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)

  x = numpy.logspace(-3, 3, num=60, endpoint=False)
  F = F_fun(x)
  H = mcfit.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 = mcfit.Hankel(y, lowring=True)
  x, F = H_inv(G, extrap=True)
  numpy.allclose(F, F_fun(x), rtol=1e-10, atol=1e-10)

To use JAX instead of the default numpy backend:

  H = mcfit.Hankel(x, lowring=True, backend='jax')  # do not jit
  H_jit = jax.jit(functools.partial(H, extrap=True))
  y, G = H_jit(F)

It is not necessary to apply jit or other JAX transforms to the constructor, and one should not do it because of the scipy special functions in there.

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

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

and the other way around

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

Similarly for the quadrupoles

  k, P2 = numpy.loadtxt('P2.txt', unpack=True)
  r, xi2 = mcfit.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.

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