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')
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()
#plot.sho()
# In[ ]:
w_list = np.logspace(0.5, 3.5, 200)
mu_list = np.linspace(1 / 50, 1, 50)
lti_w_mu = np.zeros((len(w_list), len(mu_list)))
for i, mu in enumerate(mu_list):
lti_w_mu[:, i] = spla.norm(lti.freq_resp(w_list, mu=mu), axis=(1, 2))
# In[ ]:
fig, ax = plt.subplots()