Feynman integral reduction: balanced reconstruction of sparse rational functions and implementation on supercomputers in a co-design approach

Abstract

Integration-by-parts (IBP) reduction is one of the essential steps in evaluating Feynman integrals. A modern approach to IBP reduction uses modular arithmetic evaluations with parameters set to numerical values at sample points, followed by reconstruction of the analytic rational coefficients. Due to the large number of sample points needed, problems at the frontier of science require an application of supercomputers. In this article, we present a rational function reconstruction method that fully takes advantage of sparsity, combining the balanced reconstruction method and the Zippel method. Additionally, to improve the efficiency of the finite-field IBP reduction runs, at each run several numerical probes are computed simultaneously, which allows to decrease the resource overhead. We describe what performance issues one encounters on the way to an efficient implementation on supercomputers, and how one should co-design the algorithm and the supercomputer infrastructure. Benchmarks are presented for IBP reductions for massless two-loop four- and five-point integrals using a development version of FIRE, as well as synthetic examples mimicking the coefficients involved in scattering amplitudes for post-Minkowskian gravitational binary dynamics.