Optimal importance sampling for the approximation of integrals

Authors:
Aicke Hinrichs
Affiliations:
Mathematisches Institut, Universität Jena, Ernst-Abbe-Platz 2, 07740 Jena, Germany
Venue:
Journal of Complexity
Year:
2010

Citing 3
Cited 3

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Lower bounds for the complexity of linear functionals in the randomized setting

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Tractability of infinite-dimensional integration in the worst case and randomized settings

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Abstract

We consider optimal importance sampling for approximating integrals I(f)=@!"Df(x)@r(x)dx of functions f in a reproducing kernel Hilbert space H@?L"1(@r) where @r is a given probability density on D@?R^d. We show that there exists another density @w such that the worst case error of importance sampling with density function @w is of order n^-^1^/^2. As a result, for multivariate problems generated from nonnegative kernels we prove strong polynomial tractability of the integration problem in the randomized setting. The density function @w is obtained from the application of change of density results used in the geometry of Banach spaces in connection with a theorem of Grothendieck concerning 2-summing operators.