2015/07/29 by Dirk Nuyens, Nuyens, Dirk, Gowri Suryanarayana +3
Mathematics · #FOS: Mathematics #Mathematical Approximation and Integration #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.1507.08084
openalex publication_date 2015/07/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study multivariate integration of functions that are invariant under the\npermutation (of a subset) of their arguments. Recently, in Nuyens,\nSuryanarayana, and Weimar (Adv. Comput. Math. (2016), 42(1):55--84), the\nauthors derived an upper estimate for the nth minimal worst case error for\nsuch problems, and showed that under certain conditions this upper bound only\nweakly depends on the dimension. We extend these results by proposing two\n(semi-) explicit construction schemes. We develop a component-by-component\nalgorithm to find the generating vector for a shifted rank-1 lattice rule\nthat obtains a rate of convergence arbitrarily close to\n\O(n-\α), where \α>1/2 denotes the smoothness of our\nfunction space and n is the number of cubature nodes. Further, we develop a\nsemi-constructive algorithm that builds on point sets which can be used to\napproximate the integrands of interest with a small error; the cubature error\nis then bounded by the error of approximation. Here the same rate of\nconvergence is achieved while the dependence of the error bounds on the\ndimension d is significantly improved.\n