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()