We study the $$L_2$$ L 2 -approximation of functions from a Hilbert space and compare the sampling numbers with the approximation numbers. The sampling number $$e_n$$ e n is the minimal worst-case error that can be achieved with n function values, whereas the approximation number $$a_n$$ a n is the minimal worst-case error that can be achieved with n pieces of arbitrary linear information (like derivatives or Fourier coefficients). We show that $$\begin{aligned} e_n \,\lesssim \, \sqrt{\frac{1}{k_n} \sum _{j\ge k_n} a_j^2}, \end{aligned}$$ e n ≲ 1 k n ∑ j ≥ k n a j 2 , where $$k_n \asymp n/\log (n)$$ k n ≍ n / log ( n ) . This proves that the sampling numbers decay with the same polynomial rate as the approximation numbers and therefore that function values are basically as powerful as arbitrary linear information if the approximation numbers are square-summable. Our result applies, in particular, to Sobolev spaces $$H^s_\mathrm{mix}(\mathbb {T}^d)$$ H mix s ( T d ) with dominating mixed smoothness $$s>1/2$$ s > 1 / 2 and dimension $$d\in \mathbb {N}$$ d ∈ N , and we obtain $$\begin{aligned} e_n \,\lesssim \, n^{-s} \log ^{sd}(n). \end{aligned}$$ e n ≲ n - s log sd ( n ) . For $$d>2s+1$$ d > 2 s + 1 , this improves upon all previous bounds and disproves the prevalent conjecture that Smolyak’s (sparse grid) algorithm is optimal.

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 rapid mixing of SB are equivalent. Additionally, the spectral gap of SW and SB on planar graphs is bounded from above and from below by the spectral gap of these dynamics on the corresponding dual graph with suitably changed temperature.As a consequence we obtain rapid mixing of the Swendsen-Wang dynamics for the Potts model on the two-dimensional square lattice at all non-critical temperatures as well as rapid mixing for the two-dimensional Ising model at all temperatures. Furthermore, we obtain new results for general graphs at high or low enough temperatures.Acknowledgements. First and foremost, I would like to thank the advisor of my PhD thesis (the present paper is based on it), Erich Novak, for his constant support and the many discussions we had, not only about mathematics. This has certainly led to the development of my present interest in and fun with mathematics.I am also grateful to Aicke Hinrichs and Daniel Rudolf for helpful comments and suggestions during the work on my thesis, as well as for the valuable discussions we had about various topics.Finally, I want to express my thanks to my colleagues and friends Erich, Aicke, Daniel, Henning Kempka, Lev Markhasin, Philipp Rudolph, Winfried Sickel, Markus Weimar and Heidi Weyhausen who were (more or less regular) members of our daily coffee break that served with its friendly and relaxed atmosphere as a source of extra energy every day.

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 admissible parameters (s ≥ 1/p). In particular, we obtain upper bounds for the difficult the case of small smoothness which is given for Triebel-Lizorkin spaces F s p,θ in case 1 < θ < p < ∞ with 1/p < s ≤ 1/θ. The presented upper bounds on the worst-case error show a completely different behavior compared to "large" smoothness s > 1/θ. In the latter case the presented upper bounds are optimal, i.e., they can not be improved by any other cubature formula. The optimality for "small" smoothness is open.

We prove the curse of dimensionality in the worst case setting for numerical integration for a number of classes of smooth d-variate functions. Roughly speaking, we consider different bounds for the directional or partial derivatives of f ∈ C k (D d ) and ask whether the curse of dimensionality holds for the respective classes of functions. We always assume that D d ⊂ R d has volume one and we often assume additionally that D d is either convex or that its radius is proportional to √ d. In particular, D d can be the unit cube. We consider various values of k including the case k = ∞ which corresponds to infinitely differentiable functions. We obtain necessary and sufficient conditions, and in some cases a full characterization for the curse of dimensionality. For infinitely differentiable functions we prove the curse if the bounds on the successive derivatives are appropriately large. The proof technique is based on a volume estimate of a neighborhood of the convex hull of n points which decays exponentially fast in d. For k = ∞, we also study conditions for quasi-polynomial, weak and uniform weak tractability. In particular, weak tractability holds if all directional derivatives are bounded by one. It is still an open problem if weak tractability holds if all partial derivatives are bounded by one.