class ParabolicRBReductor(InstationaryRBReductor):
r"""Reduced Basis Reductor for parabolic equations.
This reductor uses :class:`~pymor.reductors.basic.InstationaryRBReductor` for the actual
RB-projection. The only addition is the assembly of an error estimator which
bounds the discrete l2-in time / energy-in space error similar to [GP05]_, [HO08]_
as follows:
.. math::
\left[ C_a^{-1}(\mu)\|e_N(\mu)\|^2 + \sum_{n=1}^{N} \Delta t\|e_n(\mu)\|^2_e \right]^{1/2}
\leq \left[ C_a^{-2}(\mu)\Delta t \sum_{n=1}^{N}\|\mathcal{R}^n(u_n(\mu), \mu)\|^2_{e,-1}
+ C_a^{-1}(\mu)\|e_0\|^2 \right]^{1/2}
Here, :math:`\|\cdot\|` denotes the norm induced by the problem's mass matrix
(e.g. the L^2-norm) and :math:`\|\cdot\|_e` is an arbitrary energy norm w.r.t.
which the space operator :math:`A(\mu)` is coercive, and :math:`C_a(\mu)` is a
lower bound for its coercivity constant. Finally, :math:`\mathcal{R}^n` denotes
the implicit Euler timestepping residual for the (fixed) time step size :math:`\Delta t`,
.. math::
\mathcal{R}^n(u_n(\mu), \mu) :=
f - M \frac{u_{n}(\mu) - u_{n-1}(\mu)}{\Delta t} - A(u_n(\mu), \mu),
where :math:`M` denotes the mass operator and :math:`f` the source term.
The dual norm of the residual is computed using the numerically stable projection
from [BEOR14]_.
Parameters
----------
fom
The |InstationaryModel| which is to be reduced.
RB
|VectorArray| containing the reduced basis on which to project.
product
The energy inner product |Operator| w.r.t. which the reduction error is
estimated and `RB` is orthonormalized.
coercivity_estimator
`None` or a |Parameterfunctional| returning a lower bound :math:`C_a(\mu)`
for the coercivity constant of `fom.operator` w.r.t. `product`.
"""
def __init__(self,
fom,
RB=None,
product=None,
coercivity_estimator=None,
check_orthonormality=None,
check_tol=None):
if not isinstance(fom.time_stepper, ImplicitEulerTimeStepper):
raise NotImplementedError
if fom.mass is not None and fom.mass.parametric and 't' in fom.mass.parameters:
raise NotImplementedError
super().__init__(fom,
RB,
product=product,
check_orthonormality=check_orthonormality,
check_tol=check_tol)
self.coercivity_estimator = coercivity_estimator
self.residual_reductor = ImplicitEulerResidualReductor(
self.bases['RB'],
fom.operator,
fom.mass,
fom.T / fom.time_stepper.nt,
rhs=fom.rhs,
product=product)
self.initial_residual_reductor = ResidualReductor(
self.bases['RB'],
IdentityOperator(fom.solution_space),
fom.initial_data,
product=fom.l2_product,
riesz_representatives=False)
def assemble_estimator(self):
residual = self.residual_reductor.reduce()
initial_residual = self.initial_residual_reductor.reduce()
estimator = ParabolicRBEstimator(
residual, self.residual_reductor.residual_range_dims,
initial_residual,
self.initial_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)
class ParabolicRBReductor(GenericRBReductor):
r"""Reduced Basis Reductor for parabolic equations.
This reductor uses :class:`~pymor.reductors.basic.GenericRBReductor` for the actual
RB-projection. The only addition is the assembly of an error estimator which
bounds the discrete l2-in time / energy-in space error similar to [GP05]_, [HO08]_
as follows:
.. math::
\left[ C_a^{-1}(\mu)\|e_N(\mu)\|^2 + \sum_{n=1}^{N} \Delta t\|e_n(\mu)\|^2_e \right]^{1/2}
\leq \left[ C_a^{-1}(\mu)\Delta t \sum_{n=1}^{N}\|\mathcal{R}^n(u_n(\mu), \mu)\|^2_{e,-1}
+ C_a^{-1}(\mu)\|e_0\|^2 \right]^{1/2}
Here, :math:`\|\cdot\|` denotes the norm induced by the problem's mass matrix
(e.g. the L^2-norm) and :math:`\|\cdot\|_e` is an arbitrary energy norm w.r.t.
which the space operator :math:`A(\mu)` is coercive, and :math:`C_a(\mu)` is a
lower bound for its coercivity constant. Finally, :math:`\mathcal{R}^n` denotes
the implicit Euler timestepping residual for the (fixed) time step size :math:`\Delta t`,
.. math::
\mathcal{R}^n(u_n(\mu), \mu) :=
f - M \frac{u_{n}(\mu) - u_{n-1}(\mu)}{\Delta t} - A(u_n(\mu), \mu),
where :math:`M` denotes the mass operator and :math:`f` the source term.
