def test_pinv(self):
"""test pinv"""
# construct non-operator tensor train
cores = [self.cores[i][:, :, 0:1, :] for i in range(self.order)]
t = TT(cores)
# compute pseudoinverse
t_pinv = TT.pinv(t, self.order - 1)
# matricize tensor trains
t = t.full().reshape([np.prod(self.row_dims[:-1]), self.row_dims[-1]])
t_pinv = t_pinv.full().reshape(
[np.prod(self.row_dims[:-1]), self.row_dims[-1]]).transpose()
# compute relative errors
rel_err_1 = np.linalg.norm(t.dot(t_pinv).dot(t) -
t) / np.linalg.norm(t)
rel_err_2 = np.linalg.norm(t_pinv.dot(t).dot(t_pinv) -
t_pinv) / np.linalg.norm(t_pinv)
rel_err_3 = np.linalg.norm((t.dot(t_pinv)).transpose() -
t.dot(t_pinv)) / np.linalg.norm(
t.dot(t_pinv))
rel_err_4 = np.linalg.norm((t_pinv.dot(t)).transpose() -
t_pinv.dot(t)) / np.linalg.norm(
t_pinv.dot(t))
# check if relative errors are smaller than tolerance
self.assertLess(rel_err_1, self.tol)
self.assertLess(rel_err_2, self.tol)
self.assertLess(rel_err_3, self.tol)
self.assertLess(rel_err_4, self.tol)
def mandy_fm(x, y, psi, threshold=0, add_one=True):
"""Multidimensional Approximation of Nonlinear Dynamics (MANDy)
Function-major approach for construction of the tensor train xi. See [1]_ for details.
Parameters
----------
x: ndarray
snapshot matrix of size d x m (e.g., coordinates)
y: ndarray
corresponding snapshot matrix of size d x m (e.g., derivatives)
psi: list of lambda functions
list of basis functions
threshold: float, optional
threshold for SVDs, default is 0
add_one: bool, optional
whether to add the basis function 1 to the cores or not, default is True
Returns
-------
xi: instance of TT class
tensor train of coefficients for chosen basis functions
References
----------
.. [1] P. Gelß, S. Klus, J. Eisert, C. Schütte, "Multidimensional Approximation of Nonlinear Dynamical Systems",
arXiv:1809.02448, 2018
"""
# parameters
d = x.shape[0]
m = x.shape[1]
p = len(psi)
# define cores as empty arrays
cores = [np.zeros([1, d + add_one, 1, m])
] + [np.zeros([m, d + add_one, 1, m]) for _ in range(1, p)]
# insert elements of first core
if add_one is True:
for j in range(m):
cores[0][0, 0, 0, j] = 1
cores[0][0, 1:, 0,
j] = np.array([psi[0](x[k, j]) for k in range(d)])
else:
for j in range(m):
cores[0][0, :, 0,
j] = np.array([psi[0](x[k, j]) for k in range(d)])
# insert elements of subsequent cores
for i in range(1, p):
if add_one is True:
for j in range(m):
cores[i][j, 0, 0, j] = 1
cores[i][j, 1:, 0,
j] = np.array([psi[i](x[k, j]) for k in range(d)])
else:
for j in range(m):
cores[0][0, :, 0,
j] = np.array([psi[0](x[k, j]) for k in range(d)])
# append core containing unit vectors
cores.append(np.eye(m).reshape(m, m, 1, 1))
# construct tensor train
xi = TT(cores)
# compute pseudoinverse of xi
xi = xi.pinv(p, threshold=threshold, ortho_r=False)
# multiply last core with y
xi.cores[p] = (
xi.cores[p].reshape([xi.ranks[p], m]) @ y.transpose()).reshape(
xi.ranks[p], d, 1, 1)
# set new row dimension
xi.row_dims[p] = d
return xi
