Tent-transformed lattice rules for integration and approximation of multivariate non-periodic functions

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• 作者：Ronald Cools^a ; ^{ronald.cools@cs.kuleuven.be" class="auth_mail" title="E-mail the corresponding author} ; Frances Y. Kuo^b ; ^{f.kuo@unsw.edu.au" class="auth_mail" title="E-mail the corresponding author} ; Dirk Nuyens^a ; ^{dirk.nuyens@cs.kuleuven.be" class="auth_mail" title="E-mail the corresponding author} ; Gowri Suryanarayana^a ; ^{gowri.suryanarayana@cs.kuleuven.be" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Quasi-Monte Carlo methods ; Cosine series ; Function approximation ; Hyperbolic crosses ; Rank-11 lattice rules ; Component-by-component construction
• 刊名：Journal of Complexity
• 出版年：2016
• 出版时间：October 2016
• 年：2016
• 卷：36
• 期：Complete
• 页码：166-181
• 全文大小：488 K

文摘

We develop algorithms for multivariate integration and approximation in the weighted half-period cosine space of smooth non-periodic functions. We use specially constructed tent-transformed rank-1$1$ lattice points as cubature nodes for integration and as sampling points for approximation. For both integration and approximation, we study the connection between the worst-case errors of our algorithms in the cosine space and the worst-case errors of some related algorithms in the well-known weighted Korobov space of smooth periodic functions. By exploiting this connection, we are able to obtain constructive worst-case error bounds with good convergence rates for the cosine space.