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In this paper, we study the approximation problem for functions in the Gaussian-weighted Sobolev space $W^\alpha_p(\mathbb{R}^d, \gamma)$ of mixed smoothness $\alpha \in \mathbb{N}$ with error measured in the Gaussian-weighted space $L_q(\mathbb{R}^d, \gamma)$. We obtain the exact asymptotic order of pseudo $s$-numbers for the cases $1 \leq q< p < \infty$ and $p=q=2$. Additionally, we also obtain an upper bound and a lower bound for pseudo $s$-numbers of the embedding of $W^\alpha_2(\mathbb{R}^d, \gamma)$ into $L_{\infty}^{\sqrt{g}}(\mathbb{R}^d)$. Our result is an extension of that obtained in Dinh D\~ung and Van Kien Nguyen (IMA Journal of Numerical Analysis, 2023) for approximation and Kolmogorov numbers., Comment: 17 pages

We investigate the numerical approximation of integrals over $\mathbb{R}^d$ equipped with the standard Gaussian measure $\gamma$ for integrands belonging to the Gaussian-weighted Sobolev spaces $W^\alpha_p(\mathbb{R}^d, \gamma)$ of mixed smoothness $\alpha \in \mathbb{N}$ for $1 < p < \infty$. We prove the asymptotic order of the convergence of optimal quadratures based on $n$ integration nodes and propose a novel method for constructing asymptotically optimal quadratures. As for related problems, we establish by a similar technique the asymptotic order of the linear, Kolmogorov and sampling $n$-widths in the Gaussian-weighted space $L_q(\mathbb{R}^d, \gamma)$ of the unit ball of $W^\alpha_p(\mathbb{R}^d, \gamma)$ for $1 \leq q < p < \infty$ and $q=p=2$., Comment: 23 pages

In the present paper, we study the analyticity of the leftmost eigenvalue of the linear elliptic partial differential operator with random coefficient and analyze the convergence rate of the quasi-Monte Carlo method for approximation of the expectation of this quantity. The random coefficient is assumed to be represented by an affine expansion $a_0(\boldsymbol{x})+\sum_{j\in \mathbb{N}}y_ja_j(\boldsymbol{x})$, where elements of the parameter vector $\boldsymbol{y}=(y_j)_{j\in \mathbb{N}}\in U^\infty$ are independent and identically uniformly distributed on $U:=[-\frac{1}{2},\frac{1}{2}]$. Under the assumption $ \|\sum_{j\in \mathbb{N}}\rho_j|a_j|\|_{L_\infty(D)} <\infty$ with some positive sequence $(\rho_j)_{j\in \mathbb{N}}\in \ell_p(\mathbb{N})$ for $p\in (0,1]$ we show that for any $\boldsymbol{y}\in U^\infty$, the elliptic partial differential operator has a countably infinite number of eigenvalues $(\lambda_j(\boldsymbol{y}))_{j\in \mathbb{N}}$ which can be ordered non-decreasingly. Moreover, the spectral gap $\lambda_2(\boldsymbol{y})-\lambda_1(\boldsymbol{y})$ is uniformly positive in $U^\infty$. From this, we prove the holomorphic extension property of $\lambda_1(\boldsymbol{y})$ to a complex domain in $\mathbb{C}^\infty$ and estimate mixed derivatives of $\lambda_1(\boldsymbol{y})$ with respect to the parameters $\boldsymbol{y}$ by using Cauchy's formula for analytic functions. Based on these bounds we prove the dimension-independent convergence rate of the quasi-Monte Carlo method to approximate the expectation of $\lambda_1(\boldsymbol{y})$., Comment: 18 pages

