def to_lti(self):
"""Convert model to |LTIModel|.
This method interprets the given model as an |LTIModel|
in the following way::
- self.operator -> A
self.rhs -> B
self.outputs -> C
None -> D
self.mass -> E
"""
if len(self.outputs) == 0:
raise ValueError('No outputs defined.')
if len(self.outputs) > 1:
raise NotImplementedError('Only one output supported.')
A = -self.operator
B = self.rhs
C = next(iter(self.outputs.values()))
E = self.mass
if not all(op.linear for op in [A, B, C, E]):
raise ValueError('Operators not linear.')
from pymor.models.iosys import LTIModel
return LTIModel(A, B, C, E=E, visualizer=self.visualizer)
def create_cl_fom(Re=110, level=2, palpha=1e-3, control='bc'):
"""Create model which is used to evaluate the H2-Gap norm."""
setup_str = 'lvl_' + str(level) + ('_' + control if control is not None else '') \
+ '_re_' + str(Re) + ('_palpha_' + str(palpha) if control == 'bc' else '')
fom = load_fom(Re, level, palpha, control)
Bra = fom.B.as_range_array()
Cva = fom.C.as_source_array()
Z = solve_ricc_lrcf(fom.A, fom.E, Bra, Cva, trans=False)
K = fom.E.apply(Z).lincomb(Z.dot(Cva).T)
KC = LowRankOperator(K, np.eye(len(K)), Cva)
mKB = cat_arrays([-K, Bra]).to_numpy().T
mKBop = NumpyMatrixOperator(mKB)
mKBop_proj = LerayProjectedOperator(mKBop,
fom.A.source.G,
fom.A.source.E,
projection_space='range')
cl_fom = LTIModel(fom.A - KC, mKBop_proj, fom.C, None, fom.E)
with open(setup_str + '/cl_fom', 'wb') as cl_fom_file:
pickle.dump({'cl_fom': cl_fom}, cl_fom_file)
def get_gap_rom(rom):
"""Based on a rom, create model which is used to evaluate H2-Gap norm."""
A = to_matrix(rom.A, format='dense')
B = to_matrix(rom.B, format='dense')
C = to_matrix(rom.C, format='dense')
if isinstance(rom.E, IdentityOperator):
P = spla.solve_continuous_are(A.T,
C.T,
B.dot(B.T),
np.eye(len(C)),
balanced=False)
F = P @ C.T
else:
E = to_matrix(rom.E, format='dense')
P = spla.solve_continuous_are(A.T,
C.T,
B.dot(B.T),
np.eye(len(C)),
e=E.T,
balanced=False)
F = E @ P @ C.T
AF = A - F @ C
mFB = np.concatenate((-F, B), axis=1)
return LTIModel.from_matrices(
AF, mFB, C, E=None if isinstance(rom.E, IdentityOperator) else E)
def to_lti(self):
"""Convert model to |LTIModel|.
This method interprets the given model as an |LTIModel|
in the following way::
- self.operator -> A
self.rhs -> B
self.output_functional -> C
None -> D
self.mass -> E
"""
if self.output_functional is None:
raise ValueError('No output defined.')
A = -self.operator
B = self.rhs
C = self.output_functional
E = self.mass
if not all(op.linear for op in [A, B, C, E]):
raise ValueError('Operators not linear.')
from pymor.models.iosys import LTIModel
return LTIModel(A,
B,
C,
E=E,
parameter_space=self.parameter_space,
visualizer=self.visualizer)
def _poles_b_c_to_lti(poles, b, c):
r"""Create an |LTIModel| from poles and residue rank-1 factors.
Returns an |LTIModel| with real matrices such that its transfer
function is
.. math::
\sum_{i = 1}^r \frac{c_i b_i^T}{s - \lambda_i}
where :math:`\lambda_i, b_i, c_i` are the poles and residue rank-1
factors.
Parameters
----------
poles
Sequence of poles.
b
|VectorArray| of right residue rank-1 factors.
c
|VectorArray| of left residue rank-1 factors.
Returns
-------
|LTIModel|.
"""
A, B, C = [], [], []
for i, pole in enumerate(poles):
if pole.imag == 0:
A.append(pole.real)
B.append(b[i].to_numpy().real)
C.append(c[i].to_numpy().real.T)
elif pole.imag > 0:
A.append([[pole.real, pole.imag],
[-pole.imag, pole.real]])
bi = b[i].to_numpy()
B.append(np.vstack([2 * bi.real, -2 * bi.imag]))
ci = c[i].to_numpy()
C.append(np.hstack([ci.real.T, ci.imag.T]))
A = spla.block_diag(*A)
B = np.vstack(B)
C = np.hstack(C)
return LTIModel.from_matrices(A, B, C)
def _poles_b_c_to_lti(poles, b, c):
r"""Create an |LTIModel| from poles and residue rank-1 factors.