The dual norm of the residual is computed using the numerically stable projection
from [BEOR14]_.
.. warning::
The reduced basis `RB` is required to be orthonormal w.r.t. the given
energy product. If not, the projection of the initial values will be
computed incorrectly.
.. [GP05] M. A. Grepl, A. T. Patera, A Posteriori Error Bounds For Reduced-Basis
Approximations Of Parametrized Parabolic Partial Differential Equations,
M2AN 39(1), 157-181, 2005.
.. [HO08] B. Haasdonk, M. Ohlberger, Reduced basis method for finite volume
approximations of parametrized evolution equations,
M2AN 42(2), 277-302, 2008.
Parameters
----------
d
The |InstationaryDiscretization| which is to be reduced.
RB
|VectorArray| containing the reduced basis on which to project.
product
The energy inner product |Operator| w.r.t. the reduction error is estimated.
RB must be to be orthonomrmal w.r.t. this product!
coercivity_estimator
`None` or a |Parameterfunctional| returning a lower bound :math:`C_a(\mu)`
for the coercivity constant of `d.operator` w.r.t. `product`.
"""
def __init__(self, d, RB=None, product=None, coercivity_estimator=None):
assert isinstance(d.time_stepper, ImplicitEulerTimeStepper)
super().__init__(d, RB, product=product)
self.coercivity_estimator = coercivity_estimator
self.residual_reductor = ImplicitEulerResidualReductor(
self.RB,
d.operator,
d.mass,
d.T / d.time_stepper.nt,
functional=d.rhs,
product=product
)
self.initial_residual_reductor = ResidualReductor(
self.RB,
IdentityOperator(d.solution_space),
d.initial_data,
product=d.l2_product
)
def _reduce(self):
with self.logger.block('RB projection ...'):
rd = super()._reduce()
with self.logger.block('Assembling error estimator ...'):
residual = self.residual_reductor.reduce()
initial_residual = self.initial_residual_reductor.reduce()
estimator = ParabolicRBEstimator(residual, self.residual_reductor.residual_range_dims,
initial_residual, self.initial_residual_reductor.residual_range_dims,
self.coercivity_estimator)
rd = rd.with_(estimator=estimator)
return rd
class ParabolicRBReductor(GenericRBReductor):
r"""Reduced Basis Reductor for parabolic equations.
This reductor uses :class:`~pymor.reductors.basic.GenericRBReductor` for the actual
RB-projection. The only addition is the assembly of an error estimator which
bounds the discrete l2-in time / energy-in space error similar to [GP05]_, [HO08]_
as follows:
.. math::
\left[ C_a^{-1}(\mu)\|e_N(\mu)\|^2 + \sum_{n=1}^{N} \Delta t\|e_n(\mu)\|^2_e \right]^{1/2}
\leq \left[ C_a^{-1}(\mu)\Delta t \sum_{n=1}^{N}\|\mathcal{R}^n(u_n(\mu), \mu)\|^2_{e,-1}
+ C_a^{-1}(\mu)\|e_0\|^2 \right]^{1/2}
Here, :math:`\|\cdot\|` denotes the norm induced by the problem's mass matrix
(e.g. the L^2-norm) and :math:`\|\cdot\|_e` is an arbitrary energy norm w.r.t.
which the space operator :math:`A(\mu)` is coercive, and :math:`C_a(\mu)` is a
lower bound for its coercivity constant. Finally, :math:`\mathcal{R}^n` denotes
the implicit Euler timestepping residual for the (fixed) time step size :math:`\Delta t`,
.. math::
\mathcal{R}^n(u_n(\mu), \mu) :=
f - M \frac{u_{n}(\mu) - u_{n-1}(\mu)}{\Delta t} - A(u_n(\mu), \mu),
where :math:`M` denotes the mass operator and :math:`f` the source term.
