2,111 research outputs found
Tractability of multivariate analytic problems
In the theory of tractability of multivariate problems one usually studies
problems with finite smoothness. Then we want to know which -variate
problems can be approximated to within by using, say,
polynomially many in and function values or arbitrary
linear functionals.
There is a recent stream of work for multivariate analytic problems for which
we want to answer the usual tractability questions with
replaced by . In this vein of research, multivariate
integration and approximation have been studied over Korobov spaces with
exponentially fast decaying Fourier coefficients. This is work of J. Dick, G.
Larcher, and the authors. There is a natural need to analyze more general
analytic problems defined over more general spaces and obtain tractability
results in terms of and .
The goal of this paper is to survey the existing results, present some new
results, and propose further questions for the study of tractability of
multivariate analytic questions
Hot new directions for quasi-Monte Carlo research in step with applications
This article provides an overview of some interfaces between the theory of
quasi-Monte Carlo (QMC) methods and applications. We summarize three QMC
theoretical settings: first order QMC methods in the unit cube and in
, and higher order QMC methods in the unit cube. One important
feature is that their error bounds can be independent of the dimension
under appropriate conditions on the function spaces. Another important feature
is that good parameters for these QMC methods can be obtained by fast efficient
algorithms even when is large. We outline three different applications and
explain how they can tap into the different QMC theory. We also discuss three
cost saving strategies that can be combined with QMC in these applications.
Many of these recent QMC theory and methods are developed not in isolation, but
in close connection with applications
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
Rank-1 lattice rules for multivariate integration in spaces of permutation-invariant functions: Error bounds and tractability
We study multivariate integration of functions that are invariant under
permutations (of subsets) of their arguments. We find an upper bound for the
th minimal worst case error and show that under certain conditions, it can
be bounded independent of the number of dimensions. In particular, we study the
application of unshifted and randomly shifted rank- lattice rules in such a
problem setting. We derive conditions under which multivariate integration is
polynomially or strongly polynomially tractable with the Monte Carlo rate of
convergence . Furthermore, we prove that those tractability
results can be achieved with shifted lattice rules and that the shifts are
indeed necessary. Finally, we show the existence of rank- lattice rules
whose worst case error on the permutation- and shift-invariant spaces converge
with (almost) optimal rate. That is, we derive error bounds of the form
for all , where denotes
the smoothness of the spaces.
Keywords: Numerical integration, Quadrature, Cubature, Quasi-Monte Carlo
methods, Rank-1 lattice rules.Comment: 26 pages; minor changes due to reviewer's comments; the final
publication is available at link.springer.co
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