QMC integration based on arbitrary (t,m,s)-nets yields optimal convergence rates on several scales of function spaces
Michael Gnewuch, Josef Dick, Lev Markhasin, Winfried Sickel
TL;DR
The paper studies high-dimensional QMC integration across Banach spaces of fractional and dominating mixed smoothness, proving that point sets based on arbitrary $(t,m,s)$-nets yield the optimal worst-case convergence rates. It first analyzes Haar wavelet spaces to obtain matching upper and lower bounds, then leverages a network of embeddings to transfer these rates to Besov and Sobolev spaces, including fractional smoothness spaces with sharp fractional Koksma-Hlawka inequalities. The results show that net-based QMC achieves the best possible convergence rates not only in Haar spaces but also in periodic and non-periodic Sobolev/Besov spaces, with rates of the form $e^{wor}(Q_P,X) = O\big(N^{-\\alpha}\log(N)^{(s-1)/2}\big)$ in suitable settings. These findings provide precise guidance on the design and evaluation of QMC point sets for high-dimensional integration across a broad spectrum of function spaces, underpinned by tensor-product Banach space theory.
Abstract
We study the integration problem over the $s$-dimensional unit cube on four types of Banach spaces of integrands. First we consider Haar wavelet spaces, consisting of functions whose Haar wavelet coefficients exhibit a certain decay behavior measured by a parameter $α>0$. We study the worst case error of integration over the norm unit ball and provide upper error bounds for quasi-Monte Carlo (QMC) cubature rules based on arbitrary $(t,m,s)$-nets as well as matching lower error bounds for arbitrary cubature rules. These results show that using arbitrary $(t,m,s)$-nets as sample points yields the best possible rate of convergence. Afterwards we study spaces of integrands of fractional smoothness $α\in (0,1)$ and state a sharp Koksma-Hlawka-type inequality. More precisely, we show that on those spaces the worst case error of integration is equal to the corresponding fractional discrepancy. Those spaces can be continuously embedded into tensor product Bessel potential spaces, also known as Sobolev spaces of dominated mixed smoothness, with the same set of parameters. The latter spaces can be embedded into suitable Besov spaces of dominating mixed smoothness $α$, which in turn can be embedded into the Haar wavelet spaces with the same set of parameters. Therefore our upper error bounds on Haar wavelet spaces for QMC cubatures based on $(t,m,s)$-nets transfer (with possibly different constants) to the corresponding spaces of integrands of fractional smoothness and to Sobolev and Besov spaces of dominating mixed smoothness. Moreover, known lower error bounds for periodic Sobolev and Besov spaces of dominating mixed smoothness show that QMC integration based on arbitrary $(t,m,s)$-nets yields the best possible convergence rate on periodic as well as on non-periodic Sobolev and Besov spaces of dominating smoothness.