# This file is part of the pyMOR project (http://www.pymor.org).
# Copyright 2013-2020 pyMOR developers and contributors. All rights reserved.
# License: BSD 2-Clause License (http://opensource.org/licenses/BSD-2-Clause)
import numpy as np
import scipy.linalg as spla
from pymor.algorithms.gram_schmidt import gram_schmidt, gram_schmidt_biorth
from pymor.algorithms.riccati import solve_ricc_lrcf, solve_pos_ricc_lrcf
from pymor.core.base import BasicObject
from pymor.models.iosys import LTIModel
from pymor.operators.constructions import IdentityOperator
from pymor.parameters.base import Mu
from pymor.reductors.basic import LTIPGReductor
[docs]class GenericBTReductor(BasicObject):
"""Generic Balanced Truncation reductor.
Parameters
----------
fom
The full-order |LTIModel| to reduce.
mu
|Parameter values|.
"""
def __init__(self, fom, mu=None):
assert isinstance(fom, LTIModel)
if not isinstance(mu, Mu):
mu = fom.parameters.parse(mu)
assert fom.parameters.assert_compatible(mu)
self.fom = fom
self.mu = mu
self.V = None
self.W = None
self._pg_reductor = None
self._sv_U_V_cache = None
def _gramians(self):
"""Return low-rank Cholesky factors of Gramians."""
raise NotImplementedError
def _sv_U_V(self):
"""Return singular values and vectors."""
if self._sv_U_V_cache is None:
cf, of = self._gramians()
U, sv, Vh = spla.svd(self.fom.E.apply2(of, cf, mu=self.mu), lapack_driver='gesvd')
self._sv_U_V_cache = (sv, U.T, Vh)
return self._sv_U_V_cache
[docs] def error_bounds(self):
"""Returns error bounds for all possible reduced orders."""
raise NotImplementedError
[docs] def reduce(self, r=None, tol=None, projection='bfsr'):
"""Generic Balanced Truncation.
Parameters
----------
r
Order of the reduced model if `tol` is `None`, maximum order if `tol` is specified.
tol
Tolerance for the error bound if `r` is `None`.
projection
Projection method used:
- `'sr'`: square root method
- `'bfsr'`: balancing-free square root method (default, since it avoids scaling by
singular values and orthogonalizes the projection matrices, which might make it more
accurate than the square root method)
- `'biorth'`: like the balancing-free square root method, except it biorthogonalizes the
projection matrices (using :func:`~pymor.algorithms.gram_schmidt.gram_schmidt_biorth`)
Returns
-------
rom
Reduced-order model.
"""
assert r is not None or tol is not None
assert r is None or 0 < r < self.fom.order
assert projection in ('sr', 'bfsr', 'biorth')
cf, of = self._gramians()
sv, sU, sV = self._sv_U_V()
# find reduced order if tol is specified
if tol is not None:
error_bounds = self.error_bounds()
r_tol = np.argmax(error_bounds <= tol) + 1
r = r_tol if r is None else min(r, r_tol)
if r > min(len(cf), len(of)):
raise ValueError('r needs to be smaller than the sizes of Gramian factors.')
# compute projection matrices
self.V = cf.lincomb(sV[:r])
self.W = of.lincomb(sU[:r])
if projection == 'sr':
alpha = 1 / np.sqrt(sv[:r])
self.V.scal(alpha)
self.W.scal(alpha)
elif projection == 'bfsr':
gram_schmidt(self.V, atol=0, rtol=0, copy=False)
gram_schmidt(self.W, atol=0, rtol=0, copy=False)
elif projection == 'biorth':
gram_schmidt_biorth(self.V, self.W, product=self.fom.E, copy=False)
# find reduced-order model
if self.fom.parametric:
fom_mu = self.fom.with_(**{op: getattr(self.fom, op).assemble(mu=self.mu)
for op in ['A', 'B', 'C', 'D', 'E']})
else:
fom_mu = self.fom
self._pg_reductor = LTIPGReductor(fom_mu, self.W, self.V, projection in ('sr', 'biorth'))
rom = self._pg_reductor.reduce()
return rom
[docs] def reconstruct(self, u):
"""Reconstruct high-dimensional vector from reduced vector `u`."""
return self._pg_reductor.reconstruct(u)
[docs]class BTReductor(GenericBTReductor):
"""Standard (Lyapunov) Balanced Truncation reductor.
See Section 7.3 in [A05]_.
Parameters
----------
fom
The full-order |LTIModel| to reduce.
mu
|Parameter values|.
"""
def _gramians(self):
return self.fom.gramian('c_lrcf', mu=self.mu), self.fom.gramian('o_lrcf', mu=self.mu)
[docs] def error_bounds(self):
sv = self._sv_U_V()[0]
return 2 * sv[:0:-1].cumsum()[::-1]
[docs]class LQGBTReductor(GenericBTReductor):
r"""Linear Quadratic Gaussian (LQG) Balanced Truncation reductor.
See Section 3 in [MG91]_.
Parameters
----------
fom
The full-order |LTIModel| to reduce.
mu
|Parameter values|.
solver_options
The solver options to use to solve the Riccati equations.
"""
def __init__(self, fom, mu=None, solver_options=None):
super().__init__(fom, mu=mu)
self.solver_options = solver_options
def _gramians(self):
A, B, C, E = (getattr(self.fom, op).assemble(mu=self.mu)
for op in ['A', 'B', 'C', 'E'])
if isinstance(E, IdentityOperator):
E = None
options = self.solver_options
cf = solve_ricc_lrcf(A, E, B.as_range_array(), C.as_source_array(),
trans=False, options=options)
of = solve_ricc_lrcf(A, E, B.as_range_array(), C.as_source_array(),
trans=True, options=options)
return cf, of
[docs] def error_bounds(self):
sv = self._sv_U_V()[0]
return 2 * (sv[:0:-1] / np.sqrt(1 + sv[:0:-1]**2)).cumsum()[::-1]
[docs]class BRBTReductor(GenericBTReductor):
r"""Bounded Real (BR) Balanced Truncation reductor.
See [A05]_ (Section 7.5.3) and [OJ88]_.
Parameters
----------
fom
The full-order |LTIModel| to reduce.
gamma
Upper bound for the :math:`\mathcal{H}_\infty`-norm.
mu
|Parameter values|.
solver_options
The solver options to use to solve the positive Riccati equations.
"""
def __init__(self, fom, gamma=1, mu=None, solver_options=None):
super().__init__(fom, mu=mu)
self.gamma = gamma
self.solver_options = solver_options
def _gramians(self):
A, B, C, E = (getattr(self.fom, op).assemble(mu=self.mu)
for op in ['A', 'B', 'C', 'E'])
if isinstance(E, IdentityOperator):
E = None
options = self.solver_options
cf = solve_pos_ricc_lrcf(A, E, B.as_range_array(), C.as_source_array(),
R=self.gamma**2 * np.eye(self.fom.output_dim) if self.gamma != 1 else None,
trans=False, options=options)
of = solve_pos_ricc_lrcf(A, E, B.as_range_array(), C.as_source_array(),
R=self.gamma**2 * np.eye(self.fom.input_dim) if self.gamma != 1 else None,
trans=True, options=options)
return cf, of
[docs] def error_bounds(self):
sv = self._sv_U_V()[0]
return 2 * sv[:0:-1].cumsum()[::-1]