Locally Corrected Multidimensional Quadrature Rules for Singular Functions

Accurate numericalintegrationof singularfunctions usually requireseither adaptivity or product integration. Both interfere with fast summation techniques and thus hamper large-scale computations. This paper presents a method for computing highly accurate quadrature formulas for singular functions which combine well with fast summation methods. Given the singularity and the N nodes, we rst construct weights which integrate smooth functions with order-k accuracy. Then we locally correct a small number of weights near the singularity, to achieve order-k accuracy on singular functions as well. The method is highly eecient and runs in O(N k 2d +N log 2 N) time and O(k 2d +N) space. We derive precise error bounds and time estimates and connrm them with numerical results which demonstrate the accuracy and eeciency of the method in large-scale computations. As part of our implementation, we also construct a new adaptive multidimensional product Gauss quadrature routine with an eeective error estimate, and compare it with a standard package. The approach generalizes to interpolate and diierentiate scattered data and to integrate singular functions over curves and surfaces in several dimensions.