from __future__ import print_function
import numpy as nm
from sfepy.base.base import pause, Struct
from sfepy.homogenization.utils import integrate_in_time
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def compute_mean_decay(coef):
r"""
Compute mean decay approximation of a non-scalar fading memory
coefficient.
"""
n_step = coef.shape[0]
weights = nm.abs(coef[int(nm.fix(n_step / 2))])
weights /= weights.sum()
## coef_flat = nm.reshape(coef, (n_step, nm.prod(coef.shape[1:])))
## maxs = nm.abs(coef_flat).max(axis=-1)
## decay = avgs + maxs
aux = weights[None,...] * coef
aux_flat = nm.reshape(aux, (n_step, nm.prod(coef.shape[1:])))
avgs = aux_flat.sum(axis=-1)
decay = avgs # Gives better results than the above.
decay /= decay[0]
return decay
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def eval_exponential(coefs, x):
c1, c2 = coefs
return c1 * nm.exp(-c2 * x)
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def approximate_exponential(x, y):
r"""
Approximate :math:`y = f(x)` by :math:`y_a = c_1 exp(- c_2 x)`.
Initial guess is given by assuming y has already the required exponential
form.
"""
from scipy.optimize import leastsq
## weights = nm.abs(y)
## weights = weights / weights.sum()
weights = nm.ones_like(y)
def fun(c, x, y):
val = weights * (y - eval_exponential(c, x))
return val
c1 = y[0]
c2 = - nm.log(y[1] / c1) / (x[1] - x[0])
coefs, ier = leastsq(fun, nm.array([c1, c2]), args=(x, y))
if ier != 1:
print(c1, c2)
print(coefs, ier)
pause('exponential fit failed!')
return coefs
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def fit_exponential(x, y, return_coefs=False):
"""
Evaluate :math:`y = f(x)` after approximating :math:`f` by an exponential.
"""
coefs = approximate_exponential(x, y)
if return_coefs:
return coefs, eval_exponential(coefs, x)
else:
return eval_exponential(coefs, x)
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class ConvolutionKernel(Struct):
r"""
The convolution kernel with exponential synchronous decay approximation
approximating the original kernel represented by the array :math:`c[i]`,
:math:`i = 0, 1, \dots`.
.. math::
\begin{split}
& c_0 \equiv c[0] \;, c_{e0} \equiv c_0 c^e_0 \;, \\
& c(t) \approx c_0 d(t) \approx c_0 e(t) = c_{e0} e_n(t) \;,
\end{split}
where :math:`d(0) = e_n(0) = 1`, :math:`d` is the synchronous decay and
:math:`e` its
exponential approximation, :math:`e = c^e_0 exp(-c^e_1 t)`.
"""
def __init__(self, name, times, kernel, decay=None,
exp_coefs=None, exp_decay=None):
Struct.__init__(self, name=name, times=times, c=kernel,
d=decay, ec=exp_coefs, e=exp_decay)
if decay is None:
self.d = compute_mean_decay(kernel)
if exp_decay is None:
self.ec, self.e = fit_exponential(times, self.d,
return_coefs=True)
self.en = self.e / self.e[0]
self.c0 = self.c[0]
self.e_c0 = self.c0 * self.e[0]
self.e_d1 = self.e[1] / self.e[0]
self.c_slice = (slice(None),) + ((None,) * (self.c.ndim - 1))
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def diff_dt(self, use_exp=False):
"""
The derivative of the kernel w.r.t. time.
"""
if use_exp:
val = - self.ec[1] * self.c0 * self.e[self.c_slice]
else:
zz = nm.zeros(self.c[0:1].shape, dtype=self.c.dtype)
val = nm.r_[nm.diff(self.c, axis=0) /
nm.diff(self.times, axis=0)[self.c_slice], zz]
return val
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def int_dt(self, use_exp=False):
"""
The integral of the kernel in time.
"""
if use_exp:
val = (self.e_c0/self.ec[1]) * (1.0 - self.en[-1])
else:
val = integrate_in_time(self.c, self)
return val
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def get_exp(self):
"""
Get the exponential synchronous decay kernel approximation.
"""
return self.c0 * self.e[self.c_slice]
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def get_full(self):
"""
Get the original (full) kernel.
"""
return self.c
def __call__(self, use_exp=False):
"""
Get the kernel or its approximation.
"""
if use_exp:
return self.get_exp()
else:
return self.get_full()