We establish sparsity and summability results for coefficient sequences of Wiener-Hermite polynomial chaos expansions of countably-parametric solutions of linear elliptic and parabolic divergence-form partial differential equations with Gaussian random field inputs. The novel proof technique developed here is based on analytic continuation of parametric solutions into the complex domain. It differs from previous works that used bootstrap arguments and induction on the differentiation order of solution derivatives with respect to the parameters. The present holomorphy-based argument allows a unified, ``differentiation-free'' proof of sparsity (expressed in terms of $\ell^p$-summability or weighted $\ell^2$-summability) of sequences of Wiener-Hermite coefficients in polynomial chaos expansions in various scales of function spaces. The analysis also implies corresponding analyticity and sparsity results for posterior densities in Bayesian inverse problems subject to Gaussian priors on uncertain inputs from function spaces. Our results furthermore yield dimension-independent convergence rates of various \emph{constructive} high-dimensional deterministic numerical approximation schemes such as single-level and multi-level versions of Hermite-Smolyak anisotropic sparse-grid interpolation and quadrature in both forward and inverse computational uncertainty quantification., Comment: 165 pages

We investigate non-adaptive methods of deep ReLU neural network approximation in Bochner spaces $L_2({\mathbb U}^\infty, X, \mu)$ of functions on ${\mathbb U}^\infty$ taking values in a separable Hilbert space $X$, where ${\mathbb U}^\infty$ is either ${\mathbb R}^\infty$ equipped with the standard Gaussian probability measure, or ${\mathbb I}^\infty:= [-1,1]^\infty$ equipped with the Jacobi probability measure. Functions to be approximated are assumed to satisfy a certain weighted $\ell_2$-summability of the generalized chaos polynomial expansion coefficients with respect to the measure $\mu$. We prove the convergence rate of this approximation in terms of the size of approximating deep ReLU neural networks. These results then are applied to approximation of the solution to parametric elliptic PDEs with random inputs for the lognormal and affine cases., Comment: arXiv admin note: substantial text overlap with arXiv:2111.05504

In this paper we give exact values of the best $n$-term approximation widths of diagonal operators between $\ell_p(\mathbb{N})$ and $\ell_q(\mathbb{N})$ with $0

The purpose of the present paper is to study the computation complexity of deep ReLU neural networks to approximate functions in H\"older-Nikol'skii spaces of mixed smoothness $H_\infty^\alpha(\mathbb{I}^d)$ on the unit cube $\mathbb{I}^d:=[0,1]^d$. In this context, for any function $f\in H_\infty^\alpha(\mathbb{I}^d)$, we explicitly construct nonadaptive and adaptive deep ReLU neural networks having an output that approximates $f$ with a prescribed accuracy $\varepsilon$, and prove dimension-dependent bounds for the computation complexity of this approximation, characterized by the size and the depth of this deep ReLU neural network, explicitly in $d$ and $\varepsilon$. Our results show the advantage of the adaptive method of approximation by deep ReLU neural networks over nonadaptive one., Comment: 30 pages. arXiv admin note: text overlap with arXiv:2007.08729

We study high-dimensional nonlinear approximation of functions in H\"older-Nikol'skii spaces $H^\alpha_\infty(\mathbb{I}^d)$ on the unit cube $\mathbb{I}^d:=[0,1]^d$ having mixed smoothness, by parametric manifolds. The approximation error is measured in the $L_\infty$-norm. In this context, we explicitly constructed methods of nonlinear approximation, and give dimension-dependent estimates of the approximation error explicitly in dimension $d$ and number $N$ measuring computation complexity of the parametric manifold of approximants. For $d=2$, we derived a novel right asymptotic order of noncontinuous manifold $N$-widths of the unit ball of $H^\alpha_\infty(\mathbb{I}^2)$ in the space $L_\infty(\mathbb{I}^2)$. In constructing approximation methods, the function decomposition by the tensor product Faber series and special representations of its truncations on sparse grids play a central role., Comment: 25 pages

In this paper we study the asymptotic behavior of Kolmogorov, approximation, Bernstein and Weyl numbers of embeddings $ \mathcal{A}^{s,r}_{\rm mix}(\mathbb{T}^d) \to L_2(\mathbb{T}^d)$ and $\mathcal{A}^{s,r}_{\rm mix}(\mathbb{T}^d) \to \mathcal{A}(\mathbb{T}^d)$, where $\mathcal{A}^{s,r}_{\rm mix}(\mathbb{T}^d)$ is a weighted Wiener algebra of mixed smoothness $s$ and $\mathcal{A}(\mathbb{T}^d)$ is the Wiener algebra itself, both defined on the $d$-dimensional torus $\mathbb{T}^d$. Our main interest consists in the calculation of the associated asymptotic constants., Comment: 25 pages

