Quasi-random sequences and their discrepancies
SIAM Journal on Scientific Computing
A generalized discrepancy and quadrature error bound
Mathematics of Computation
When are quasi-Monte Carlo algorithms efficient for high dimensional integrals?
Journal of Complexity
The error bounds and tractability of quasi-Monte Carlo algorithms in infinite dimension
Mathematics of Computation
Journal of Complexity
Deterministic multi-level algorithms for infinite-dimensional integration on RN
Journal of Complexity
Error reduction techniques in quasi-monte carlo integration
Mathematical and Computer Modelling: An International Journal
Asymptotic behavior of average Lp -discrepancies
Journal of Complexity
A New Randomized Algorithm to Approximate the Star Discrepancy Based on Threshold Accepting
SIAM Journal on Numerical Analysis
Weighted discrepancy and numerical integration in function spaces
Journal of Complexity
On weighted Hilbert spaces and integration of functions of infinitely many variables
Journal of Complexity
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We extend the notion of L"2-B-discrepancy introduced in [E. Novak, H. Wozniakowski, L"2 discrepancy and multivariate integration, in: W.W.L. Chen, W.T. Gowers, H. Halberstam, W.M. Schmidt, and R.C. Vaughan (Eds.), Analytic Number Theory. Essays in Honour of Klaus Roth, Cambridge University Press, Cambridge, 2009, pp. 359-388] to what we shall call weighted geometric L"2-discrepancy. This extension enables us to consider weights in order to moderate the importance of different groups of variables, as well as to consider volume measures different from the Lebesgue measure and classes of test sets different from measurable subsets of Euclidean spaces. We relate the weighted geometric L"2-discrepancy to numerical integration defined over weighted reproducing kernel Hilbert spaces and settle in this way an open problem posed by Novak and Wozniakowski. Furthermore, we prove an upper bound for the numerical integration error for cubature formulas that use admissible sample points. The set of admissible sample points may actually be a subset of the integration domain of measure zero. We illustrate that particularly in infinite-dimensional numerical integration it is crucial to distinguish between the whole integration domain and the set of those sample points that actually can be used by the algorithms.