Tino Ullrich

Dinh Dũng, Vladimir N. Temlyakov, Tino Ullrich

Hyperbolic cross approximation is a special type of multivariate approximation. Recently, driven by applications in engineering, biology, medicine and other areas of science new challenging problems have appeared. The common feature of these problems is high dimensions. We present here a survey on classical methods developed in multivariate approximation theory, which are known to work very well for moderate dimensions and which have potential for applications in really high dimensions. The theory of hyperbolic cross approximation and related theory of functions with mixed smoothness are under detailed study for more than 50 years. It is now well understood that this theory is important both for theoretical study and for practical applications. It is also understood that both theoretical analysis and construction of practical algorithms are very difficult problems. This explains why many fundamental problems in this area are still unsolved. Only a few survey papers and monographs on the topic are published. This and recently discovered deep connections between the hyperbolic cross approximation (and related sparse grids) and other areas of mathematics such as probability, discrepancy, and numerical integration motivated us to write this survey. We try to put emphases on the development of ideas and methods rather than list all the known results in the area. We formulate many problems, which, to our knowledge, are open problems. We also include some very recent results on the topic, which sometimes highlight new interesting directions of research. We hope that this survey will stimulate further active research in this fascinating and challenging area of approximation theory and numerical analysis.

Glenn Byrenheid, Dinh Dũng, Winfried Sickel et al.

We investigate the rate of convergence of linear sampling numbers of the embedding $H^{α,β} (\mathbb{T}^d) \hookrightarrow H^γ(\mathbb{T}^d)$. Here $α$ governs the mixed smoothness and $β$ the isotropic smoothness in the space $H^{α,β}(\mathbb{T}^d)$ of hybrid smoothness, whereas $H^γ(\mathbb{T}^d)$ denotes the isotropic Sobolev space. If $γ>β$ we obtain sharp polynomial decay rates for the first embedding realized by sampling operators based on "energy-norm based sparse grids" for the classical trigonometric interpolation. This complements earlier work by Griebel, Knapek and Dũng, Ullrich, where general linear approximations have been considered. In addition, we study the embedding $H^α_{mix} (\mathbb{T}^d) \hookrightarrow H^γ_{mix}(\mathbb{T}^d)$ and achieve optimality for Smolyak's algorithm applied to the classical trigonometric interpolation. This can be applied to investigate the sampling numbers for the embedding $H^α_{mix} (\mathbb{T}^d) \hookrightarrow L_q(\mathbb{T}^d)$ for $2<q\leq \infty$ where again Smolyak's algorithm yields the optimal order. The precise decay rates for the sampling numbers in the mentioned situations always coincide with those for the approximation numbers, except probably in the limiting situation $β= γ$ (including the embedding into $L_2(\mathbb{T}^d)$). The best what we could prove there is a (probably) non-sharp results with a logarithmic gap between lower and upper bound.

Mario Ullrich, Tino Ullrich

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 $\mathbf{B}^s_{p,θ}$ and Triebel-Lizorkin spaces $\mathbf{F}^s_{p,θ}$ and our results treat the whole range of admissible parameters $(s\geq 1/p)$. In particular, we obtain upper bounds for the difficult the case of small smoothness which is given for Triebel-Lizorkin spaces $\mathbf{F}^s_{p,θ}$ in case $1<θ<p<\infty$ with $1/p<s\leq 1/θ$. 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.

Van Kien Nguyen, Mario Ullrich, Tino Ullrich

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 integration of multivariate functions with non-trivial boundary data, in particular non-periodic functions, arises. In this paper we give a general result that the asymptotic rate of the minimal worst-case integration error is not affected by boundary conditions in the above mentioned spaces. In fact, we propose two tailored modifications of Frolov's cubature formulae suitable for functions supported on the cube (not in the cube) which provide the same minimal worst-case error up to a constant. This constant involves the norms of a ``change of variable'' and a ``pointwise multiplication'' mapping, respectively, between the function spaces of interest. In fact, we complement, extend and improve classical results by Bykovskii, Dubinin and Temlyakov on the boundedness of change of variable mappings in Besov-Sobolev spaces of mixed smoothness. Our proof technique relies on a new characterization via integral means of mixed differences and maximal function techniques, general enough to treat Besov and Triebel-Lizorkin spaces at once. The second modification, which only tackles the case of periodic functions, is based on a pointwise multiplication and is therefore most likely more suitable for applications than the (traditional) ``change of variable'' approach. These new theoretical insights are expected to be useful for the design of new (and robust) cubature rules for multivariate functions on the cube.

