""" Methods to integrate functions """
#################################################
# Imports
################################################
import numpy as np
from waveformtools.waveformtools import message
##################################################
# Fixed frequency integration
##################################################
[docs]def fixed_frequency_integrator(udata_time, delta_t, utilde_conven=None, omega0=0, order=1, zero_mode=0):
"""Fixed frequency integrator as presented in Reisswig et. al.
Parameters
----------
udata_time: 1d array
The input data in time.
delta_t: float
The time stepping.
utilde_conven: 1d array, optional
The conventional FFT of the samples udata_time.
omega0: float, optional
The cutoff angular frequency in the integration.
Must be lower than the starting angular frequency
of the input waveform. All frequencies whose absolute
value is below this value will be neglected.
The default cutoff-value is 0.
order: int, optional
The number of times to integrate
the integrand in time. Defaults to 1.
zero_mode: float, optional
The zero mode amplitude of the FFT required.
Defaults to 0 i.e. the zero mode is removed.
Returns
-------
u_integ_n_time: 1d array
The input waveform in time-space, integrated in frequency space using FFI.
u_integ_integ_n: 1d array
The integrated u samples in Fourier space.
"""
if not utilde_conven:
# Compute the FFT of data
from numpy.fft import ifft
from waveformtools.transforms import compute_fft, unset_fft_conven
# from waveformtools import taper
# udata_x_re = taper(u_time.real, delta_t=delta_t)
# udata_x_im = taper(u_time.imag, delta_t=delta_t)
# udata_x = np.array(udata_x_re) + 1j * np.array(udata_x_im)
# x_axis = udata_x_re.sample_times
# udata_x = np.array(udata_x)
freq_axis, utilde_conven = compute_fft(udata_time, delta_t)
# Find the length of the input data.
Nlen = len(udata_time)
else:
Nlen = len(utilde_conven)
# Find the location of the zero index.
if Nlen % 2 == 0:
zero_index = int(Nlen / 2)
else:
zero_index = int((Nlen + 1) / 2)
# Construct the angular frequency axis.
omega_axis = 2 * np.pi * freq_axis
# print("The chosen cutoff angular frequency is", omega0)
if omega0 > 0:
for index, element in enumerate(omega_axis):
# Loop over the samples.
# Skip the zero index
if index != zero_index:
# print(freq_integ[index])
try:
# Get the sign of the angular frequency.
sign = int(element / abs(element))
except Exception as excep:
message(excep)
sign = 1
# print(sign)
# Change the angular frequency if its magnitude is below a given omega0.
if abs(element) < omega0:
omega_axis[index] = sign * omega0
# Set the zero frequency element separately.
if not zero_mode:
utilde_conven[zero_index] = 0
else:
utilde_conven[zero_index] = zero_mode
# Integrate in frequency space
utilde_integ_n = np.power((-1j / omega_axis), order) * utilde_conven
# Get the inverse fft
utilde_integ_n_orig = unset_fft_conven(utilde_integ_n)
u_integ_n_time = ifft(utilde_integ_n_orig)
return u_integ_n_time, utilde_integ_n
#############################################
# 2D integrals
#############################################
[docs]def TwoDIntegral(func, info, method="DH"):
"""Integrate a function over a sphere.
Parameters
----------
func : function
The function to be integrated
NTheta, NPhi : int
The number of grid points in the theta and phi directions.
Note that NTheta must be even.
ht, hp : float
The grid spacings.
method : string
The method to use for the integration. Options are DH (Driscoll Healy), SP (Simpson's), MP (Midpoint).
Returns
-------
integ : float
The function f integrated over the sphere.
"""
# NTheta = info.ntheta_act
# NPhi = info.nphi_act
# NTheta, NPhi = func.shape
# ht = info.dtheta
# hp = info.dphi
if method == "DH":
integral = DriscollHealy2DInteg(func, info)
elif method == "MP":
integral = MidPoint2DInteg(func, info)
elif method == "SP":
integral = Simpson2DInteg(func, info)
elif method == "GL":
integral = GaussLegendre2DInteg(func, info)
else:
raise ValueError("Unknown method!")
return integral
[docs]def MidPoint2DInteg(func, info):
"""Evaulate the 2D surface integral using the midpoint rule.
Parameters
----------
func : ndarray
The data to be integrated
info : surface_grid_info
An instance of the surface grid info class containing information about the grid.
Returns
-------
integ : float
The function f integrated over the sphere.
"""
ht = info.dtheta
hp = info.dphi
theta_grid, _ = info.meshgrid
integral = np.sum(func) * ht * hp
return integral
[docs]def DriscollHealy2DInteg(func, info):
"""Implementation of the Driscoll Healy 2D integration that
exhibits near spectral convergence.
Parameters
----------
func : function
The function to be integrated
NTheta, NPhi : int
The number of grid points in the theta and phi directions.
Note that NTheta must be even.
ht, hp : float
The grid spacings.
Returns
-------
integ : float
The function f integrated over the sphere.