We study the computation complexity of deep ReLU (Rectified Linear Unit) neural networks for the approximation of functions from the H\"older-Zygmund space of mixed smoothness defined on the $d$-dimensional unit cube when the dimension $d$ may be very large. The approximation error is measured in the norm of isotropic Sobolev space. For every function $f$ from the H\"older-Zygmund space of mixed smoothness, we explicitly construct a deep ReLU neural network having an output that approximates $f$ with a prescribed accuracy $\varepsilon$, and prove tight dimension-dependent upper and lower bounds of the computation complexity of this approximation, characterized as the size and the depth of this deep ReLU neural network, explicitly in $d$ and $\varepsilon$. The proof of these results are in particular, relied on the approximation by sparse-grid sampling recovery based on the Faber series., Comment: 5 figures

The periodization of a stationary Gaussian random field on a sufficiently large torus comprising the spatial domain of interest is the basis of various efficient computational methods, such as the classical circulant embedding technique using the fast Fourier transform for generating samples on uniform grids. For the family of Mat\'ern covariances with smoothness index $\nu$ and correlation length $\lambda$, we analyse the nonsmooth periodization (corresponding to classical circulant embedding) and an alternative procedure using a smooth truncation of the covariance function. We solve two open problems: the first concerning the $\nu$-dependent asymptotic decay of eigenvalues of the resulting circulant in the nonsmooth case, the second concerning the required size in terms of $\nu$, $\lambda$ of the torus when using a smooth periodization. In doing this we arrive at a complete characterisation of the performance of these two approaches. Both our theoretical estimates and the numerical tests provided here show substantial advantages of smooth truncation., Comment: 24 pages, 3 figures

The problem of recovering a diffusion coefficient $a$ in a second-order elliptic partial differential equation from a corresponding solution $u$ for a given right-hand side $f$ is considered, with particular focus on the case where $f$ is allowed to take both positive and negative values. Identifiability of $a$ from $u$ is shown under mild smoothness requirements on $a$, $f$, and on the spatial domain $D$, assuming that either the gradient of $u$ is nonzero almost everywhere, or that $f$ as a distribution does not vanish on any open subset of $D$. Further results of this type under essentially minimal regularity conditions are obtained for the case of $D$ being an interval, including detailed information on the continuity properties of the mapping from $u$ to $a$., Comment: 14 pages

Let $X$ be a Banach space and $\mathcal{K}$ be a compact subset in $X$. We consider a greedy algorithm for finding an $n$-dimensional subspace $V_n\subset X$ which can be used to approximate the elements of $\mathcal{K}$. We are interested in how well the space $V_n$ approximates the elements of $\mathcal{K}$. For this purpose we compare the performance of greedy algorithm measured by $\sigma_n(\mathcal{K})_X:=\text{dist}(\mathcal{K},V_n)_X$ with the Kolmogorov width $d_n(\mathcal{K})_X$ which is the best possible error one can achieve when approximating $\mathcal{K}$ by $n$-dimensional subspaces. Various results in this direction have been given, e.g., in Binev et al. (SIAM J. Math. Anal. (2011)), DeVore et al. (Constr. Approx. (2013)) and Wojtaszczyk (J. Math. Anal. Appl. (2015)). The purpose of the present paper is to continue this line. We shall show that there exists a constant $C>0$ such that $$ \sigma_n(\mathcal{K})_X\leq C n^{-s+\mu}\big(\log(n+2)\big)^{\min(s,1/2)}, \quad \ n\geq 1\,, $$ if Kolmogorov widths $d_n(\mathcal{K})_X$ decay as $n^{-s}$ and the Banach-Mazur distance between an arbitrary $n$-dimensional subspace $V_n \subset X$ and $\ell_2^n$ satisfies $d(V_n,\ell_2^n)\leq C_1 n^\mu$. In particular, when some additional information about the set $\mathcal{K}$ is given then there is no logarithmic factor in this estimate., Comment: 14 pages

