Multiplicatively convolutional fast integral transforms


Keywords
numerical, integral, transform, FFTLog, cosmology, integral-transform, numerical-integration
License
GPL-3.0
Install
pip install mcfit==0.0.17

Documentation

Multiplicatively Convolutional Fast Integral Transforms

Features

  • Compute integral transforms
  • Inverse transform without analytic inversion
  • Integral kernels as derivatives
  • Transform input array along any axis
  • Output the matrix form
  • 1-to-n transform for multiple kernels (TODO)
  • Easily extensible for other kernels

Installation

pip install mcfit

Algorithm

mcfit computes integral transforms of the form

G(y) = \int_0^\infty F(x) K(xy) \frac{dx}x

where F(x) is the input function, G(y) is the output function, and K(xy) is the integral kernel. One is free to scale all three functions by a power law

g(y) = \int_0^\infty f(x) k(xy) \frac{dx}x

where f(x)=x^{-q}F(x), g(y)=y^q G(y), and k(t)=t^q K(t). And q is a tilt parameter serving to shift power of x between the input function and the kernel.

mcfit implements the FFTLog algorithm. The idea is to take advantage of the convolution theorem in \ln x and \ln y. It approximates the input function with truncated Fourier series over one period of a periodic approximant, and use the exact Fourier transform of the kernel. One can calculate the latter analytically as a Mellin transform. This algorithm is optimal when the input function is smooth in \ln x, and is ideal for oscillatory kernels with input spanning a wide range in \ln x.

Examples

One can perform the following pair of Hankel transforms

e^{-y} &= \int_0^\infty (1+x^2)^{-\frac32} J_0(xy) x dx
(1+y^2)^{-\frac32} &= \int_0^\infty e^{-x} J_0(xy) x dx

easily as follows

def F_fun(x): return 1 / (1 + x*x)**1.5
def G_fun(y): return numpy.exp(-y)

from mcfit import Hankel

x = numpy.logspace(-3, 3, num=60, endpoint=False)
F = F_fun(x)
H = Hankel(x, lowring=True)
y, G = H(F, extrap=True)
numpy.allclose(G, G_fun(y), rtol=1e-8, atol=1e-8)

y = numpy.logspace(-4, 2, num=60, endpoint=False)
G = G_fun(y)
H_inv = Hankel(y, lowring=True)
x, F = H_inv(G, extrap=True)
numpy.allclose(F, F_fun(x), rtol=1e-10, atol=1e-10)

Cosmologists often need to transform a power spectrum to its correlation function

from mcfit import P2xi
k, P = numpy.loadtxt('P.txt', unpack=True)
r, xi = P2xi(k)(P)

and the other way around

from mcfit import xi2P
r, xi = numpy.loadtxt('xi.txt', unpack=True)
k, P = xi2P(r)(xi)

Similarly for the quadrupoles

k, P2 = numpy.loadtxt('P2.txt', unpack=True)
r, xi2 = P2xi(k, l=2)(P2)

Also useful to the cosmologists is the tool below that computes the variance of the overdensity field as a function of radius, from which \sigma_8 can be interpolated.

from mcfit import TophatVar
R, var = TophatVar(k, lowring=True)(P, extrap=True)
from scipy.interpolate import CubicSpline
varR = CubicSpline(R, var)
sigma8 = numpy.sqrt(varR(8))