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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.

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 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)

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)

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)

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)

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)

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)

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)

- MARIO ULLRICH
- 2010

The Ising model is often referred to as the most studied model of statistical physics. It describes the behavior of ferro-magnetic material at different temperatures. It is an interesting model also for mathematicians, because although the Boltzmann distribution is continuous in the temperature parameter, the behavior of the usual single-spin dynamics to… (More)