"""
NTheta = info.ntheta_act
NPhi = info.nphi_act
# NTheta, NPhi = func.shape
ht = info.dtheta
hp = info.dphi
if NTheta < 0:
raise ValueError("N_x is negative!")
elif NPhi < 0:
raise ValueError("N_y is negative!")
if (NTheta % 2) != 0:
raise ValueError("NTheta must be even!")
integrand_sum = 0.0
# Skip the poles (ix=0 and ix=NTheta), as the weight there is zero
for index_theta in range(1, NTheta):
# These weights lead to an almost spectral convergence
this_theta = np.pi * index_theta / NTheta
weight = 0
# CCTK_REAL const theta = M_PI * ix / NTheta;
# CCTK_REAL weight = 0.0;
for ell in range(int(NTheta / 2)):
# for (int l = 0; l < NTheta/2; ++ l)
weight += np.sin((2 * ell + 1) * this_theta) / (2 * ell + 1)
weight *= 4.0 / np.pi
latitude_sum = 0
# CCTK_REAL local_sum = 0.0;
# Skip the last point (iy=NPhi), since we assume periodicity and
# therefore it has the same value as the first point. We don't use
# weights in this direction, which leads to spectral convergence.
# (Yay periodicity!)
for index_phi in range(NPhi):
# for (int iy = 0; iy < NPhi; ++ iy)
latitude_sum += func[index_theta, index_phi]
integrand_sum += weight * latitude_sum
return ht * hp * integrand_sum
[docs]def Simpson2DInteg(func, info):
"""Implementation of Simpson's 2D integration
scheme.
Parameters
----------
func : function
The function to be integrated
NTheta, NPhi : int
The number of grid points in the theta and phi directions.
Note that NTheta must be even.
ht, hp : float
The grid spacings.
Returns
-------
integ : float
The function f integrated over the sphere.
"""
NTheta = info.ntheta_act
NPhi = info.nphi_act
# NTheta, NPhi = func.shape
ht = info.dtheta
hp = info.dphi
integrand_sum = 0
index_theta = 0
index_phi = 0
assert NTheta > 0
assert NPhi > 0
# assert(func.all())
assert NTheta % 2 == 0
assert NPhi % 2 == 0
Px = int(NTheta / 2)
Py = int(NPhi / 2)
# Around corners
integrand_sum += func[0, 0] + func[NTheta - 1, 0] + func[0, NPhi - 1] + func[NTheta - 1, NPhi - 1]
# Arount edges
for index_phi in range(1, Py):
integrand_sum += 4 * func[0, 2 * index_phi - 1] + 4 * func[NTheta - 1, 2 * index_phi - 1]
# for (iy = 1; iy <= py-1; iy++)
for index_phi in range(1, Py - 1):
integrand_sum += 2 * func[0, 2 * index_phi] + 2 * func[NTheta - 1, 2 * index_phi]
# for (ix = 1; ix <= px; ix++)
for index_theta in range(1, Px):
integrand_sum += 4 * func[2 * index_theta - 1, 0] + 4 * func[2 * index_theta - 1, NPhi - 1]
# for (ix = 1; ix <= px-1; ix++)
for index_theta in range(1, Px - 1):
integrand_sum += 2 * func[2 * index_theta, 0] + 2 * func[2 * index_theta, NPhi - 1]
# In the Interiors
# for (iy = 1; iy <= py; iy++)
for index_phi in range(1, Py):
# for (ix = 1; ix <= px; ix++)
for index_theta in range(1, Px):
integrand_sum += 16 * func[2 * index_theta - 1, 2 * index_phi - 1]
# for (iy = 1; iy <= py-1; iy++)
for index_phi in range(1, Py - 1):
# for (ix = 1; ix <= px; ix++)
for index_theta in range(Px):
integrand_sum += 8 * func[2 * index_theta - 1, 2 * index_phi]
# for (iy = 1; iy <= py; iy++)
for index_phi in range(1, Py):
# for (ix = 1; ix <= px-1; ix++)
for index_theta in range(1, Px - 1):
integrand_sum += 8 * func[2 * index_theta, 2 * index_phi - 1]
# for (iy = 1; iy <= py-1; iy++)
for index_phi in range(1, Py - 1):
# for (ix = 1; ix <= px-1; ix++)
for index_theta in range(1, Px - 1):
integrand_sum += 4 * func[2 * index_theta, 2 * index_phi]
return (1 / 9) * ht * hp * integrand_sum
[docs]def GaussLegendre2DInteg(func, info):
"""Evaulate the 2D surface integral using the Gauss-Legendre rule.
Parameters
----------
func : ndarray
The data to be integrated
info : surface_grid_info
An instance of the surface grid info class containing information about the grid.
Returns
-------
integ : float
The function f integrated over the sphere.
"""
# NTheta = info.ntheta_act
# NPhi = info.nphi_act
# NTheta, NPhi = func.shape
# ht = info.dtheta
# hp = info.dphi
# theta_grid, _ = info.meshgrid
# Norm = 1#(info.L +1)/4
integral = np.sum(func * info.weights_grid) * info.dphi
return integral