def kedmd(X, Y, k, epsilon=0, evs=5, operator='P'):
'''
Kernel EDMD for the Koopman or Perron-Frobenius operator. The matrices X and Y
contain the input data.
:param k: kernel, see d3s.kernels
:param epsilon: regularization parameter
:param evs: number of eigenvalues/eigenvectors
:param operator: 'K' for Koopman or 'P' for Perron-Frobenius (note that the default is P here)
:return: eigenvalues d and eigenfunctions evaluated in X
'''
if isinstance(X, list): # e.g., for strings
n = len(X)
else:
n = X.shape[1]
G_0 = _kernels.gramian(X, k)
G_1 = _kernels.gramian2(X, Y, k)
if operator == 'K': G_1 = G_1.T
A = _sp.linalg.pinv(G_0 + epsilon * _np.eye(n), rcond=1e-15) @ G_1
d, V = sortEig(A, evs)
if operator == 'K': V = G_0 @ V
return (d, V)
k = kernels.gaussianKernel(0.5)
#%% apply kernel generator EDMD
epsilon = 0.1
S = np.einsum('ijk,ljk->ilk', Z, Z) # sigma \cdot sigma^T
G_00 = kernels.gramian(X, k)
G_10 = np.zeros((m, m))
for i in range(m):
for j in range(m):
G_10[i, j] = Y[:, i].T @ k.diff(X[:, i], X[:, j]) + 0.5 * np.sum(
S[:, :, i] * k.ddiff(X[:, i], X[:, j]), axis=(0, 1))
A, _, _, _ = np.linalg.lstsq(G_00, G_10, rcond=1e-12)
d, V = algorithms.sortEig(A, evs=m, which='LM')
W = kernels.gramian2(Omega.midpointGrid(), X, k) @ V
#%% plot eigenvalues
plt.figure()
plt.plot(d, '.')
plt.title('Spectrum')
#%% plot eigenfunctions
ind, = np.where(d < -0.1)
for i in ind[:6]:
plt.figure()
Omega.plot(np.real(W[:, i] / np.amax(abs(W[:, i]))))
plt.ylim((-1, 1))
plt.title('lambda = %f' % np.real(d[i]))
#%% direct kernelization for kernels with finite-dimensional feature space
k = kernels.gaussianKernel(0.3)
#%% apply kernel generator EDMD
evs = 4
G_00 = kernels.gramian(X, k)
G_10 = np.zeros((m, m))
for i in range(m):
for j in range(m):
G_10[i, j] = c0(X[:, i]) * k(X[:, i], X[:, j]) + np.sum(
c2(X[:, i]) * k.ddiff(X[:, i], X[:, j]), axis=(0, 1))
A, _, _, _ = np.linalg.lstsq(G_00, G_10, rcond=1e-12)
d, V = algorithms.sortEig(A, evs=evs, which='SR')
c = Omega.midpointGrid()
W = kernels.gramian2(c, X, k) @ V
#%% plot spectrum
plt.figure()
plt.plot(np.real(d), '.')
plt.title('Spectrum')
printVector(np.real(d), 'd')
#%% plot eigenfunctions
for i in range(evs):
plt.figure()
j = evs - i - 1
Omega.plot(np.real(W[:, j] / np.amax(abs(W[:, j]))))
wa = psi[i](c)
N = G_00[isBoundary, :].T
#%% solve BVP
epsilon = 1e-12
G_10 = np.zeros((m, m))
for i in range(m):
for j in range(m):
G_10[i, j] = -h**2/(2*m0) * k_a.laplace(X[:, i], X[:, j]) # no term related to the potential since V is 0 inside the box
d, V = ceig.hcgeig(G_10, G_00 + epsilon*np.eye(m), N)
# evaluate eigenfunctions in midpoints of the grid
Omega2 = domain.discretization(bounds, np.array([30, 30]))
c, _ = Omega2.vertexGrid()
W = kernels.gramian2(c, X, k_a) @ V
#%% plot spectrum
plt.figure(2)
plt.clf()
plt.plot(d, '.')
plt.title('Spectrum')
#%% plot eigenfunctions
ind, = np.where(abs(d) > 1.0)
fig = 3
for i in ind[-3:][::-1]:
#plt.figure(fig)
plt.figure()
plt.clf()
fig += 1