Cubature using Stroud rules and their extensions


  • Alias: None

  • Arguments: INTEGER


Multi-dimensional integration by Stroud cubature rules [Str71] and extensions [Xiu08], as specified with cubature_integrand. A total-order expansion is used, where the isotropic order p of the expansion is half of the integrand order, rounded down. The total number of terms N for an isotropic total-order expansion of order p over n variables is given by

\[N~=~1 + P ~=~1 + \sum_{s=1}^{p} {\frac{1}{s!}} \prod_{r=0}^{s-1} (n + r) ~=~\frac{(n+p)!}{n!p!}\]

Since the maximum integrand order is currently five for normal and uniform and two for all other types, at most second- and first-order expansions, respectively, will be used. As a result, cubature is primarily useful for global sensitivity analysis, where the Sobol’ indices will provide main effects and, at most, two-way interactions. In addition, the random variable set must be independent and identically distributed ( iid), so the use of askey or wiener transformations may be required to create iid variable sets in the transformed space (as well as to allow usage of the higher order cubature rules for normal and uniform). Note that global sensitivity analysis often assumes uniform bounded regions, rather than precise probability distributions, so the iid restriction would not be problematic in that case.