Glenn Byrenheid, Tino Ullrich

In this paper we consider the $L_q$-approximation of multivariate periodic functions $f$ with $L_p$-bounded mixed derivative (difference). The (possibly non-linear) reconstruction algorithm is supposed to recover the function from function values, sampled on a discrete set of $n$ sampling nodes. The general performance is measured in terms of (non-)linear sampling widths $\varrho_n$. We conduct a systematic analysis of Smolyak type interpolation algorithms in the framework of Besov-Lizorkin-Triebel spaces of dominating mixed smoothness based on specifically tailored discrete Littlewood-Paley type characterizations. As a consequence, we provide sharp upper bounds for the asymptotic order of the (non-)linear sampling widths in various situations and close some gaps in the existing literature. For example, in case $2\leq p<q<\infty$ and $r>1/p$ the linear sampling widths $\varrho_n^{\text{lin}}(S^r_pW(\mathbb{T}^d),L_q(\mathbb{T}^d))$ and $\varrho^{\text{lin}}_n(S^r_{p,\infty}B(\mathbb{T}^d),L_q(\mathbb{T}^d))$ show the asymptotic behavior of the corresponding Gelfand $n$-widths, whereas in case $1 < p < q \leq 2$ and $r>1/p$ the linear sampling widths match the corresponding linear widths. In the mentioned cases linear Smolyak interpolation based on univariate classical trigonometric interpolation turns out to be optimal.

Christopher Kacwin, Jens Oettershagen, Tino Ullrich

We deal with lattices that are generated by the Vandermonde matrices associated to the roots of Chebyshev-polynomials. If the dimension $d$ of the lattice is a power of two, i.e. $d=2^m, m \in \mathbb{N}$, the resulting lattice is an admissible lattice in the sense of Skriganov. Those are related to the Frolov cubature formulas, which recently drew attention due to their optimal convergence rates in a broad range of function spaces with mixed smoothness. We prove that the resulting lattices are orthogonal and possess a lattice representation matrix with entries not larger than $2$ (in modulus). This allows for an efficient enumeration of the Frolov cubature nodes in the $d$-cube $[-1/2,1/2]^d$ up to dimension $d=16$.

Thomas Kühn, Sebastian Mayer, Tino Ullrich

In this paper, we reveal a new connection between approximation numbers of periodic Sobolev type spaces, where the smoothness weights on the Fourier coefficients are induced by a (quasi-)norm $\|\cdot\|$ on $\mathbb{R}^d$, and entropy numbers of the embedding $\textrm{id}: \ell_{\|\cdot\|}^d \to \ell_\infty^d$. This connection yields preasymptotic error bounds for approximation numbers of isotropic Sobolev spaces, spaces of analytic functions, and spaces of Gevrey type in $L_2$ and $H^1$, which find application in the context of Galerkin methods. Moreover, we observe that approximation numbers of certain Gevrey type spaces behave preasymptotically almost identical to approximation numbers of spaces of dominating mixed smoothness. This observation can be exploited, for instance, for Galerkin schemes for the electronic Schrödinger equation, where mixed regularity is present.

Glenn Byrenheid, Lutz Kämmerer, Tino Ullrich et al.

We consider the approximate recovery of multivariate periodic functions from a discrete set of function values taken on a rank-$s$ integration lattice. The main result is the fact that any (non-)linear reconstruction algorithm taking function values on a rank-$s$ lattice of size $M$ has a dimension-independent lower bound of $2^{-(α+1)/2} M^{-α/2}$ when considering the optimal worst-case error with respect to function spaces of (hybrid) mixed smoothness $α>0$ on the $d$-torus. We complement this lower bound with upper bounds that coincide up to logarithmic terms. These upper bounds are obtained by a detailed analysis of a rank-1 lattice sampling strategy, where the rank-1 lattices are constructed by a component-by-component (CBC) method. This improves on earlier results obtained in [25] and [27]. The lattice (group) structure allows for an efficient approximation of the underlying function from its sampled values using a single one-dimensional fast Fourier transform. This is one reason why these algorithms keep attracting significant interest. We compare our results to recent (almost) optimal methods based upon samples on sparse grids.

Aicke Hinrichs, Lev Markhasin, Jens Oettershagen et al.

We investigate quasi-Monte Carlo rules for the numerical integration of multivariate periodic functions from Besov spaces $S^r_{p,q}B(\mathbb{T}^d)$ with dominating mixed smoothness $1/p<r<2$. We show that order 2 digital nets achieve the optimal rate of convergence $N^{-r} (\log N)^{(d-1)(1-1/q)}$. The logarithmic term does not depend on $r$ and hence improves the known bound provided by J. Dick for the special case of Sobolev spaces $H^r_{\text{mix}}(\mathbb{T}^d)$. Secondly, the rate of convergence is independent of the integrability $p$ of the Besov space, which allows for sacrificing integrability in order to gain Besov regularity. Our method combines characterizations of periodic Besov spaces with dominating mixed smoothness via Faber bases with sharp estimates of Haar coefficients for the discrepancy function of higher order digital nets. Moreover, we provide numerical computations which indicate that this bound also holds for the case $r=2$.