2011/09/23 by Josef Dick, Dick, Josef
Mathematics · #65D30 #65D32 #Analytic and geometric function theory #FOS: Mathematics #Holomorphic and Operator Theory #Mathematical Approximation and Integration #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.1109.4998
openalex publication_date 2011/09/23 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
In this paper we study lattice rules which are cubature formulae to\napproximate integrands over the unit cube [0,1]s from a weighted reproducing\nkernel Hilbert space. We assume that the weights are independent random\nvariables with a given mean and variance for two reasons stemming from\npractical applications: (i) It is usually not known in practice how to choose\nthe weights. Thus by assuming that the weights are random variables, we obtain\nrobust constructions (with respect to the weights) of lattice rules. This, to\nsome extend, removes the necessity to carefully choose the weights. (ii) In\npractice it is convenient to use the same lattice rule for many different\nintegrands. The best choice of weights for each integrand may vary to some\ndegree, hence considering the weights random variables does justice to how\nlattice rules are used in applications.\n We also study a generalized version which uses r constraints which we call\nthe cbcrc (component-by-component with r constraints) algorithm. We show\nthat lattice rules generated by the cbcrc algorithm simultaneously work well\nfor all weights in a subspace spanned by the chosen weights\n boldsymbol\γ(1),..., boldsymbol\γ(r). Thus, in\napplications, instead of finding one set of weights, it is enough to find an\nr dimensional convex polytope in which the optimal weights lie. The price for\nthis method is a factor r in the upper bound on the error and in the\nconstruction cost of the lattice rule. Thus the burden of determining one set\nof weights very precisely can be shifted to the construction of good lattice\nrules.\n