We continue the research on the asymptotic and preasymptotic decay of singular numbers for tensor product Hilbert-Sobolev type embeddings in high dimensions with special emphasis on the influence of the underlying dimension $d$. The main focus in this paper lies on tensor products involving univariate Sobolev type spaces with different smoothness. We study the embeddings into $L_2$ and $H^1$. In other words, we investigate the worst-case approximation error measured in $L_2$ and $H^1$ when only $n$ linear samples of the function are available. Recent progress in the field shows that accurate bounds on the singular numbers are essential for recovery bounds using only function values. The asymptotic bounds in our setting are known for a long time. In this paper we contribute the correct asymptotic constant and explicit bounds in the preasymptotic range for $n$. We complement and improve on several results in the literature. In addition, we refine the error bounds coming from the setting where the smoothness vector is moderately increasing, which has been already studied by Papageorgiou and Wo{ź}niakowski.
Let $Ω_i\subset\mathbb{R}^{n_i}$, $i=1,\ldots,m$, be given domains. In this article, we study the low-rank approximation with respect to $L^2(Ω_1\times\dots\timesΩ_m)$ of functions from Sobolev spaces with dominating mixed smoothness. To this end, we first estimate the rank of a bivariate approximation, i.e., the rank of the continuous singular value decomposition. In comparison to the case of functions from Sobolev spaces with isotropic smoothness, compare \cite{GH14,GH19}, we obtain improved results due to the additional mixed smoothness. This convergence result is then used to study the ten…
Monte Carlo Methods for Uniform Approximation on Periodic Sobolev Spaces with Mixed Smoothness
Hausdorff-Center for Mathematics（豪斯多夫数学中心）；Universität Osnabrück, Institut für Mathematik（奥斯纳布吕克大学数学研究所）；Department of Mathematics, University of Transport and Communications（交通与通信大学数学系）
In this paper, we study multivariate approximation defined over weighted anisotropic Sobolev spaces which depend on two sequences ${\bf a}=\{a_j\}_{j\geq1}$ and ${\bf b}=\{b_j\}_{j\geq1}$ of positive numbers. We obtain strong equivalences of the approximation numbers, and necessary and sufficient conditions on ${\bf a}$, ${\bf b}$ to achieve various notions of tractability of the weighted anisotropic Sobolev embeddings.
The present paper analyzes a spectral regularization of a time-reversed reaction-diffusion problem with globally and locally Lipschitz nonlinearities. This type of inverse and ill-posed problems arises in a variety of real-world applications concerning heat conduction and tumour source localization. In accordance with the weak solvability result for the forward problem, we focus on the inverse problem with high-order Sobolev-Gevrey smoothness and with Sobolev measurements. As expected from the well-known results for the linear case, we prove that this nonlinear spectral regularization possesse…
We prove that a variant of the classical Sobolev space of first-order dominating mixed smoothness is equivalent (under a certain condition) to the unanchored ANOVA space on $\mathbb{R}^d$, for $d \geq 1$. Both spaces are Hilbert spaces involving weight functions, which determine the behaviour as different variables tend to $\pm \infty$, and weight parameters, which represent the influence of different subsets of variables. The unanchored ANOVA space on $\mathbb{R}^d$ was initially introduced by Nichols & Kuo in 2014 to analyse the error of quasi-Monte Carlo (QMC) approximations for integrals o…
We consider a space-time variational formulation of parabolic initial-boundary value problems in anisotropic Sobolev spaces in combination with a Hilbert-type transformation. This variational setting is the starting point for the space-time Galerkin finite element discretization that leads to a large global linear system of algebraic equations. We propose and investigate new efficient direct solvers for this system. In particular, we use a tensor-product approach with piecewise polynomial, globally continuous ansatz and test functions. The developed solvers are based on the Bartels-Stewart met…
QMC integration based on arbitrary (t,m,s)-nets yields optimal convergence rates on several scales of function spaces
Institut für Mathematik, Universität Osnabrück（奥斯纳布吕克大学）；School of Mathematics and Statistics, University of New South Wales（新南威尔士大学）；Stuttgart Technology Center, Sony Semiconductor Solutions Europe（索尼半导体解决方案欧洲斯图加特技术中心）；Mathematisches Institut, Friedrich-Schiller-Universität Jena（耶拿大学）
We examine the necessary and sufficient complexity of neural networks to approximate functions from different smoothness spaces under the restriction of encodable network weights. Based on an entropy argument, we start by proving lower bounds for the number of nonzero encodable weights for neural network approximation in Besov spaces, Sobolev spaces and more. These results are valid for all sufficiently smooth activation functions. Afterwards, we provide a unifying framework for the construction of approximate partitions of unity by neural networks with fairly general activation functions. Thi…
Embeddings for Infinite-Dimensional Integration and $L_2$-Approximation with Increasing Smoothness
Christian-Albrechts-Universität zu Kiel（基尔大学）；Technische Universität Kaiserslautern（凯泽斯劳滕工业大学）；Johannes-Kepler-Universität Linz（林茨约翰内斯开普勒大学）；University of Kentucky（肯塔基大学）
We study fractional variants of the quasi-norms introduced by Brezis, Van Schaftingen, and Yung in the study of the Sobolev space $\dot W^{1,p}$. The resulting spaces are identified as a special class of real interpolation spaces of Sobolev-Slobodecki\uı spaces. We establish the equivalence between Fourier analytic definitions and definitions via difference operators acting on measurable functions. We prove various new results on embeddings and non-embeddings, and give applications to harmonic and caloric extensions. For suitable wavelet bases we obtain a characterization of the approximation …
