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Vegas Revisited: Adaptive Monte Carlo Integration Beyond Factorization

Thorsten Ohl

TL;DR

The paper addresses the challenge of integrating ill-behaved high-dimensional integrands with non-factorizable singularities by extending the Vegas algorithm with multichannel sampling across multiple coordinate maps. It introduces a framework where the integral is decomposed into a sum of factorizable channels, each sampled with its own adaptive grid, and the channel weights α_i are optimized to minimize variance. The approach yields substantial improvements in accuracy over classic Vegas for many physics-related integrals and enables unweighted event generation, with a practical reduction in error demonstrated on representative test functions. The method offers a path toward automated, parameter-dependent phase-space integration in scattering cross-section calculations, with avenues for further refinement and alternative implementations.

Abstract

We present a new adaptive Monte Carlo integration algorithm for ill-behaved integrands with non-factorizable singularities. The algorithm combines Vegas with multi channel sampling and performs significantly better than Vegas for a large class of integrals appearing in physics.

Vegas Revisited: Adaptive Monte Carlo Integration Beyond Factorization

TL;DR

The paper addresses the challenge of integrating ill-behaved high-dimensional integrands with non-factorizable singularities by extending the Vegas algorithm with multichannel sampling across multiple coordinate maps. It introduces a framework where the integral is decomposed into a sum of factorizable channels, each sampled with its own adaptive grid, and the channel weights α_i are optimized to minimize variance. The approach yields substantial improvements in accuracy over classic Vegas for many physics-related integrals and enables unweighted event generation, with a practical reduction in error demonstrated on representative test functions. The method offers a path toward automated, parameter-dependent phase-space integration in scattering cross-section calculations, with avenues for further refinement and alternative implementations.

Abstract

We present a new adaptive Monte Carlo integration algorithm for ill-behaved integrands with non-factorizable singularities. The algorithm combines Vegas with multi channel sampling and performs significantly better than Vegas for a large class of integrals appearing in physics.

Paper Structure

This paper contains 8 sections, 25 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Maps
  3. Multichannel
  4. Performance
  5. Costs
  6. Gains
  7. A Cheaper Alternative
  8. Conclusions

Figures (2)

  • Figure 1: $e^+e^-\to q\bar{q}g$
  • Figure 2: Comparison of the sampling error for the integral of $f$ in (\ref{['eq:f(x)']}) as a function of the width parameter $b$ for the two algorithms at comparable computational costs.