Source code for sfepy.homogenization.convolutions

from __future__ import print_function
import numpy as nm

from sfepy.base.base import pause, Struct
from sfepy.homogenization.utils import integrate_in_time

[docs]def compute_mean_decay(coef): r""" Compute mean decay approximation of a non-scalar fading memory coefficient. """ n_step = coef.shape 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 return decay
[docs]def eval_exponential(coefs, x): c1, c2 = coefs return c1 * nm.exp(-c2 * x)
[docs]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 c2 = - nm.log(y / c1) / (x - x) 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
[docs]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)
[docs]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 \;, 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 self.c0 = self.c self.e_c0 = self.c0 * self.e self.e_d1 = self.e / self.e self.c_slice = (slice(None),) + ((None,) * (self.c.ndim - 1))
[docs] def diff_dt(self, use_exp=False): """ The derivative of the kernel w.r.t. time. """ if use_exp: val = - self.ec * 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
[docs] def int_dt(self, use_exp=False): """ The integral of the kernel in time. """ if use_exp: val = (self.e_c0/self.ec) * (1.0 - self.en[-1]) else: val = integrate_in_time(self.c, self) return val
[docs] def get_exp(self): """ Get the exponential synchronous decay kernel approximation. """ return self.c0 * self.e[self.c_slice]
[docs] 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()