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Tino Ullrich

Publications and source records attributed to Tino Ullrich.

At least 55 records · Page 3Linked to original sources
arXiv2017

Hyperbolic Cross Approximation

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.

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arXiv2017

Optimal sampling recovery of mixed order Sobolev embeddings via discrete Littlewood-Paley type characterizations

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 $2\leq p 1/p$ 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.

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arXiv2017

On uniform boundedness of dyadic averaging operators in spaces of Hardy-Sobolev type

We give an alternative proof of recent results by the authors on uniform boundedness of dyadic averaging operators in (quasi-)Banach spaces of Hardy-Sobolev and Triebel-Lizorkin type. This result served as the main tool to establish Schauder basis properties of suitable enumerations of the univariate Haar system in the mentioned spaces. The rather elementary proof here is based on characterizations of the respective spaces in terms of orthogonal compactly supported Daubechies wavelets.

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arXiv2016

Counting via entropy: new preasymptotics for the approximation numbers of Sobolev embeddings

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.

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arXiv2016

General coorbit space theory for quasi-Banach spaces and inhomogeneous function spaces with variable smoothness and integrability

In this paper we propose a general coorbit space theory suitable to define coorbits of quasi-Banach spaces using an abstract continuous frame, indexed by a locally compact Hausdorff space, and an associated generalized voice transform. The proposed theory realizes a further step in the development of a universal abstract theory towards various function spaces and their atomic decompositions which has been initiated by Feichtinger and Gr{ö}chenig in the late 1980ies. We combine the recent approaches in Rauhut, Ullrich and Rauhut to identify, in particular, various inhomogeneous (quasi-Banach) spaces of Besov-Lizorkin-Triebel type. To prove the potential of our new theory we apply it to spaces with variable smoothness and integrability which have attracted significant interest in the last 10 years. From the abstract discretization machinery we obtain atomic decompositions as well as wavelet frame isomorphisms for these spaces.

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arXiv2016

Tight error bounds for rank-1 lattice sampling in spaces of hybrid mixed smoothness

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.

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arXiv2016

On the orthogonality of the Chebyshev-Frolov lattice and applications

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 $d=16$.

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arXiv2016

The role of Frolov's cubature formula for functions with bounded mixed derivative

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 $1<θ 1/θ$. 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.

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arXiv2015

Change of variable in spaces of mixed smoothness and numerical integration of multivariate functions on the unit cube

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.

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arXiv2015

Lower bounds for Haar projections: deterministic examples

In a previous paper by the authors the existence of Haar projections with growing norms in Sobolev-Triebel-Lizorkin spaces has been shown via a probabilistic argument. This existence was sufficient to determine the precise range of Triebel-Lizorkin spaces for which the Haar system is an unconditional basis. The aim of the present paper is to give simple deterministic examples of Haar projections that show this growth behavior in the respective range of parameters.

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arXiv2015

Optimal quasi-Monte Carlo rules on order 2 digital nets for the numerical integration of multivariate periodic functions

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 $r=2$.

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arXiv2014

Sampling on energy-norm based sparse grids for the optimal recovery of Sobolev type functions in $H^γ$

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.

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arXiv2014

A new framework for generalized Besov-type and Triebel-Lizorkin-type spaces

In this paper, the authors propose a new framework under which a theory of generalized Besov-type and Triebel-Lizorkin-type function spaces is developed. Many function spaces appearing in harmonic analysis fall under the scope of this new framework. Among others, the boundedness of the Hardy-Littlewood maximal operator or the related vector-valued maximal function on any of these function spaces is not required to construct these generalized scales of smoothness spaces. Instead of this, a key idea used in this framework is an application of the Peetre maximal function. This idea originates from recent findings in the abstract coorbit space theory obtained by Holger Rauhut and Tino Ullrich. Under this new setting, the authors establish the boundedness of pseudo-differential operators based on atomic and molecular characterizations and also the boundedness of the Fourier multipliers. The characterizations of these function spaces by means of differences and oscillations are also established. As further applications of this new framework, the authors reexamine and polish some existing results for many different scales of function spaces.

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arXiv2014

Lower bounds for the integration error for multivariate functions with mixed smoothness and optimal Fibonacci cubature for functions on the square

We prove lower bounds for the error of optimal cubature formulae for $d$-variate functions from Besov spaces of mixed smoothness $B^α_{p,θ}({\mathbb G}^d)$ in the case $0 < p, θ\le \infty$ and $α> 1/p$, where ${\mathbb G}^d$ is either the $d$-dimensional torus ${\mathbb T}^d$ or the $d$-dimensional unit cube ${\mathbb I}^d$. We prove upper bounds for QMC methods of integration on the Fibonacci lattice for bivariate periodic functions from $B^α_{p,θ}({\mathbb T}^2)$ in the case $1\leq p \leq \infty$, $0 < θ\leq \infty$, $α>1/p$. A non-periodic modification of this classical formula yields upper bounds for $B^α_{p,θ}({\mathbb I}^2)$ if $1/p<α<1+1/p$. In combination these results yield the correct asymptotic error of optimal cubature formulae for functions from $B^α_{p,θ}({\mathbb G}^2)$ and indicate that a corresponding result is most likely also true in case $d>2$. This is compared to the correct asymptotic of optimal cubature formulae on Smolyak grids which results in the observation that any cubature formula on Smolyak grids is never optimal for the general setting.

