294 research outputs found
Discrepancy of Symmetric Products of Hypergraphs
For a hypergraph , its --fold symmetric
product is . We give
several upper and lower bounds for the -color discrepancy of such products.
In particular, we show that the bound proven for all in [B. Doerr, A. Srivastav, and P.
Wehr, Discrepancy of {C}artesian products of arithmetic progressions, Electron.
J. Combin. 11(2004), Research Paper 5, 16 pp.] cannot be extended to more than
colors. In fact, for any and such that does not divide
, there are hypergraphs having arbitrary large discrepancy and
. Apart
from constant factors (depending on and ), in these cases the symmetric
product behaves no better than the general direct product ,
which satisfies .Comment: 12 pages, no figure
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
Discrepancy Bounds for Mixed Sequences
A mixed sequence is a sequence in the -dimensional unit cube
which one obtains by concatenating a -dimensional low-discrepancy
sequence with an -dimensional random sequence.
We discuss some probabilistic bounds on the star discrepancy of
mixed sequences
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