Numerical integration

• R. Cools
• A. Haegemans
• P. Verlinden

A multivariate integral is approximated by a cubature formula, i.e., a weighted sum of integrand evaluations. A cubature formula is efficient if it is exact for a large class of integrands (all polynomials up to a certain degree) and if it requires few integrand evaluations.

The construction of an efficient cubature formula requires the solution of a large system of polynomial equations. Imposing symmetry on the cubature formula reduces the size of the system and often decomposes the cubature problem into a sequence of smaller cubature or quadrature problems. Expertise on constructing cubature formulas is acquired during many years. Also pairs of cubature formulas were constructed which give both an approximation of the integral and an estimate of the approximation error.

For the numerical integration of a periodic function over a multi-dimensional cube, cubature formulas are used which are exact for a class of trigonometric polynomials. Lattice rules are such cubature formulas. They can also be applied to approximate the integral of a non-periodic function after a periodizing transformation. Practical experience with lattice rules is not yet widely available. We constructed new lattice rules ad investigated criteria to study their quality.

Quadrature or cubature rules generally converge to the integral as the number of integrand evaluation increases. We analyse this asymptotic behavior for integrands which present a singularity. For composite rules, based on a subdivision of the integration domain, this asymptotic behavior can generally be described by an asymptotic expansion, which explains the acceleration of the convergence by extrapolation techniques. For Gauss-type quadrature formulas, the asymptotic analysis involves the asymptotic behavior of orthogonal polynomials in the presence of a singularity.

All these are building blocks for a package of routines for automatic multivariate integration.