Convergence estimates in probability and in expectation for discrete least squares with noisy evaluations at random points
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ArticleKAUST Department
Applied Mathematics and Computational Science ProgramCenter for Uncertainty Quantification in Computational Science and Engineering (SRI-UQ)
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Date
2015-08-28Online Publication Date
2015-08-28Print Publication Date
2015-12Permanent link to this record
http://hdl.handle.net/10754/576075
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We study the accuracy of the discrete least-squares approximation on a finite dimensional space of a real-valued target function from noisy pointwise evaluations at independent random points distributed according to a given sampling probability measure. The convergence estimates are given in mean-square sense with respect to the sampling measure. The noise may be correlated with the location of the evaluation and may have nonzero mean (offset). We consider both cases of bounded or square-integrable noise / offset. We prove conditions between the number of sampling points and the dimension of the underlying approximation space that ensure a stable and accurate approximation. Particular focus is on deriving estimates in probability within a given confidence level. We analyze how the best approximation error and the noise terms affect the convergence rate and the overall confidence level achieved by the convergence estimate. The proofs of our convergence estimates in probability use arguments from the theory of large deviations to bound the noise term. Finally we address the particular case of multivariate polynomial approximation spaces with any density in the beta family, including uniform and Chebyshev.Citation
Convergence estimates in probability and in expectation for discrete least squares with noisy evaluations at random points 2015 Journal of Multivariate AnalysisPublisher
Elsevier BVJournal
Journal of Multivariate AnalysisAdditional Links
http://linkinghub.elsevier.com/retrieve/pii/S0047259X15001931ae974a485f413a2113503eed53cd6c53
10.1016/j.jmva.2015.08.009