# Intractability Results for Integration and Discrepancy

@article{Novak2001IntractabilityRF, title={Intractability Results for Integration and Discrepancy}, author={Erich Novak and Henryk Wozniakowski}, journal={J. Complex.}, year={2001}, volume={17}, pages={388-441} }

We mainly study multivariate (uniform or Gaussian) integration defined for integrand spaces Fd such as weighted Sobolev spaces of functions of d variables with smooth mixed derivatives. The weight ?j moderates the behavior of functions with respect to the jth variable. For ?j?1, we obtain the classical Sobolev spaces whereas for decreasing ?j's the weighted Sobolev spaces consist of functions with diminishing dependence on the jth variables. We study the minimal errors of quadratures that use n…

## 76 Citations

Tractability of Multivariate Integration for Periodic Functions

- MathematicsJ. Complex.
- 2001

It is shown that tractability and strong tractability in the worst case setting hold under the same assumptions on the weights of the Korobov space as for the restricted classes of quadrature rules.

Tractability of Approximation and Integrationfor Weighted Tensor Product Problems overUnbounded

- Mathematics
- 2002

We study tractability and strong tractability of multivariate approximation and integration in the worst case deterministic setting. Tractability means that the number of functional evaluations…

Tractability of Approximation and Integration for Weighted Tensor Product Problems over Unbounded Domains

- Mathematics
- 2002

We study tractability and strong tractability of multivariate approximation and integration in the worst case deterministic setting. Tractability means that the number of functional evaluations…

Finite-order weights imply tractability of multivariate integration

- Computer Science, MathematicsJ. Complex.
- 2004

Uniform weak tractability of multivariate problems with increasing smoothness

- MathematicsJ. Complex.
- 2014

Periodization strategy may fail in high dimensions

- MathematicsNumerical Algorithms
- 2007

A number of cases are presented suggesting that this conjecture that all lattice rules fail for large d are indeed true, but the most interesting case, when the sum of the weights of the corresponding Sobolev space is bounded in d, remains open.

Tractability of multivariate approximation over a weighted unanchored Sobolev space: Smoothness sometimes hurts

- Mathematics
- 2008

We study d-variate L2-approximation for a weighted unanchored Sobolev space having smoothness m ≥ 1. Folk wisdom would lead us to believe that this problem should become easier as its smoothness…

Strong tractability of multivariate integration using quasi-Monte Carlo algorithms

- Mathematics, Computer ScienceMath. Comput.
- 2003

It is proved in a constructive way that multivariate integration in some weighted Sobolev spaces is strongly tractable with e-exponent equal to 1, which is the best possible value under a stronger assumption than Sloan and Wozniakowski's assumption.

The error bounds and tractability of quasi-Monte Carlo algorithms in infinite dimension

- Mathematics, Computer ScienceMath. Comput.
- 2002

It is shown constructively using the Halton sequence that the e-exponent of tractability is 1, which implies that infinite dimensional integration is no harder than one-dimensional integration.

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