On optimal grid construction in numerical integration

This paper presents an optimization approach to the construction of optimal grids for numerical integration in one dimension by some well-known quadrature rules. In this approach, a conventional numerical integration problem is posed as an optimization problem so that the solution of the latter yields vertices of the optimal grid. Numerical experiments are performed to verify the effectiveness of the approach. The numerical results show that, for a fixed number of mesh nodes, the numerical integrals on a grid obtained from the present method are at least one order of magnitude more accurate than those on the uniform grid.