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arXiv2013

Approximation of mixed order Sobolev functions on the $d$-torus -- Asymptotics, preasymptotics and $d$-dependence

We investigate the approximation of $d$-variate periodic functions in Sobolev spaces of dominating mixed (fractional) smoothness $s>0$ on the $d$-dimensional torus, where the approximation error is measured in the $L_2-$norm. In other words, we study the approximation numbers of the Sobolev embeddings $H^s_{\rm mix}(\mathbb{T}^d)\hookrightarrow L_2(\mathbb{T}^d)$, with particular emphasis on the dependence on the dimension $d$. For any fixed smoothness $s>0$, we find the exact asymptotic behavior of the constants as $d\to\infty$. We observe super-exponential decay of the constants in $d$, if $n$, the number of linear samples of $f$, is large. In addition, motivated by numerical implementation issues, we also focus on the error decay that can be achieved by low rank approximations. We present some surprising results for the so-called ``preasymptotic'' decay and point out connections to the recently introduced notion of quasi-polynomial tractability of approximation problems.

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arXiv2013

Entropy and sampling numbers of classes of ridge functions

We study properties of ridge functions $f(x)=g(a\cdot x)$ in high dimensions $d$ from the viewpoint of approximation theory. The considered function classes consist of ridge functions such that the profile $g$ is a member of a univariate Lipschitz class with smoothness $α> 0$ (including infinite smoothness), and the ridge direction $a$ has $p$-norm $\|a\|_p \leq 1$. First, we investigate entropy numbers in order to quantify the compactness of these ridge function classes in $L_{\infty}$. We show that they are essentially as compact as the class of univariate Lipschitz functions. Second, we examine sampling numbers and face two extreme cases. In case $p=2$, sampling ridge functions on the Euclidean unit ball faces the curse of dimensionality. It is thus as difficult as sampling general multivariate Lipschitz functions, a result in sharp contrast to the result on entropy numbers. When we additionally assume that all feasible profiles have a first derivative uniformly bounded away from zero in the origin, then the complexity of sampling ridge functions reduces drastically to the complexity of sampling univariate Lipschitz functions. In between, the sampling problem's degree of difficulty varies, depending on the values of $α$ and $p$. Surprisingly, we see almost the entire hierarchy of tractability levels as introduced in the recent monographs by Novak and Woźniakowski.

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arXiv2012

New Characterizations of Besov-Triebel-Lizorkin-Hausdorff Spaces Including Coorbits and Wavelets

In this paper, the authors establish new characterizations of the recently introduced Besov-type spaces $\dot{B}^{s,τ}_{p,q}({\mathbb R}^n)$ and Triebel-Lizorkin-type spaces $\dot{F}^{s,τ}_{p,q}({\mathbb R}^n)$ with $p\in (0,\infty]$, $s\in{\mathbb R}$, $τ\in [0,\infty)$, and $q\in (0,\infty]$, as well as their preduals, the Besov-Hausdorff spaces $B\dot{H}^{s,τ}_{p,q}(\R^n)$ and Triebel-Lizorkin-Hausdorff spaces $F\dot{H}^{s,τ}_{p,q}(\R^n)$, in terms of the local means, the Peetre maximal function of local means, and the tent space (the Lusin area function) in both discrete and continuous types. As applications, the authors then obtain interpretations as coorbits in the sense of H. Rauhut in [Studia Math. 180 (2007), 237-253] and discretizations via the biorthogonal wavelet bases for the full range of parameters of these function spaces. Even for some special cases of this setting such as $\dot F^s_{\infty,q}({\mathbb R}^n)$ for $s\in{\mathbb R}$, $q\in (0,\infty]$ (including $\mathop\mathrm{BMO} ({\mathbb R}^n)$ when $s=0$, $q=2$), the $Q$ space $Q_α({\mathbb R}^n)$, the Hardy-Hausdorff space $HH_{-α}({\mathbb R}^n)$ for $α\in (0,\min\{\frac n2,1\})$, the Morrey space ${\mathcal M}^u_p({\mathbb R}^n)$ for $1<p\le u<\infty$, and the Triebel-Lizorkin-Morrey space $\dot{\mathcal{E}}^s_{upq}({\mathbb R}^n)$ for $0<p\le u<\infty$, $s\in{\mathbb R}$ and $q\in(0,\infty]$, some of these results are new.

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