The dual norm of the residual is computed using the numerically stable projection
from [BEOR14]_.
Parameters
----------
d
The |InstationaryDiscretization| which is to be reduced.
RB
|VectorArray| containing the reduced basis on which to project.
basis_is_orthonormal
Indicate whether or not the basis is orthonormal w.r.t. `product`.
product
The energy inner product |Operator| w.r.t. which the reduction error is
estimated and `RB` is orthonormalized.
coercivity_estimator
`None` or a |Parameterfunctional| returning a lower bound :math:`C_a(\mu)`
for the coercivity constant of `d.operator` w.r.t. `product`.
"""
def __init__(self, d, RB=None, basis_is_orthonormal=None,
product=None, coercivity_estimator=None):
assert isinstance(d.time_stepper, ImplicitEulerTimeStepper)
super().__init__(d, RB, basis_is_orthonormal=basis_is_orthonormal, product=product)
self.coercivity_estimator = coercivity_estimator
self.residual_reductor = ImplicitEulerResidualReductor(
self.RB,
d.operator,
d.mass,
d.T / d.time_stepper.nt,
rhs=d.rhs,
product=product
)
self.initial_residual_reductor = ResidualReductor(
self.RB,
IdentityOperator(d.solution_space),
d.initial_data,
product=d.l2_product,
riesz_representatives=False
)
def _reduce(self):
with self.logger.block('RB projection ...'):
rd = super()._reduce()
with self.logger.block('Assembling error estimator ...'):
residual = self.residual_reductor.reduce()
initial_residual = self.initial_residual_reductor.reduce()
estimator = ParabolicRBEstimator(residual, self.residual_reductor.residual_range_dims,
initial_residual, self.initial_residual_reductor.residual_range_dims,
self.coercivity_estimator)
rd = rd.with_(estimator=estimator)
return rd
class ParabolicRBReductor(GenericRBReductor):
r"""Reduced Basis Reductor for parabolic equations.
This reductor uses :class:`~pymor.reductors.basic.GenericRBReductor` for the actual
RB-projection. The only addition is the assembly of an error estimator which
bounds the discrete l2-in time / energy-in space error similar to [GP05]_, [HO08]_
as follows:
.. math::
\left[ C_a^{-1}(\mu)\|e_N(\mu)\|^2 + \sum_{n=1}^{N} \Delta t\|e_n(\mu)\|^2_e \right]^{1/2}
\leq \left[ C_a^{-1}(\mu)\Delta t \sum_{n=1}^{N}\|\mathcal{R}^n(u_n(\mu), \mu)\|^2_{e,-1}
+ C_a^{-1}(\mu)\|e_0\|^2 \right]^{1/2}
Here, :math:`\|\cdot\|` denotes the norm induced by the problem's mass matrix
(e.g. the L^2-norm) and :math:`\|\cdot\|_e` is an arbitrary energy norm w.r.t.
which the space operator :math:`A(\mu)` is coercive, and :math:`C_a(\mu)` is a
lower bound for its coercivity constant. Finally, :math:`\mathcal{R}^n` denotes
the implicit Euler timestepping residual for the (fixed) time step size :math:`\Delta t`,
.. math::
\mathcal{R}^n(u_n(\mu), \mu) :=
f - M \frac{u_{n}(\mu) - u_{n-1}(\mu)}{\Delta t} - A(u_n(\mu), \mu),
where :math:`M` denotes the mass operator and :math:`f` the source term.
The dual norm of the residual is computed using the numerically stable projection
from [BEOR14]_.
.. warning::
The reduced basis `RB` is required to be orthonormal w.r.t. the given
energy product. If not, the projection of the initial values will be
computed incorrectly.
.. [GP05] M. A. Grepl, A. T. Patera, A Posteriori Error Bounds For Reduced-Basis
Approximations Of Parametrized Parabolic Partial Differential Equations,
M2AN 39(1), 157-181, 2005.
.. [HO08] B. Haasdonk, M. Ohlberger, Reduced basis method for finite volume
approximations of parametrized evolution equations,
M2AN 42(2), 277-302, 2008.
Parameters
----------
discretization
The |InstationaryDiscretization| which is to be reduced.
RB
|VectorArray| containing the reduced basis on which to project.
product
The energy inner product |Operator| w.r.t. the reduction error is estimated.
