def test_als_eig(self):
"""test for ALS with solver='eig'"""
# solve eigenvalue problem in TT format (number_ev=1 and operator_gevp is defined as identity tensor)
_, eigentensor = evp.als(self.operator_tt,
self.initial_tt,
operator_gevp=self.operator_gevp,
repeats=10)
self.assertTrue(isinstance(eigentensor, TT))
# solve eigenvalue problem in TT format (number_ev=self.number_ev)
eigenvalues_tt, eigenvectors_tt = evp.als(self.operator_tt,
self.initial_tt,
number_ev=self.number_ev,
repeats=10)
# solve eigenvalue problem in matrix format
eigenvalues_mat, eigenvectors_mat = splin.eigs(self.operator_mat,
k=self.number_ev)
eigenvalues_mat = np.real(eigenvalues_mat)
eigenvectors_mat = np.real(eigenvectors_mat)
idx = eigenvalues_mat.argsort()[::-1]
eigenvalues_mat = eigenvalues_mat[idx]
eigenvectors_mat = eigenvectors_mat[:, idx]
# compute relative error between exact and approximate eigenvalues
rel_err_val = []
for i in range(self.number_ev):
rel_err_val.append(
np.abs(eigenvalues_mat[i] - eigenvalues_tt[i]) /
eigenvalues_mat[i])
# compute relative error between exact and approximate eigenvectors
rel_err_vec = []
for i in range(self.number_ev):
norm_1 = np.linalg.norm(eigenvectors_mat[:, i] +
eigenvectors_tt[i].matricize())
norm_2 = np.linalg.norm(eigenvectors_mat[:, i] -
eigenvectors_tt[i].matricize())
rel_err_vec.append(
np.amin([norm_1, norm_2]) /
np.linalg.norm(eigenvectors_mat[:, i]))
# check if relative errors are smaller than tolerance
for i in range(self.number_ev):
self.assertLess(rel_err_val[i], self.tol_eigval)
self.assertLess(rel_err_vec[i], self.tol_eigvec)
def test_als_eigh(self):
"""test for ALS with solver='eigh'"""
# make problem symmetric
self.operator_tt = 0.5 * (self.operator_tt +
self.operator_tt.transpose())
self.operator_mat = 0.5 * (self.operator_mat +
self.operator_mat.transpose())
# solve eigenvalue problem in TT format (number_ev=self.number_ev)
eigenvalues_tt, eigenvectors_tt = evp.als(self.operator_tt,
self.initial_tt,
number_ev=self.number_ev,
repeats=10,
solver='eigh')
# solve eigenvalue problem in matrix format
eigenvalues_mat, eigenvectors_mat = splin.eigs(self.operator_mat,
k=self.number_ev)
eigenvalues_mat = np.real(eigenvalues_mat)
eigenvectors_mat = np.real(eigenvectors_mat)
idx = eigenvalues_mat.argsort()[::-1]
eigenvalues_mat = eigenvalues_mat[idx]
eigenvectors_mat = eigenvectors_mat[:, idx]
# compute relative error between exact and approximate eigenvalues
rel_err_val = []
for i in range(self.number_ev):
rel_err_val.append(
np.abs(eigenvalues_mat[i] - eigenvalues_tt[i]) /
eigenvalues_mat[i])
# compute relative error between exact and approximate eigenvectors
rel_err_vec = []
for i in range(self.number_ev):
norm_1 = np.linalg.norm(eigenvectors_mat[:, i] +
eigenvectors_tt[i].matricize())
norm_2 = np.linalg.norm(eigenvectors_mat[:, i] -
eigenvectors_tt[i].matricize())
rel_err_vec.append(
np.amin([norm_1, norm_2]) /
np.linalg.norm(eigenvectors_mat[:, i]))
# check if relative errors are smaller than tolerance
for i in range(self.number_ev):
self.assertLess(rel_err_val[i], self.tol_eigval)
self.assertLess(rel_err_vec[i], self.tol_eigvec)
print('----------------------------------------------------------')
print('p_CO in atm Method TT ranks Closeness CPU time')
print('----------------------------------------------------------')
# construct eigenvalue problems to find the stationary distributions
# ------------------------------------------------------------------
for i in range(8):
# construct and solve eigenvalue problem for current CO pressure
operator = mdl.co_oxidation(
20, 10**(8 + p_CO_exp[i])).ortho_left().ortho_right()
initial_guess = tt.ones(operator.row_dims, [1] * operator.order,
ranks=R[i]).ortho_left().ortho_right()
with utl.timer() as time:
eigenvalues, solution = evp.als(tt.eye(operator.row_dims) + operator,
initial_guess,
repeats=10,
solver='eigs')
solution = (1 / solution.norm(p=1)) * solution
# compute turn-over frequency of CO2 desorption
tof.append(utl.two_cell_tof(solution, [2, 1], 1.7e5))
