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- Aicke Hinrichs, Erich Novak, Mario Ullrich, Henryk Wozniakowski
- J. Complexity
- 2014

We prove the curse of dimensionality for multivariate integration of C k functions. The proofs are based on volume estimates for k = 1 together with smoothing by convolution. This allows us to obtain smooth fooling functions for k > 1.

- Mario Ullrich
- Random Struct. Algorithms
- 2013

We prove that the spectral gap of the Swendsen-Wang process for the Potts model on graphs with bounded degree is bounded from below by some constant times the spectral gap of any single-spin dynamics. This implies rapid mixing for the two-dimensional Potts model at all temperatures above the critical one, as well as rapid mixing at the critical temperature… (More)

- Mario Ullrich
- SIAM J. Discrete Math.
- 2014

We prove that the spectral gap of the Swendsen-Wang dynamics for the random-cluster model is larger than the spectral gap of a single-bond dynamics , that updates only a single edge per step. For this we give a representation of the algorithms on the joint (Potts/random-cluster) model. Furthermore we obtain upper and lower bounds on the mixing time of the… (More)

- Mario Ullrich
- ArXiv
- 2012

We prove that the spectral gap of the Swendsen-Wang dynamics for the random-cluster model on arbitrary graphs with m edges is bounded above by 16m log m times the spectral gap of the single-bond (or heat-bath) dynamics. This and the corresponding lower bound (from [U12]) imply that rapid mixing of these two dynamics is equivalent. Using the known lower… (More)

- Aicke Hinrichs, Erich Novak, Mario Ullrich
- Journal of Approximation Theory
- 2014

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… (More)

We prove comparison results for the Swendsen-Wang (SW) dynamics, the heat-bath (HB) dynamics for the Potts model and the single-bond (SB) dynamics for the random-cluster model on arbitrary graphs. In particular, we prove that rapid (i.e. polynomial) mixing of HB implies rapid mixing of SW on graphs with bounded maximum degree and that rapid mixing of SW and… (More)

- Mario Ullrich, Tino Ullrich
- SIAM J. Numerical Analysis
- 2016

We prove upper bounds on the order of convergence of Frolov's cubature formula for numerical integration in function spaces of dominating mixed smoothness on the unit cube with homogeneous boundary condition. More precisely, we study worst-case integration errors for Besov B s p,θ and Triebel-Lizorkin spaces F s p,θ and our results treat the whole range of… (More)

We prove positivity of the Markov operators that correspond to the hit-and-run algorithm , random scan Gibbs sampler, slice sampler and Metropolis algorithm with positive proposal. In particular, the results show that it is not necessary to consider the lazy versions of these Markov chains. The proof relies on a well known lemma which relates the positivity… (More)

- Erich Novak, Mario Ullrich, Henryk Wozniakowski
- J. Complexity
- 2015

We analyze univariate oscillatory integrals for the standard Sobolev spaces H s of periodic and non-periodic functions with an arbitrary integer s ≥ 1. We find matching lower and upper bounds on the minimal worst case error of algorithms that use n function or derivative values. We also find sharp bounds on the information complexity which is the minimal n… (More)

In a recent article by two of the present authors it turned out that Frolov's cubature formulae are optimal and universal for various settings (Besov-Triebel-Lizorkin spaces) of functions with dominating mixed smoothness. Those cubature formulae go well together with functions supported inside the unit cube [0, 1] d. The question for the optimal numerical… (More)