1,012 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
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
Lattice rules with random achieve nearly the optimal error independently of the dimension
We analyze a new random algorithm for numerical integration of -variate
functions over from a weighted Sobolev space with dominating mixed
smoothness and product weights
, where the functions are continuous and
periodic when . The algorithm is based on rank- lattice rules
with a random number of points~. For the case , we prove that
the algorithm achieves almost the optimal order of convergence of
, where the implied constant is independent of
the dimension~ if the weights satisfy . The same rate of convergence holds for the more
general case by adding a random shift to the lattice rule with
random . This shows, in particular, that the exponent of strong tractability
in the randomized setting equals , if the weights decay fast
enough. We obtain a lower bound to indicate that our results are essentially
optimal. This paper is a significant advancement over previous related works
with respect to the potential for implementation and the independence of error
bounds on the problem dimension. Other known algorithms which achieve the
optimal error bounds, such as those based on Frolov's method, are very
difficult to implement especially in high dimensions. Here we adapt a
lesser-known randomization technique introduced by Bakhvalov in 1961. This
algorithm is based on rank- lattice rules which are very easy to implement
given the integer generating vectors. A simple probabilistic approach can be
used to obtain suitable generating vectors.Comment: 17 page
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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