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Van Kien Nguyen

Publications and source records attributed to Van Kien Nguyen.

30 records · Page 2Linked to original sources
arXiv2017

Monte Carlo Methods for Uniform Approximation on Periodic Sobolev Spaces with Mixed Smoothness

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.

math.NA↗
arXiv2017

Pointwise Multipliers for Besov Spaces of Dominating Mixed Smoothness - II

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 $S^r_{p,q}B(\mathbb{R}^d)$.

math.FA↗
arXiv2016

Gelfand Numbers of Embeddings of Mixed Besov Spaces

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,Λ^{\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,Λ^{\rm all})$ when we only allow linear algorithms.

math.FA↗
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.

math.NA↗