Numerical quadrature for high-dimensional singular integrals over parallelotopes

Abstract

We introduce and analyze a family of algorithms for an efficient numerical approximation of integrals of the form I=∫C(1)∫C(2)F(x,y,y−x)dydx where C(1), C(2) are d-dimensional parallelotopes (i.e. affine images of d-hypercubes) and F has a singularity at y−x=0. Such integrals appear in Galerkin discretization of integral operators in Rd. We construct a family of quadrature rules QN with N function evaluations for a class of integrands F which may have algebraic singularities at y−x=0 and are Gevrey-δ regular for y−x≠0. The main tool is an explicit regularizing coordinate transformation, simultaneously simplifying the singular support and the domain of integration. For the full tensor product variant of the suggested quadrature family we prove that QN achieves the exponential convergence rate O(exp(−rNγ)) with the exponent γ=1/(2dδ+1). In the special case of a singularity of the form ‖y−x‖α with real α we prove that the improved convergence rate of γ=1/(2dδ) is achieved if a certain modified one-dimensional Gauss–Jacobi quadrature rule is used in the singular direction. We give numerical results for various types of the quadrature rules, in particular based on tensor product rules, standard (Smolyak), optimized and adaptive sparse grid quadratures and Sobol’ sequences.