A numerical adaptive higher order integration (AHOI) method is proposed to refine the computationally intensive numerical integrations while utilizing the previous results. Typically, to increase the accuracy of numerical integrations using methods which cannot be nested, such as Gaussian quadrature, the integrand is evaluated over a larger disjoint set of abscissas without using the previous integrand evaluations. However, the AHOI method is able to add any number of abscissas to the existing quadrature and reevaluate all the associated weights. For this end, a global optimization technique is used to minimize the sum of squared numerical integration error for a set of training functions which is dependent on the unknown weights and new abscissas. The training functions must have a known integral over the domain, D, in n-dimensional real space bounded by -1 and 1 in each dimension for the optimization. Also, the abscissas are constrained to the domain D, and the weights are constrained to be positive and less than or equal to 2 to the nth power. The new optimal abscissas are then added to the previous set of abscissas, and the optimization process is carried out until convergence is achieved. In order to assess the applicability of the AHOI, it was implemented for numerical integration of analytical problems with different number of variables. Furthermore, the flexibility of the AHOI was tested by applying it to the Polynomial Chaos Expansion (PCE) of a stochastic analytical problem by using Galerkin Projection approach. The results obtained from the preliminary studies of PCE with AHOI showed that it is capable of yielding higher accuracies and still has some room for improvement.
展开▼
