dt, product):
super().__init__(operator, mass, rhs, dt)
self.product = product

[docs]    def apply(self, U, U_old, mu=None):
R = super().apply(U, U_old, mu=mu)
if self.product:
R_riesz = self.product.apply_inverse(R)
# divide by norm, except when norm is zero:
inversel2 = 1./R_riesz.l2_norm()
inversel2 = np.nan_to_num(inversel2)
R_riesz.scal(np.sqrt(R_riesz.pairwise_dot(R)) * inversel2)
return R_riesz
else:
return R

def projected_to_subbasis(self, dim_range=None, dim_source=None, name=None):
return self.with_(operator=project_to_subbasis(self.operator, None, dim_source),
mass=project_to_subbasis(self.mass, None, dim_source))