Multiplicatively Convolutional Fast Integral Transforms

Features

• Compute integral transforms: $$G(y) = \int_0^\infty F(x) K(xy) \frac{dx}x;$$
• Inverse transform without analytic inversion;
• Integral kernels as derivatives: $$G(y) = \int_0^\infty F(x) K'(xy) \frac{dx}x;$$
• Transform input array along any axis of ndarray;
• Output the matrix form;
• 1-to-n transform for multiple kernels (TODO): $$G(y_1, \cdots, y_n) = \int_0^\infty \frac{dx}x F(x) \prod_{a=1}^n K_a(xy_a);$$
• Easily extensible for other kernels;
• Support NumPy and JAX.

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 a partial sum of the 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 $\ln x$.

Installation

  pip install mcfit

Documentation

See docstring of mcfit.mcfit, which also applies to other subclasses of transformations. Also see doc/mcfit.tex for more explanations.

Examples

One can perform the following pair of Hankel transforms $$e^{-y} = \int_0^\infty (1+x^2)^{-\frac32} J_0(xy) x dx, \quad (1+y^2)^{-\frac32} = \int_0^\infty e^{-x} J_0(xy) x dx$$ 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))