Very low truncation dimension for high dimensional integration under modest error demand

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• 作者：Peter Kritzer^a ; ^b ; ^{peter.kritzer@oeaw.ac.at" class="auth_mail" title="E-mail the corresponding author} ; Friedrich Pillichshammer^a ; ^{friedrich.pillichshammer@jku.at" class="auth_mail" title="E-mail the corresponding author} ; G.W. Wasilkowski^c ; ^{greg@cs.uky.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Numerical integration ; Weighted anchored and ANOVA Sobolev spaces ; Truncation dimension
• 刊名：Journal of Complexity
• 出版年：2016
• 出版时间：August 2016
• 年：2016
• 卷：35
• 期：Complete
• 页码：63-85
• 全文大小：470 K

文摘

We consider the problem of numerical integration for weighted anchored and ANOVA Sobolev spaces of s$s$-variate functions. Here s$s$ is large including s=∞$s=\infty$. Under the assumption of sufficiently fast decaying weights, we prove in a constructive way that such integrals can be approximated by quadratures for functions f_k$f_k$ with only k$k$ variables, where k=k(ε)$k=k\left(\epsilon \right)$ depends solely on the error demand ε$\epsilon$ and is surprisingly small when s$s$ is sufficiently large relative to ε$\epsilon$. This holds, in particular, for s=∞$s=\infty$ and arbitrary ε$\epsilon$ since then k(ε)<∞$k\left(\epsilon \right)<\infty$ for all ε$\epsilon$. Moreover k(ε)$k\left(\epsilon \right)$ does not depend on the function being integrated, i.e., is the same for all functions from the unit ball of the space.