RB must be to be orthonomrmal w.r.t. this product!
coercivity_estimator
`None` or a |Parameterfunctional| returning a lower bound :math:`C_a(\mu)`
for the coercivity constant of `discretization.operator` w.r.t. `product`.
"""
def __init__(self, d, RB=None, product=None, coercivity_estimator=None):
assert isinstance(d.time_stepper, ImplicitEulerTimeStepper)
super().__init__(d, RB, product=product)
self.coercivity_estimator = coercivity_estimator
self.residual_reductor = ImplicitEulerResidualReductor(
self.RB,
d.operator,
d.mass,
d.T / d.time_stepper.nt,
functional=d.rhs,
product=product)
self.initial_residual_reductor = ResidualReductor(
self.RB,
IdentityOperator(d.solution_space),
d.initial_data,
product=d.l2_product)
def _reduce(self):
with self.logger.block('RB projection ...'):
rd = super()._reduce()
with self.logger.block('Assembling error estimator ...'):
residual = self.residual_reductor.reduce()
initial_residual = self.initial_residual_reductor.reduce()
estimator = ParabolicRBEstimator(
residual, self.residual_reductor.residual_range_dims,
initial_residual,
self.initial_residual_reductor.residual_range_dims,
self.coercivity_estimator)
rd = rd.with_(estimator=estimator)
return rd
class ParabolicRBReductor(InstationaryRBReductor):
r"""Reduced Basis Reductor for parabolic equations.
This reductor uses :class:`~pymor.reductors.basic.InstationaryRBReductor` for the actual
RB-projection. The only addition is the assembly of an error estimator which
bounds the discrete l2-in time / energy-in space error similar to [GP05]_, [HO08]_
as follows:
.. math::
\left[ C_a^{-1}(\mu)\|e_N(\mu)\|^2 + \sum_{n=1}^{N} \Delta t\|e_n(\mu)\|^2_e \right]^{1/2}
\leq \left[ C_a^{-1}(\mu)\Delta t \sum_{n=1}^{N}\|\mathcal{R}^n(u_n(\mu), \mu)\|^2_{e,-1}
+ C_a^{-1}(\mu)\|e_0\|^2 \right]^{1/2}
Here, :math:`\|\cdot\|` denotes the norm induced by the problem's mass matrix
(e.g. the L^2-norm) and :math:`\|\cdot\|_e` is an arbitrary energy norm w.r.t.
which the space operator :math:`A(\mu)` is coercive, and :math:`C_a(\mu)` is a
lower bound for its coercivity constant. Finally, :math:`\mathcal{R}^n` denotes
the implicit Euler timestepping residual for the (fixed) time step size :math:`\Delta t`,
.. math::
\mathcal{R}^n(u_n(\mu), \mu) :=
f - M \frac{u_{n}(\mu) - u_{n-1}(\mu)}{\Delta t} - A(u_n(\mu), \mu),
where :math:`M` denotes the mass operator and :math:`f` the source term.
The dual norm of the residual is computed using the numerically stable projection
from [BEOR14]_.
Parameters
----------
fom
The |InstationaryModel| which is to be reduced.
RB
|VectorArray| containing the reduced basis on which to project.
product
The energy inner product |Operator| w.r.t. which the reduction error is
estimated and `RB` is orthonormalized.
coercivity_estimator
`None` or a |Parameterfunctional| returning a lower bound :math:`C_a(\mu)`
for the coercivity constant of `fom.operator` w.r.t. `product`.
"""
def __init__(self, fom, RB=None, product=None, coercivity_estimator=None,
check_orthonormality=None, check_tol=None):
assert isinstance(fom.time_stepper, ImplicitEulerTimeStepper)
super().__init__(fom, RB, product=product,
check_orthonormality=check_orthonormality, check_tol=check_tol)
self.coercivity_estimator = coercivity_estimator
self.residual_reductor = ImplicitEulerResidualReductor(
self.bases['RB'],
fom.operator,
fom.mass,
fom.T / fom.time_stepper.nt,
rhs=fom.rhs,
product=product
)
self.initial_residual_reductor = ResidualReductor(
self.bases['RB'],
IdentityOperator(fom.solution_space),
fom.initial_data,
product=fom.l2_product,
riesz_representatives=False
)
def assemble_estimator(self):
residual = self.residual_reductor.reduce()
initial_residual = self.initial_residual_reductor.reduce()
estimator = ParabolicRBEstimator(residual, self.residual_reductor.residual_range_dims,
initial_residual, self.initial_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)