30 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
L_p- and S_{p,q}^rB-discrepancy of (order 2) digital nets
Dick proved that all order digital nets satisfy optimal upper bounds of
the -discrepancy. We give an alternative proof for this fact using Haar
bases. Furthermore, we prove that all digital nets satisfy optimal upper bounds
of the -discrepancy for a certain parameter range and enlarge that
range for order digitals nets. -, - and -discrepancy is considered as well
-Sampling recovery for non-compact subclasses of
In this paper we study the sampling recovery problem for certain relevant
multivariate function classes which are not compactly embedded into .
Recent tools relating the sampling numbers to the Kolmogorov widths in the
uniform norm are therefore not applicable. In a sense, we continue the research
on the small smoothness problem by considering "very" small smoothness in the
context of Besov and Triebel-Lizorkin spaces with dominating mixed regularity.
There is not much known on the recovery of such functions except of an old
result by Oswald in the univariate situation. As a first step we prove the
uniform boundedness of the -norm of the Faber-Schauder coefficients in
a fixed level. Using this we are able to control the error made by a (Smolyak)
truncated Faber-Schauder series in with . It turns out that the
main rate of convergence is sharp. As a consequence we obtain results also for
, a space which is ``close'' to the space
which is important in numerical analysis, especially
numerical integration, but has rather bad Fourier analytic properties