# print results
string_p_CO = '10^' + (p_CO_exp[i] >= 0) * '+' + str("%.1f" % p_CO_exp[i])
string_method = ' ' * 9 + 'EVP' + ' ' * 9
string_rank = (R[i] < 10) * ' ' + str(R[i]) + ' ' * 8
string_closeness = str("%.2e" % (operator @ solution).norm()) + ' ' * 6
string_time = (time.elapsed < 10) * ' ' + str("%.2f" % time.elapsed) + 's'
print(string_p_CO + string_method + string_rank + string_closeness +
string_time)
utl.progress('Load data', 100, dots=39)
# construct TT operator
# ---------------------
utl.progress('Construct operator', 0, dots=30)
operator = pf.perron_frobenius_2d(transitions, [n_states, n_states],
simulations)
utl.progress('Construct operator', 100, dots=30)
# approximate leading eigenfunctions of the Perron-Frobenius and Koopman operator
# -------------------------------------------------------------------------------
initial = tt.ones(operator.row_dims, [1] * operator.order, ranks=11)
utl.progress('Approximate eigenfunctions in the TT format', 0, dots=5)
eigenvalues_pf, eigenfunctions_pf = evp.als(operator, initial, number_ev, 3)
utl.progress('Approximate eigenfunctions in the TT format', 50, dots=5)
eigenvalues_km, eigenfunctions_km = evp.als(operator.transpose(), initial,
number_ev, 2)
utl.progress('Approximate eigenfunctions in the TT format', 100, dots=5)
# compute exact eigenvectors
# --------------------------
utl.progress('Compute exact eigenfunctions in matrix format', 0)
eigenvalues_pf_exact, eigenfunctions_pf_exact = splin.eigs(
operator.matricize(), k=number_ev)
utl.progress('Compute exact eigenfunctions in matrix format', 100)
# convert results to matrices
# ---------------------------
transitions = io.loadmat("data/QuadrupleWell3D_25x25x25_100.mat")["indices"] # load data
utl.progress('Load data', 100, dots=39)
# construct TT operator
# ---------------------
utl.progress('Construct operator', 0, dots=30)
operator = pf.perron_frobenius_3d(transitions, [n_states] * 3, simulations)
utl.progress('Construct operator', 100, dots=30)
# approximate leading eigenfunctions
# ----------------------------------
utl.progress('Approximate eigenfunctions in the TT format', 0, dots=5)
initial = tt.uniform(operator.row_dims, ranks=[1, 20, 10, 1])
eigenvalues, eigenfunctions = evp.als(operator, initial, number_ev, 2)
utl.progress('Approximate eigenfunctions in the TT format', 100, dots=5)
# compute exact eigenvectors
# --------------------------
utl.progress('Compute exact eigenfunctions in matrix format', 0)
eigenvalues_exact, eigenfunctions_exact = splin.eigs(operator.matricize(), k=number_ev)
idx = eigenvalues_exact.argsort()[::-1]
eigenvalues_exact = eigenvalues_exact[idx]
eigenfunctions_exact = eigenfunctions_exact[:, idx]
utl.progress('Compute exact eigenfunctions in matrix format', 100)
# convert results to matrices
# ----------------------------------
'/data/quadruple_well_transitions.npz')['transitions']
# construct TT operator
# ---------------------
start_time = utl.progress('Construct operator', 0)
operator = ulam.ulam_3d(transitions, [n_states] * 3, simulations)
utl.progress('Construct operator', 100, cpu_time=_time.time() - start_time)
# approximate leading eigenfunctions
# ----------------------------------
start_time = utl.progress('Approximate eigenfunctions in the TT format', 0)
initial = tt.uniform(operator.row_dims, ranks=[1, 20, 10, 1])
eigenvalues, eigenfunctions = evp.als(operator,
initial,
number_ev=number_ev,
repeats=2)
utl.progress('Approximate eigenfunctions in the TT format',
100,
cpu_time=_time.time() - start_time)
# compute exact eigenvectors
# --------------------------
start_time = utl.progress('Compute exact eigenfunctions in matrix format', 0)
eigenvalues_exact, eigenfunctions_exact = splin.eigs(operator.matricize(),
k=number_ev)
idx = eigenvalues_exact.argsort()[::-1]
eigenvalues_exact = eigenvalues_exact[idx]
eigenfunctions_exact = eigenfunctions_exact[:, idx]
utl.progress('Compute exact eigenfunctions in matrix format',