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 g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0885064X16000145&_mathId=si1.gif&_user=111111111&_pii=S0885064X16000145&_rdoc=1&_issn=0885064X&md5=f8618a50cace5423b26b8044d1ac1cfd" title="Click to view the MathML source">sgif" overflow="scroll">s-variate functions. Here g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0885064X16000145&_mathId=si1.gif&_user=111111111&_pii=S0885064X16000145&_rdoc=1&_issn=0885064X&md5=f8618a50cace5423b26b8044d1ac1cfd" title="Click to view the MathML source">sgif" overflow="scroll">s is large including g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0885064X16000145&_mathId=si3.gif&_user=111111111&_pii=S0885064X16000145&_rdoc=1&_issn=0885064X&md5=fe1007b6cb91368e68ac8c5817f0a16c" title="Click to view the MathML source">s=∞gif" overflow="scroll">s=∞. Under the assumption of sufficiently fast decaying weights, we prove in a constructive way that such integrals can be approximated by quadratures for functions g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0885064X16000145&_mathId=si4.gif&_user=111111111&_pii=S0885064X16000145&_rdoc=1&_issn=0885064X&md5=74f997ec196e4bffb4cc200252a27eb5" title="Click to view the MathML source">f_kgif" overflow="scroll">fk with only g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0885064X16000145&_mathId=si5.gif&_user=111111111&_pii=S0885064X16000145&_rdoc=1&_issn=0885064X&md5=0d18c77bf1d6916d5f6650c63cfa5de9" title="Click to view the MathML source">kgif" overflow="scroll">k variables, where g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0885064X16000145&_mathId=si6.gif&_user=111111111&_pii=S0885064X16000145&_rdoc=1&_issn=0885064X&md5=e9bcf60e2e8824b8539abdbe56bf616d" title="Click to view the MathML source">k=k(ε)gif" overflow="scroll">k=k(ε) depends solely on the error demand g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0885064X16000145&_mathId=si7.gif&_user=111111111&_pii=S0885064X16000145&_rdoc=1&_issn=0885064X&md5=d5ae9824da807af628cd73689491f95f" title="Click to view the MathML source">εgif" overflow="scroll">ε and is surprisingly small when g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0885064X16000145&_mathId=si1.gif&_user=111111111&_pii=S0885064X16000145&_rdoc=1&_issn=0885064X&md5=f8618a50cace5423b26b8044d1ac1cfd" title="Click to view the MathML source">sgif" overflow="scroll">s is sufficiently large relative to g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0885064X16000145&_mathId=si7.gif&_user=111111111&_pii=S0885064X16000145&_rdoc=1&_issn=0885064X&md5=d5ae9824da807af628cd73689491f95f" title="Click to view the MathML source">εgif" overflow="scroll">ε. This holds, in particular, for g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0885064X16000145&_mathId=si3.gif&_user=111111111&_pii=S0885064X16000145&_rdoc=1&_issn=0885064X&md5=fe1007b6cb91368e68ac8c5817f0a16c" title="Click to view the MathML source">s=∞gif" overflow="scroll">s=∞ and arbitrary g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0885064X16000145&_mathId=si7.gif&_user=111111111&_pii=S0885064X16000145&_rdoc=1&_issn=0885064X&md5=d5ae9824da807af628cd73689491f95f" title="Click to view the MathML source">εgif" overflow="scroll">ε since then g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0885064X16000145&_mathId=si12.gif&_user=111111111&_pii=S0885064X16000145&_rdoc=1&_issn=0885064X&md5=0af0a5c9f670f88d332601cc6730be3f" title="Click to view the MathML source">k(ε)<∞gif" overflow="scroll">k(ε)<∞ for all g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0885064X16000145&_mathId=si7.gif&_user=111111111&_pii=S0885064X16000145&_rdoc=1&_issn=0885064X&md5=d5ae9824da807af628cd73689491f95f" title="Click to view the MathML source">εgif" overflow="scroll">ε. Moreover g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0885064X16000145&_mathId=si14.gif&_user=111111111&_pii=S0885064X16000145&_rdoc=1&_issn=0885064X&md5=829cabb3b9f2a48fe0280acc8c76aa8b" title="Click to view the MathML source">k(ε)gif" overflow="scroll">k(ε) does not depend on the function being integrated, i.e., is the same for all functions from the unit ball of the space.