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Exponential convergence for integration over R^s

Dong Nguyen, Dirk Nuyens

Published Jul 1, 2015• 0 citations
OpenAlex

Abstract

In this talk we study multivariate integration over R^s for weighted function space of infinitely many times differentiable functions for which the Fourier transform decays exponentially fast. The weights allows us to define the different influence of each variables. We prove that we can achieve the exponential convergence rate using a classical quasi-Monte Carlo method. More specific, we prove two convergence rates of $\mathcal{O}(e^{-N^{\frac{1}{D+B}}})$ and $\mathcal{O} (e^{-\frac{N^{\frac{1}{B}}}{\ln^{\frac{1}{D}} N}})$, where D and B are respectively defined by the exponential decay of the Fourier coefficients and of the integrand, for two different function spaces. We also show some numerical results that prove our theorems.

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L∞\boldsymbol{L}_{\infty}L∞​-approximation in Korobov spaces with Exponential Weights
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