An optimization approach to numerical integration in two dimensions

摘要

In this paper, we propose an efficient approach to numerical integration of functions of two variables, where a grid set with a fixed number of vertices is to be chosen such that the error between the numerical integral and the exact integral is minimized. Two schemes are first developed for sufficiently smooth functions. One is based on barycenter rule on a rectangular partition, while the other is on a triangular partition. A scheme for non-sufficiently smooth functions is also developed. For illustration, several examples are solved by using the proposed schemes.