Source code for pymor.reductors.bt

# This file is part of the pyMOR project (http://www.pymor.org).
# Copyright 2013-2019 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.interfaces import BasicInterface
from pymor.models.iosys import LTIModel
from pymor.operators.constructions import IdentityOperator
from pymor.reductors.basic import LTIPGReductor


[docs]class GenericBTReductor(BasicInterface): """Generic Balanced Truncation reductor. Parameters ---------- fom The full-order |LTIModel| to reduce. mu |Parameter|. """ def __init__(self, fom, mu=None): assert isinstance(fom, LTIModel) self.fom = fom self.mu = fom.parse_parameter(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']}, parameter_space=None) 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|. """ 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|. 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|. 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]