Weighted Geometric Discrepancies and Numerical Integration on Reproducing Kernel Hilbert Spaces

Title:

Weighted Geometric Discrepancies and Numerical Integration on Reproducing Kernel Hilbert Spaces

Author(s):

Gnewuch, Michael

Date:

2010

Type:

Technical reports

Department:

Computer Science

Permanent URL:

http://hdl.handle.net/10022/AC:P:10523

Series:

Columbia University Computer Science Technical Reports

Part Number:

CUCS03410

Publisher:

Department of Computer Science, Columbia University

Publisher Location:

New York

Abstract:

We extend the notion of L2Bdiscrepancy introduced in [E. Novak, H. Wozniakowski, L2 discrepancy and multivariate integration, in: Analytic number theory. Essays in honour of Klaus Roth. W. W. L. Chen, W. T. Gowers, H. Halberstam, W. M. Schmidt, and R. C. Vaughan (Eds.), Cambridge University Press, Cambridge, 2009, 359–388] to what we want to call weighted geometric L2discrepancy. This extended notion allows us to consider weights to moderate the importance of different groups of variables, and additionally volume measures different from the Lebesgue measure as well as classes of test sets different from measurable subsets of Euclidean spaces. We relate the weighted geometric L2discrepancy to numerical integration defined over weighted reproducing kernel Hilbert spaces and settle in this way an open problem posed by Novak and Wozniakowski. Furthermore, we prove an upper bound for the numerical integration error for cubature formulas that use admissible sample points. The set of admissible sample points may actually be a subset of the integration domain of measure zero. We illustrate that particularly in infinite dimensional numerical integration it is crucial to distinguish between the whole integration domain and the set of those sample points that actually can be used by algorithms.

Subject(s):

Computer science
 Item views:

423
 Metadata:

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 Suggested Citation:

Michael Gnewuch,
2010,
Weighted Geometric Discrepancies and Numerical Integration on Reproducing Kernel Hilbert Spaces, Columbia University Academic Commons,
http://hdl.handle.net/10022/AC:P:10523.