Rotation method for accelerating multiple-spherical Bessel function integrals against a numerical source function. (arXiv:1912.00065v1 [astro-ph.CO])

<a href="http://arxiv.org/find/astro-ph/1/au:+Slepian_Z/0/1/0/all/0/1">Zachary Slepian</a>, <a href="http://arxiv.org/find/astro-ph/1/au:+Li_Y/0/1/0/all/0/1">Yin Li</a>, <a href="http://arxiv.org/find/astro-ph/1/au:+Schmittfull_M/0/1/0/all/0/1">Marcel Schmittfull</a>, <a href="http://arxiv.org/find/astro-ph/1/au:+Vlah_Z/0/1/0/all/0/1">Zvonimir Vlah</a>

A common problem in cosmology is to integrate the product of two or more

spherical Bessel functions (sBFs) with different configuration-space arguments

against the power spectrum or its square, weighted by powers of wavenumber.

Naively computing them scales as $N_{rm g}^{p+1}$ with $p$ the number of

configuration space arguments and $N_{rm g}$ the grid size, and they cannot be

done with Fast Fourier Transforms (FFTs). Here we show that by rewriting the

sBFs as sums of products of sine and cosine and then using the product to sum

identities, these integrals can then be performed using 1-D FFTs with $N_{rm

g} log N_{rm g}$ scaling. This “rotation” method has the potential to

accelerate significantly a number of calculations in cosmology, such as

perturbation theory predictions of loop integrals, higher order correlation

functions, and analytic templates for correlation function covariance matrices.

We implement this approach numerically both in a free-standing,

publicly-available textsc{Python} code and within the larger,

publicly-available package texttt{mcfit}. The rotation method evaluated with

direct integrations already offers a factor of 6-10$times$ speed-up over the

naive approach in our test cases. Using FFTs, which the rotation method

enables, then further improves this to a speed-up of $sim$$1000-3000times$

over the naive approach. The rotation method should be useful in light of

upcoming large datasets such as DESI or LSST. In analysing these datasets

recomputation of these integrals a substantial number of times, for instance to

update perturbation theory predictions or covariance matrices as the input

linear power spectrum is changed, will be one piece in a Monte Carlo Markov

Chain cosmological parameter search: thus the overall savings from our method

should be significant.

A common problem in cosmology is to integrate the product of two or more

spherical Bessel functions (sBFs) with different configuration-space arguments

against the power spectrum or its square, weighted by powers of wavenumber.

Naively computing them scales as $N_{rm g}^{p+1}$ with $p$ the number of

configuration space arguments and $N_{rm g}$ the grid size, and they cannot be

done with Fast Fourier Transforms (FFTs). Here we show that by rewriting the

sBFs as sums of products of sine and cosine and then using the product to sum

identities, these integrals can then be performed using 1-D FFTs with $N_{rm

g} log N_{rm g}$ scaling. This “rotation” method has the potential to

accelerate significantly a number of calculations in cosmology, such as

perturbation theory predictions of loop integrals, higher order correlation

functions, and analytic templates for correlation function covariance matrices.

We implement this approach numerically both in a free-standing,

publicly-available textsc{Python} code and within the larger,

publicly-available package texttt{mcfit}. The rotation method evaluated with

direct integrations already offers a factor of 6-10$times$ speed-up over the

naive approach in our test cases. Using FFTs, which the rotation method

enables, then further improves this to a speed-up of $sim$$1000-3000times$

over the naive approach. The rotation method should be useful in light of

upcoming large datasets such as DESI or LSST. In analysing these datasets

recomputation of these integrals a substantial number of times, for instance to

update perturbation theory predictions or covariance matrices as the input

linear power spectrum is changed, will be one piece in a Monte Carlo Markov

Chain cosmological parameter search: thus the overall savings from our method

should be significant.

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