422 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
On Weak Tractability of the Clenshaw-Curtis Smolyak Algorithm
We consider the problem of integration of d-variate analytic functions
defined on the unit cube with directional derivatives of all orders bounded by
1. We prove that the Clenshaw Curtis Smolyak algorithm leads to weak
tractability of the problem. This seems to be the first positive tractability
result for the Smolyak algorithm for a normalized and unweighted problem. The
space of integrands is not a tensor product space and therefore we have to
develop a different proof technique. We use the polynomial exactness of the
algorithm as well as an explicit bound on the operator norm of the algorithm.Comment: 18 page
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
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