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Explicit error bounds for randomized Smolyak algorithms and an application to infinite-dimensional integration

JOURNAL OF approximation THEORY 2020年 251卷 105342-105342页

作者： Gnewuch, M. Wnuk, M. Univ Osnabruck Inst Math Albrechtstr 28 A D-49076 Osnabruck Germany

Smolyak's method, also known as sparse grid method, is a powerful tool to tackle multivariate tensor product problems solely with the help of efficient algorithms for the corresponding univariate problem. In this paper we study the randomized setting, i.e., we randomize Smolyak's method. We provide upper and lower error bounds for randomized Smolyak algorithms with explicitly given dependence on the number of variables and the number of information evaluations used. The error criteria we consider are the worst case root mean square error (the typical error criterion for randomized algorithms, sometimes referred to as "randomized error",) and the root mean square worst case error (sometimes referred to as "worst-case error"). Randomized Smolyak algorithms can be used as building blocks for efficient methods such as multilevel algorithms, multivariate decomposition methods or dimension-wise quadrature methods to tackle successfully high-dimensional or even infinite-dimensional problems. As an example, we provide a very general and sharp result on the convergence rate of Nth minimal errors of infinite-dimensional integration on weighted reproducing kernel Hilbert spaces. Moreover, we are able to characterize the spaces for which randomized algorithms for infinite-dimensional integration are superior to deterministic ones. We illustrate our findings for the special instance of weighted Korobov spaces. We indicate how these results can be extended, e.g., to spaces of functions whose smooth dependence on successive variables increases ("spaces of increasing smoothness") and to the problem of L-2-approximation (function recovery). (C) 2019 Elsevier Inc. All rights reserved.

关键词： Tensor product problem Sparse grid Randomized algorithm infinite-dimensional approximation Multivariate decomposition method Changing dimension algorithm

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Multivariate two-sample tests based on nearest neighbors

引用

Journal of the American Statistical Association 1986年第395期81卷 799-806页

作者： Schilling, Mark F. Department of Mathematics California State University Northridge CA 91330 United States

A new class of simple tests is proposed for the general multivariate two-sample problem based on the (possibly weighted) proportion of all k nearest neighbor comparisons in which observations and their neighbors belong to the same sample. Large values of the test statistics give evidence against the hypothesis H of equality of the two underlying distributions. Asymptotic null distributions are explicitly determined and shown to involve certain nearest neighbor interaction probabilities. Simple infinite-dimensional approximations are supplied. The unweighted version yields a distribution-free test that is consistent against all alternatives;optimally weighted statistics are also obtained and asymptotic efficiencies are calculated. Each of the tests considered is easily adapted to a permutation procedure that conditions on the pooled sample. Power performance for finite sample sizes is assessed in simulations. © 1976 Taylor & Francis Group, LLC.

关键词： Distribution-free infinite-dimensional approximation Kth nearest neighbor

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