600 research outputs found
Some Results on the Complexity of Numerical Integration
This is a survey (21 pages, 124 references) written for the MCQMC 2014
conference in Leuven, April 2014. We start with the seminal paper of Bakhvalov
(1959) and end with new results on the curse of dimension and on the complexity
of oscillatory integrals. Some small errors of earlier versions are corrected
Optimal randomized multilevel algorithms for infinite-dimensional integration on function spaces with ANOVA-type decomposition
In this paper, we consider the infinite-dimensional integration problem on
weighted reproducing kernel Hilbert spaces with norms induced by an underlying
function space decomposition of ANOVA-type. The weights model the relative
importance of different groups of variables. We present new randomized
multilevel algorithms to tackle this integration problem and prove upper bounds
for their randomized error. Furthermore, we provide in this setting the first
non-trivial lower error bounds for general randomized algorithms, which, in
particular, may be adaptive or non-linear. These lower bounds show that our
multilevel algorithms are optimal. Our analysis refines and extends the
analysis provided in [F. J. Hickernell, T. M\"uller-Gronbach, B. Niu, K.
Ritter, J. Complexity 26 (2010), 229-254], and our error bounds improve
substantially on the error bounds presented there. As an illustrative example,
we discuss the unanchored Sobolev space and employ randomized quasi-Monte Carlo
multilevel algorithms based on scrambled polynomial lattice rules.Comment: 31 pages, 0 figure
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