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 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 build_rom(self, projected_operators, error_estimator):
return LTIModel(error_estimator=error_estimator, **projected_operators)
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()