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Embeddings of weighted Hilbert spaces and applications to multivariate and infinite-dimensional integration

Michael Gnewuch, Mario Hefter, Aicke Hinrichs +1 more

arXiv:1608.00906Journal of Approximation TheoryMay 29, 201715 citations

Infinite-dimensional integration and L2 -approximation on Hermite spaces

Michael Gnewuch, Aicke Hinrichs, Klaus Ritter +1 more

We study integration and L2-approximation of functions of infinitely many variables in the following setting: The underlying function space is the countably infinite tensor product of univariate Hermite spaces and the probability measure is the corresponding product of the standard normal distribution. The maximal domain of the functions from this tensor product space is necessarily a proper subset of the sequence space RN. We establish upper and lower bounds for the minimal worst case errors under general assumptions; these bounds do match for tensor products of well-studied Hermite spaces of functions with finite or with infinite smoothness. In the proofs we employ embedding results, and the upper bounds are attained constructively with the help of multivariate decomposition methods.We study integration and L2-approximation of functions of infinitely many variables in the following setting: The underlying function space is the countably infinite tensor product of univariate Hermite spaces and the probability measure is the corresponding product of the standard normal distribution. The maximal domain of the functions from this tensor product space is necessarily a proper subset of the sequence space RN. We establish upper and lower bounds for the minimal worst case errors under general assumptions; these bounds do match for tensor products of well-studied Hermite spaces of functions with finite or with infinite smoothness. In the proofs we employ embedding results, and the upper bounds are attained constructively with the help of multivariate decomposition methods.

arXiv:2304.01754Journal of Approximation TheoryFeb 8, 20245 citations

Tractability of $ L_2 $-approximation and integration over weighted Korobov spaces of increasing smoothness in the worst case setting

Weiran Ding, Jiansong Li, Yumeng Jia +1 more

This paper is devoted to the study of tractability of the $ L_2 $-approximation and integration from weighted Korobov spaces of increasing smoothness in the worst-case setting. The considered algorithms use information from the class $ \Lambda^{\rm all} $, including all continuous linear functionals, and from the class $ \Lambda^{\rm std} $, including function evaluations. Necessary and sufficient conditions on the weights of the function space for strong polynomial tractability, polynomial tractability, quasi-polynomial tractability, uniform weak tractability, weak tractability and $ (\sigma, \tau) $-weak tractability, are provided. Our results give a comprehensive picture of the weight conditions for all standard notions of algebraic tractability. It may be helpful to study the tractability of nonhomogeneous tensor product spaces.This paper is devoted to the study of tractability of the $ L_2 $-approximation and integration from weighted Korobov spaces of increasing smoothness in the worst-case setting. The considered algorithms use information from the class $ \Lambda^{\rm all} $, including all continuous linear functionals, and from the class $ \Lambda^{\rm std} $, including function evaluations. Necessary and sufficient conditions on the weights of the function space for strong polynomial tractability, polynomial tractability, quasi-polynomial tractability, uniform weak tractability, weak tractability and $ (\sigma, \tau) $-weak tractability, are provided. Our results give a comprehensive picture of the weight conditions for all standard notions of algebraic tractability. It may be helpful to study the tractability of nonhomogeneous tensor product spaces.

Electronic Research ArchiveJan 1, 20250 citations

Randomized Approximation of Sobolev Embeddings

Stefan Heinrich

Springer eBooksDec 29, 200714 citations

Integration and approximation in cosine spaces of smooth functions

Christian Irrgeher, Peter Kritzer, Friedrich Pillichshammer

arXiv:1511.00328Mathematics and Computers in SimulationMar 9, 20164 citations

Tractability of L2-approximation and integration in weighted Hermite spaces of finite smoothness

Gunther Leobacher, Friedrich Pillichshammer, Adrian Ebert

In this paper we consider integration and L2-approximation for functions over Rs from weighted Hermite spaces. The first part of the paper is devoted to a comparison of several weighted Hermite spaces that appear in literature, which is interesting on its own. Then we study tractability of the integration and L2-approximation problem for the introduced Hermite spaces, which describes the growth rate of the information complexity when the error threshold ε tends to 0 and the problem dimension s grows to infinity. Our main results are characterizations of tractability in terms of the involved weights, which model the importance of the successive coordinate directions for functions from the weighted Hermite spaces.In this paper we consider integration and L2-approximation for functions over Rs from weighted Hermite spaces. The first part of the paper is devoted to a comparison of several weighted Hermite spaces that appear in literature, which is interesting on its own. Then we study tractability of the integration and L2-approximation problem for the introduced Hermite spaces, which describes the growth rate of the information complexity when the error threshold ε tends to 0 and the problem dimension s grows to infinity. Our main results are characterizations of tractability in terms of the involved weights, which model the importance of the successive coordinate directions for functions from the weighted Hermite spaces.

arXiv:2212.05780Journal of ComplexityJun 1, 20237 citations

Embeddings for infinite-dimensional integration and L2 -approximation with increasing smoothness

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