282 research outputs found
Computable error bounds for quasi-Monte Carlo using points with non-negative local discrepancy
Let be a completely monotone integrand as defined by
Aistleitner and Dick (2015) and let points
have a non-negative
local discrepancy (NNLD) everywhere in . We show how to use these
properties to get a non-asymptotic and computable upper bound for the integral
of over . An analogous non-positive local discrepancy (NPLD)
property provides a computable lower bound. It has been known since Gabai
(1967) that the two dimensional Hammersley points in any base have
non-negative local discrepancy. Using the probabilistic notion of associated
random variables, we generalize Gabai's finding to digital nets in any base
and any dimension when the generator matrices are permutation
matrices. We show that permutation matrices cannot attain the best values of
the digital net quality parameter when . As a consequence the computable
absolutely sure bounds we provide come with less accurate estimates than the
usual digital net estimates do in high dimensions. We are also able to
construct high dimensional rank one lattice rules that are NNLD. We show that
those lattices do not have good discrepancy properties: any lattice rule with
the NNLD property in dimension either fails to be projection regular or
has all its points on the main diagonal
Randomized Algorithms for High-Dimensional Integration and Approximation
We prove upper and lower error bounds for error of the randomized Smolyak algorithm and provide a thorough case study of applying the randomized Smolyak algorithm with the building blocks being quadratures based on scrambled nets for integration of functions coming from Haar-wavelets spaces. Moreover, we discuss different notions of negative dependence of randomized point sets which find applications in discrepancy theory and randomized quasi-Monte Carlo integration
A universal median quasi-Monte Carlo integration
We study quasi-Monte Carlo (QMC) integration over the multi-dimensional unit
cube in several weighted function spaces with different smoothness classes. We
consider approximating the integral of a function by the median of several
integral estimates under independent and random choices of the underlying QMC
point sets (either linearly scrambled digital nets or infinite-precision
polynomial lattice point sets). Even though our approach does not require any
information on the smoothness and weights of a target function space as an
input, we can prove a probabilistic upper bound on the worst-case error for the
respective weighted function space, where the failure probability converges to
0 exponentially fast as the number of estimates increases. Our obtained rates
of convergence are nearly optimal for function spaces with finite smoothness,
and we can attain a dimension-independent super-polynomial convergence for a
class of infinitely differentiable functions. This implies that our
median-based QMC rule is universal in the sense that it does not need to be
adjusted to the smoothness and the weights of the function spaces and yet
exhibits the nearly optimal rate of convergence. Numerical experiments support
our theoretical results.Comment: Major revision, 32 pages, 4 figure
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