Returns an |LTIModel| with real matrices such that its transfer
function is
.. math::
\sum_{i = 1}^r \frac{c_i b_i^T}{s - \lambda_i}
where :math:`\lambda_i, b_i, c_i` are the poles and residue rank-1
factors.
Parameters
----------
poles
Sequence of poles.
b
|NumPy array| of shape `(rom.order, rom.dim_input)`.
c
|NumPy array| of shape `(rom.order, rom.dim_output)`.
Returns
-------
|LTIModel|.
"""
A, B, C = [], [], []
for i, pole in enumerate(poles):
if pole.imag == 0:
A.append(pole.real)
B.append(b[i].real)
C.append(c[i].real[:, np.newaxis])
elif pole.imag > 0:
A.append([[pole.real, pole.imag],
[-pole.imag, pole.real]])
B.append(np.vstack([2 * b[i].real, -2 * b[i].imag]))
C.append(np.hstack([c[i].real[:, np.newaxis], c[i].imag[:, np.newaxis]]))
A = spla.block_diag(*A)
B = np.vstack(B)
C = np.hstack(C)
return LTIModel.from_matrices(A, B, C)
def reduce(self, sigma, b, c):
"""Realization-independent tangential Hermite interpolation.
Parameters
----------
sigma
Interpolation points (closed under conjugation), list of length `r`.
b
Right tangential directions, |NumPy array| of shape `(fom.input_dim, r)`.
c
Left tangential directions, |NumPy array| of shape `(fom.output_dim, r)`.
Returns
-------
lti
The reduced-order |LTIModel| interpolating the transfer function of `fom`.
"""
r = len(sigma)
assert isinstance(b, np.ndarray) and b.shape == (self.fom.input_dim, r)
assert isinstance(c,
np.ndarray) and c.shape == (self.fom.output_dim, r)
# rescale tangential directions (to avoid overflow or underflow)
if b.shape[0] > 1:
for i in range(r):
b[:, i] /= spla.norm(b[:, i])
else:
b = np.ones((1, r))
if c.shape[0] > 1:
for i in range(r):
c[:, i] /= spla.norm(c[:, i])
else:
c = np.ones((1, r))
# matrices of the interpolatory LTI system
Er = np.empty((r, r), dtype=complex)
Ar = np.empty((r, r), dtype=complex)
Br = np.empty((r, self.fom.input_dim), dtype=complex)
Cr = np.empty((self.fom.output_dim, r), dtype=complex)
Hs = [self.fom.eval_tf(s, mu=self.mu) for s in sigma]
dHs = [self.fom.eval_dtf(s, mu=self.mu) for s in sigma]
for i in range(r):
for j in range(r):
if i != j:
Er[i, j] = -c[:, i].dot(
(Hs[i] - Hs[j]).dot(b[:, j])) / (sigma[i] - sigma[j])
Ar[i, j] = -c[:, i].dot(
(sigma[i] * Hs[i] - sigma[j] * Hs[j])).dot(b[:, j]) / (
sigma[i] - sigma[j])
else:
Er[i, i] = -c[:, i].dot(dHs[i].dot(b[:, i]))
Ar[i, i] = -c[:, i].dot(
(Hs[i] + sigma[i] * dHs[i]).dot(b[:, i]))
Br[i, :] = Hs[i].T.dot(c[:, i])
Cr[:, i] = Hs[i].dot(b[:, i])
# transform the system to have real matrices
T = np.zeros((r, r), dtype=complex)
for i in range(r):
if sigma[i].imag == 0:
T[i, i] = 1
else:
indices = np.nonzero(
np.isclose(sigma[i + 1:], sigma[i].conjugate()))[0]
if len(indices) > 0:
j = i + 1 + indices[0]
T[i, i] = 1
T[i, j] = 1
T[j, i] = -1j
T[j, j] = 1j
Er = (T.dot(Er).dot(T.conj().T)).real
Ar = (T.dot(Ar).dot(T.conj().T)).real
Br = (T.dot(Br)).real
Cr = (Cr.dot(T.conj().T)).real
return LTIModel.from_matrices(Ar,
Br,
Cr,
D=None,
E=Er,
cont_time=self.fom.cont_time)
def build_rom(self, projected_operators, error_estimator):
return LTIModel(error_estimator=error_estimator, **projected_operators)
def reduce(self, sigma, b, c):
"""Realization-independent tangential Hermite interpolation.
