We study approximations of compact linear multivariate operators defined over Hilbert spaces. We provide necessary and sufficient conditions on various notions of tractability. These conditions are mainly given in terms of sums of certain functions depending on the singular values of the multivariate problem. They do not require the ordering of these singular values which in many cases is difficult to achieve.
We study multivariate approximation in the average case setting with the error measured in the weighted $L_2$ norm. We consider algorithms that use standard information $Λ^{\rm std}$ consisting of function values or general linear information $Λ^{\rm all}$ consisting of arbitrary continuous linear functionals. We investigate the equivalences of various notions of algebraic and exponential tractability for $Λ^{\rm std}$ and $Λ^{\rm all}$ for the absolute error criterion, and show that the power of $Λ^{\rm std}$ is the same as that of $Λ^{\rm all}$ for all notions of algebraic and exponential tr…
We study the worst case tractability of multivariate linear problems defined on separable Hilbert spaces. Information about a problem instance consists of noisy evaluations of arbitrary bounded linear functionals, where the noise is either deterministic or random. The cost of a single evaluation depends on its precision and is controlled by a cost function. We establish mutual interactions between tractability of a problem with noisy information, the cost function, and tractability of the same problem, but with exact information.
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.
We study approximation of multivariate functions from a separable Hilbert space in the randomized setting with the error measured in the weighted $L_2$ norm. We consider algorithms that use standard information $Λ^{\rm std}$ consisting of function values or general linear information $Λ^{\rm all}$ consisting of arbitrary linear functionals. We use the weighted least squares regression algorithm to obtain the upper estimates of the minimal randomized error using $Λ^{\rm std}$. We investigate the equivalences of various notions of algebraic and exponential tractability for $Λ^{\rm std}$ and $Λ^{…
We study numerical integration for a weighted Korobov space of analytic periodic functions for which the Fourier coefficients decay exponentially fast. In particular, we are interested in how the error depends on the dimension $d$. Many recent papers deal with this problem or similar problems and provide matching necessary and sufficient conditions for various notions of tractability. In most cases even simple algorithms are known which allow to achieve these notions of tractability. However, there is a gap in the literature: while for the notion of exponential-weak tractability one knows matc…
Notes on $(s,t)$-weak tractability: A refined classification of problems with (sub)exponential information complexity
University of Warsaw, Faculty of Mathematics, Informatics and Mechanics（华沙大学数学、信息学与力学学院）；Philipps-University Marburg, Faculty of Mathematics and Computer Science（马尔堡菲利普斯大学数学与计算机科学系）
We develop a general framework for estimating the $L_\infty(\mathbb{T}^d)$ error for the approximation of multivariate periodic functions belonging to specific reproducing kernel Hilbert spaces (RHKS) using approximants that are trigonometric polynomials computed from sampling values. The used sampling schemes are suitable sets of rank-1 lattices that can be constructed in an extremely efficient way. Furthermore, the structure of the sampling schemes allows for fast Fourier transform (FFT) algorithms. We present and discuss one FFT algorithm and analyze the worst case $L_\infty$ error for this…
