Construction of quasi-Monte Carlo rules for multivariate integration in spaces of permutation-invariant functions
| Autor: | Markus Weimar, Gowri Suryanarayana, Dirk Nuyens |
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| Jazyk: | angličtina |
| Rok vydání: | 2016 |
| Předmět: |
Function space
General Mathematics Quadrature 010103 numerical & computational mathematics 01 natural sciences Upper and lower bounds Component-by-component construction FOS: Mathematics Mathematics - Numerical Analysis Quasi-Monte Carlo method 0101 mathematics Invariant (mathematics) Mathematics Discrete mathematics Quasi-Monte Carlo methods Numerical Analysis (math.NA) Numerical integration Quadrature (mathematics) Cubature 010101 applied mathematics Computational Mathematics Rate of convergence Bounded function Rank-1 lattice rules Analysis |
| Popis: | We study multivariate integration of functions that are invariant under the permutation (of a subset) of their arguments. Recently, in Nuyens, Suryanarayana, and Weimar [10] (Adv. Comput. Math. (2016), 42(1):55–84), the authors derived an upper estimate for the nth minimal worst case error for such problems, and showed that under certain conditions this upper bound only weakly depends on the dimension. We extend these results by proposing two (semi-) explicit construction schemes. We develop a component-by-component algorithm to find the generating vector for a shifted rank-1 lattice rule that obtains a rate of convergence arbitrarily close to O(n^−α), where α > 1/2 denotes the smoothness of our function space and n is the number of cubature nodes. Further, we develop a semi-constructive algorithm that builds on point sets which can be used to approximate the integrands of interest with a small error; the cubature error is then bounded by the error of approximation. Here the same rate of convergence is achieved while the dependence of the error bounds on the dimension d is significantly improved. ispartof: Constructive Approximation vol:45 issue:2 pages:311-344 status: published |
| Databáze: | OpenAIRE |
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