Parameters
----------
sigma
Interpolation points (closed under conjugation), sequence of
length `r`.
b
Right tangential directions, |NumPy array| of shape
`(r, fom.dim_input)`.
c
Left tangential directions, |NumPy array| of shape
`(r, fom.dim_output)`.
Returns
-------
lti
The reduced-order |LTIModel| interpolating the transfer
function of `fom`.
"""
r = len(sigma)
assert b.shape == (r, self.fom.dim_input)
assert c.shape == (r, self.fom.dim_output)
# rescale tangential directions (to avoid overflow or underflow)
b = b * (1 / np.linalg.norm(b)) if b.shape[1] > 1 else np.ones((r, 1))
c = c * (1 / np.linalg.norm(c)) if c.shape[1] > 1 else np.ones((r, 1))
# matrices of the interpolatory LTI system
Er = np.empty((r, r), dtype=np.complex_)
Ar = np.empty((r, r), dtype=np.complex_)
Br = np.empty((r, self.fom.dim_input), dtype=np.complex_)
Cr = np.empty((self.fom.dim_output, r), dtype=np.complex_)
Hs = [self.fom.eval_tf(s, mu=self.mu) for s in sigma]
dHs = [self.fom.eval_dtf(s, mu=self.mu) for s in sigma]
for i in range(r):
for j in range(r):
if i != j:
Er[i, j] = -c[i] @ (Hs[i] - Hs[j]) @ b[j] / (sigma[i] -
sigma[j])
Ar[i, j] = (
-c[i] @ (sigma[i] * Hs[i] - sigma[j] * Hs[j]) @ b[j] /
(sigma[i] - sigma[j]))
else:
Er[i, i] = -c[i] @ dHs[i] @ b[i]
Ar[i, i] = -c[i] @ (Hs[i] + sigma[i] * dHs[i]) @ b[i]
Br[i, :] = Hs[i].T @ c[i]
Cr[:, i] = Hs[i] @ b[i]
# transform the system to have real matrices
T = np.zeros((r, r), dtype=np.complex_)
for i in range(r):
if sigma[i].imag == 0:
T[i, i] = 1
else:
j = np.argmin(np.abs(sigma - sigma[i].conjugate()))
if i < j:
T[i, i] = 1
T[i, j] = 1
T[j, i] = -1j
T[j, j] = 1j
Er = (T @ Er @ T.conj().T).real
Ar = (T @ Ar @ T.conj().T).real
Br = (T @ Br).real
Cr = (Cr @ T.conj().T).real
return LTIModel.from_matrices(Ar,
Br,
Cr,
None,
Er,
cont_time=self.fom.cont_time)
def main(
n: int = Argument(100, help='Order of the FOM.'),
r: int = Argument(10, help='Order of the ROMs.'),
):
"""Synthetic parametric demo.
See the `MOR Wiki page <http://modelreduction.org/index.php/Synthetic_parametric_model>`_.
"""
# Model
# set coefficients
a = -np.linspace(1e1, 1e3, n // 2)
b = np.linspace(1e1, 1e3, n // 2)
c = np.ones(n // 2)
d = np.zeros(n // 2)
# build 2x2 submatrices
aa = np.empty(n)
aa[::2] = a
aa[1::2] = a
bb = np.zeros(n)
bb[::2] = b
# set up system matrices
Amu = sps.diags(aa, format='csc')
A0 = sps.diags([bb, -bb], [1, -1], shape=(n, n), format='csc')
B = np.zeros((n, 1))
B[::2, 0] = 2
C = np.empty((1, n))
C[0, ::2] = c
C[0, 1::2] = d
# form operators
A0 = NumpyMatrixOperator(A0)
Amu = NumpyMatrixOperator(Amu)
B = NumpyMatrixOperator(B)
C = NumpyMatrixOperator(C)
A = A0 + Amu * ProjectionParameterFunctional('mu')
# form a model
lti = LTIModel(A, B, C)
mu_list = [1 / 50, 1 / 20, 1 / 10, 1 / 5, 1 / 2, 1]
w = np.logspace(0.5, 3.5, 200)
# System poles
fig, ax = plt.subplots()
for mu in mu_list:
poles = lti.poles(mu=mu)
ax.plot(poles.real, poles.imag, '.', label=fr'$\mu = {mu}$')
ax.set_title('System poles')
ax.legend()
plt.show()
# Magnitude plot
fig, ax = plt.subplots()
for mu in mu_list:
lti.mag_plot(w, ax=ax, mu=mu, label=fr'$\mu = {mu}$')
ax.legend()
plt.show()
# Hankel singular values
fig, ax = plt.subplots()
