def add_noise(E, dt, noise, method):
"""Treatment of additive noise for ensembles.
Refs: `bib.raanes2014ext`
"""
if noise.C == 0:
return E
N, Nx = E.shape
A, mu = center(E)
Q12 = noise.C.Left
Q = noise.C.full
def sqrt_core():
T = np.nan # cause error if used
Qa12 = np.nan # cause error if used
A2 = A.copy() # Instead of using (the implicitly nonlocal) A,
# which changes A outside as well. NB: This is a bug in Datum!
if N <= Nx:
Ainv = tinv(A2.T)
Qa12 = Ainv@Q12
T = funm_psd(eye(N) + dt*(N-1)*([email protected]), sqrt)
A2 = T@A2
else: # "Left-multiplying" form
P = A2.T @ A2 / (N-1)
L = funm_psd(eye(Nx) + dt*mrdiv(Q, P), sqrt)
A2 = A2 @ L.T
E = mu + A2
return E, T, Qa12
if method == 'Stoch':
# In-place addition works (also) for empty [] noise sample.
E += sqrt(dt)*noise.sample(N)
elif method == 'none':
pass
elif method == 'Mult-1':
varE = np.var(E, axis=0, ddof=1).sum()
ratio = (varE + dt*diag(Q).sum())/varE
E = mu + sqrt(ratio)*A
E = svdi(*tsvd(E, 0.999)) # Explained in Datum
elif method == 'Mult-M':
varE = np.var(E, axis=0)
ratios = sqrt((varE + dt*diag(Q))/varE)
E = mu + A*ratios
E = svdi(*tsvd(E, 0.999)) # Explained in Datum
elif method == 'Sqrt-Core':
E = sqrt_core()[0]
elif method == 'Sqrt-Mult-1':
varE0 = np.var(E, axis=0, ddof=1).sum()
varE2 = (varE0 + dt*diag(Q).sum())
E, _, Qa12 = sqrt_core()
if N <= Nx:
A, mu = center(E)
varE1 = np.var(E, axis=0, ddof=1).sum()
ratio = varE2/varE1
E = mu + sqrt(ratio)*A
E = svdi(*tsvd(E, 0.999)) # Explained in Datum
elif method == 'Sqrt-Add-Z':
E, _, Qa12 = sqrt_core()
if N <= Nx:
Z = Q12 - A.T@Qa12
E += sqrt(dt)*([email protected](Z.shape[1], N)).T
elif method == 'Sqrt-Dep':
E, T, Qa12 = sqrt_core()
if N <= Nx:
# Q_hat12: reuse svd for both inversion and projection.
Q_hat12 = A.T @ Qa12
U, s, VT = tsvd(Q_hat12, 0.99)
Q_hat12_inv = (VT.T * s**(-1.0)) @ U.T
Q_hat12_proj = VT.T@VT
rQ = Q12.shape[1]
# Calc D_til
Z = Q12 - Q_hat12
D_hat = A.T@(T-eye(N))
Xi_hat = Q_hat12_inv @ D_hat
Xi_til = (eye(rQ) - Q_hat12_proj)@rnd.randn(rQ, N)
D_til = Z@(Xi_hat + sqrt(dt)*Xi_til)
E += D_til.T
else:
raise KeyError('No such method')
return E
# uniform (i.e. not Gaussian) random numbers.
wnumQ = 25
sample_filename = modelling.rc.dirs.samples / ('LA_Q_wnum%d.npz' % wnumQ)
try:
# Load pre-generated
L = np.load(sample_filename)['Left']
except FileNotFoundError:
# First-time use
print('Did not find sample file', sample_filename,
'for experiment initialization. Generating...')
NQ = 20000 # Must have NQ > (2*wnumQ+1)
A = sinusoidal_sample(Nx, wnumQ, NQ)
A = 1 / 10 * (A - A.mean(0)) / np.sqrt(NQ)
Q = A.T @ A
U, s, _ = tsvd(Q)
L = U * np.sqrt(s)
np.savez(sample_filename, Left=L)
X0 = modelling.GaussRV(C=modelling.CovMat(np.sqrt(5) * L, 'Left'))
###################
# Forward model #
###################
damp = 0.98
Fm = Fmat(Nx, -1, 1, tseq.dt)
def step(x, t, dt):
assert dt == tseq.dt
return x @ Fm.T