The CUBA Library

Concepts and implementation of the Cuba library for multidimensional numerical integration are elucidated. 1. Overview Cuba [1] is a library for multidimensional numerical integration. It features four integration algorithms with interfaces for Fortran, C/C++, and Mathematica. All four can integrate vector integrands and their invocation is quite similar to make it easy to switch routines for comparison. The main characteristics are summarized below. Routine: Vegas Basic integration methods available: • Sobol sample (quasi Monte Carlo) • Mersenne Twister sample (pseudo Monte Carlo) Variance reduction: importance sampling Routine: Suave Basic integration methods available: • Sobol sample (quasi Monte Carlo) • Mersenne Twister sample (pseudo Monte Carlo) Variance reduction: importance sampling combined with globally adaptive subdivision Routine: Divonne Basic integration methods available: • Korobov sample (lattice method) • Sobol sample (quasi Monte Carlo) • Mersenne Twister sample (pseudo Monte Carlo) • cubature rules (deterministic method) Variance reduction: stratified sampling, aided by methods from numerical optimization Routine: Cuhre Basic integration method available: • cubature rules (deterministic method) Variance reduction: globally adaptive subdivision Before explaining the buzzwords appearing in this list, it is perhaps worthwhile to address two frequently asked questions. First, numerical integration rapidly becomes more difficult with increasing dimension, no matter how many tricks are built into the integra-tor. To illustrate this, imagine computing the volume of the d-dim. sphere S d by integrating its characteristic function χ = θ(1 − x 2) inside the surrounding hypercube C d = [−1, 1] d. In a Monte Carlo way of thinking, then, the ratio r = Vol S d / Vol C d can be taken as the chance that a general-purpose integrator will find the sphere at all. The numbers clearly display what is often termed the 'curse of dimensionality':