for mu in mu_list:
hsv = lti.hsv(mu=mu)
ax.semilogy(range(1, len(hsv) + 1), hsv, '.-', label=fr'$\mu = {mu}$')
ax.set_title('Hankel singular values')
ax.legend()
plt.show()
# System norms
for mu in mu_list:
print(f'mu = {mu}:')
print(f' H_2-norm of the full model: {lti.h2_norm(mu=mu):e}')
if config.HAVE_SLYCOT:
print(
f' H_inf-norm of the full model: {lti.hinf_norm(mu=mu):e}')
print(f' Hankel-norm of the full model: {lti.hankel_norm(mu=mu):e}')
# Model order reduction
run_mor_method_param(lti, r, w, mu_list, BTReductor, 'BT')
run_mor_method_param(lti, r, w, mu_list, IRKAReductor, 'IRKA')
C[0, 1::2] = d
# In[ ]:
A0 = NumpyMatrixOperator(A0)
Amu = NumpyMatrixOperator(Amu)
B = NumpyMatrixOperator(B)
C = NumpyMatrixOperator(C)
# In[ ]:
A = A0 + Amu * ProjectionParameterFunctional('mu', ())
# In[ ]:
lti = LTIModel(A, B, C)
# # Magnitude plot
# In[ ]:
mu_list_short = [1 / 50, 1 / 20, 1 / 10, 1 / 5, 1 / 2, 1]
# In[ ]:
w = np.logspace(0.5, 3.5, 200)
fig, ax = plt.subplots()
for mu in mu_list_short:
lti.mag_plot(w, ax=ax, mu=mu, label=fr'$\mu = {mu}$')
ax.legend()
def reduce(self, sigma, b, c):
"""Realization-independent tangential Hermite interpolation.
Parameters
----------
sigma
Interpolation points (closed under conjugation), list of length `r`.
b
Right tangential directions, |NumPy array| of shape `(fom.input_dim, r)`.
c
Left tangential directions, |NumPy array| of shape `(fom.output_dim, r)`.
Returns
-------
lti
The reduced-order |LTIModel| interpolating the transfer function of `fom`.
"""
r = len(sigma)
assert isinstance(b, np.ndarray) and b.shape == (self.fom.input_dim, r)
assert isinstance(c, np.ndarray) and c.shape == (self.fom.output_dim, r)
# rescale tangential directions (to avoid overflow or underflow)
if b.shape[0] > 1:
for i in range(r):
b[:, i] /= spla.norm(b[:, i])
else:
b = np.ones((1, r))
if c.shape[0] > 1:
for i in range(r):
c[:, i] /= spla.norm(c[:, i])
else:
c = np.ones((1, r))
# matrices of the interpolatory LTI system
Er = np.empty((r, r), dtype=complex)
Ar = np.empty((r, r), dtype=complex)
Br = np.empty((r, self.fom.input_dim), dtype=complex)
Cr = np.empty((self.fom.output_dim, r), dtype=complex)
Hs = [self.fom.eval_tf(s) for s in sigma]
dHs = [self.fom.eval_dtf(s) for s in sigma]
for i in range(r):
for j in range(r):
if i != j:
Er[i, j] = -c[:, i].dot((Hs[i] - Hs[j]).dot(b[:, j])) / (sigma[i] - sigma[j])
Ar[i, j] = -c[:, i].dot((sigma[i] * Hs[i] - sigma[j] * Hs[j])).dot(b[:, j]) / (sigma[i] - sigma[j])
else:
Er[i, i] = -c[:, i].dot(dHs[i].dot(b[:, i]))
Ar[i, i] = -c[:, i].dot((Hs[i] + sigma[i] * dHs[i]).dot(b[:, i]))
Br[i, :] = Hs[i].T.dot(c[:, i])
Cr[:, i] = Hs[i].dot(b[:, i])
# transform the system to have real matrices
T = np.zeros((r, r), dtype=complex)
for i in range(r):
if sigma[i].imag == 0:
T[i, i] = 1
else:
indices = np.nonzero(np.isclose(sigma[i + 1:], sigma[i].conjugate()))[0]
if len(indices) > 0:
j = i + 1 + indices[0]
T[i, i] = 1
T[i, j] = 1
T[j, i] = -1j
T[j, j] = 1j
Er = (T.dot(Er).dot(T.conj().T)).real
Ar = (T.dot(Ar).dot(T.conj().T)).real
Br = (T.dot(Br)).real
Cr = (Cr.dot(T.conj().T)).real
return LTIModel.from_matrices(Ar, Br, Cr, D=None, E=Er, cont_time=self.fom.cont_time)