We consider the order of convergence for linear and nonlinear Monte Carlo approximation of compact embeddings from Sobolev spaces of dominating mixed smoothness defined on the torus $\mathbb{T}^d$ into the space $L_{\infty}(\mathbb{T}^d)$ via methods that use arbitrary linear information. These cases are interesting because we can gain a speedup of up to $1/2$ in the main rate compared to the worst case approximation. In doing so we determine the rate for some cases that have been left open by Fang and Duan.

We continue our investigations on pointwise multipliers for Besov spaces of dominating mixed smoothness. This time we study the algebra property of the classes $S^r_{p,q}B(\mathbb{R}^d)$ with respect to pointwise multiplication. In addition if $p\leq q$, we are able to describe the space of all pointwise multipliers for $S^r_{p,q}B(\mathbb{R}^d)$., Comment: 29 pages. arXiv admin note: text overlap with arXiv:1608.03414

We characterize the set of all pointwise multipliers of the Besov spaces $B^s_{p,q}(\R)$ under the restrictions $0 < p,q \le \infty$ and $s>d/p$., Comment: 23 pages

We shall compare isotropic Lizorkin-Triebel spaces with their counterparts of dominating mixed smoothness., Comment: 27 pages

Under certain restrictions we describe the set of all pointwise multipliers in case of Sobolev and Besov spaces of dominating mixed smoothness. In addition we shall give necessary and sufficient conditions for the case that these spaces form algebras with respect to pointwise multiplication.

Gelfand numbers represent a measure for the information complexity which is given by the number of information needed to approximate functions in a subset of a normed space with an error less than $\varepsilon$. More precisely, Gelfand numbers coincide up to the factor 2 with the minimal error $ e^{\rm wor}(n,\Lambda^{\rm all})$ which describes the error of the optimal (non-linear) algorithm that is based on $n$ arbitrary linear functionals. This explain the crucial role of Gelfand numbers in the study of approximation problems. Let $S^t_{p_1,p_1}B((0,1)^d)$ be the Besov spaces with dominating mixed smoothness on $(0,1)^d$. In this paper we consider the problem ${\rm App}: S^t_{p_1,p_1}B((0,1)^d) \to L_{p_2}((0,1)^d)$ and investigate the asymptotic behaviour of Gelfand numbers of this embedding. We shall give the correct order of convergence of Gelfand numbers in almost all cases. In addition we shall compare these results with the known behaviour of approximation numbers which coincide with $ e^{\rm wor-lin}(n,\Lambda^{\rm all})$ when we only allow linear algorithms., Comment: 24 pages, 1 figure

We compare Besov spaces with isotropic smoothness with Besov spaces of dominating mixed smoothness. Necessary and sufficient conditions for continuous embeddings will be given., Comment: 29 pages, 2 figures

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., Comment: 33 pages

This paper is a continuation of the papers [21] and [22]. Here we shall investigate the asymptotic behaviour of Weyl and Bernstein numbers of embeddings of Sobolev spaces with dominating mixed smoothness into Lebesgue spaces., Comment: 33 pages

The purpose of the present paper is to investigate the decay of Bernstein numbers of the embedding from $B^t_{p_1,q}((0,1)^d)$ into the space $L_{p_2}((0,1)^d) $. The asymptotic behaviour of Bernstein numbers of the identity $id: S_{p_1,p_1}^tB((0,1)^d)\rightarrow L_{p_2}((0,1)^d)$ will be also considered., Comment: 31 pages, 1 figure

In this paper we shall give two-sided sharp estimates of Kolmogorov numbers of embeddings of the Besov spaces with dominating mixed smoothness $S^t_{p,q}B((0,1)^d)$ into $ L_{\infty}((0,1)^d)$., Comment: This paper has been withdrawn by the author due an error