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- W2963757024 abstract "In this paper we give a new Koksma-Hlawka type inequality for Quasi-Monte Carlo (QMC) integration. QMC integration of a function $fcolon[0,1)^srightarrowmathbb{R}$ by a finite point set $mathcal{P}subset[0,1)^s$ is the approximation of the integral $I(f):=int_{[0,1)^s}f(mathbf{x}),dmathbf{x}$ by the average $I_{mathcal{P}}(f):=frac{1}{|mathcal{P}|}sum_{mathbf{x} in mathcal{P}}f(mathbf{x})$. We treat a certain class of point sets $mathcal{P}$ called digital nets. A Koksma-Hlawka type inequality is an inequality providing an upper bound on the integration error $text{Err}(f;mathcal{P}):=I(f)-I_{mathcal{P}}(f)$ of the form $|text{Err}(f;mathcal{P})|le Ccdot |f|cdot D(mathcal{P})$. We can obtain a Koksma-Hlawka type inequality by estimating bounds on $|hat{f}(mathbf{k})|$, where $hat{f}(mathbf{k})$ is a generalized Fourier coefficient with respect to the Walsh system. In this paper we prove bounds on the Walsh coefficients $hat{f}(mathbf{k})$ by introducing an operator called ‘dyadic difference’ $partial_{i,n}$. By converting dyadic differences $partial_{i,n}$ to derivatives $frac{partial }{partial x_i}$, we get a new bound on $|hat{f}(mathbf{k})|$ for a function $f$ whose mixed partial derivatives up to order $alpha$ in each variable are continuous. This new bound is smaller than the known bound on $|hat{f}(mathbf{k})|$ under some instances. The new Koksma-Hlawka type inequality is derived using this new bound on the Walsh coefficients." @default.
- W2963757024 created "2019-07-30" @default.
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- W2963757024 date "2017-07-01" @default.
- W2963757024 modified "2023-09-28" @default.
- W2963757024 title "Bounds on Walsh coefficients by dyadic difference and a new Koksma-Hlawka type inequality for Quasi-Monte Carlo integration" @default.
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- W2963757024 doi "https://doi.org/10.32917/hmj/1499392824" @default.
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