2023/06/02 by Takashi Goda, Goda, Takashi
Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Algebra over a field #Combinatorics #FOS: Mathematics #Mathematical Approximation and Integration #Mathematical analysis #Mathematical functions and polynomials #Mathematics #Numerical Analysis (math.NA) #Polynomial #Pure mathematics
paper · pdf · doi:10.48550/arxiv.2306.01541
published in arXiv (Cornell University) (Cornell University)
openalex publication_date 2023/06/02 · openalex created_date 2023/06/07 · openalex updated_date 2026/07/28
Building upon recent work by the author, we prove that multivariate integration in the following subspace of the Wiener algebra over [0,1)d is strongly polynomially tractable: Fd:=\ f∈ C([0,1)d) \middle| ‖f‖:=∑_\boldsymbolk∈ ℤd|f(\boldsymbolk)|max(width(supp(\boldsymbolk)),min_j∈ supp(\boldsymbolk)log |kj|)lt;∞ \, with f(\boldsymbolk) being the \boldsymbolk-th Fourier coefficient of f, supp(\boldsymbolk):=\j∈ \1,…,d\| kj≠ 0\, and width: 2^\1,…,d\→ \1,…,d\ being defined by width(u):=maxj∈ uj-minj∈ uj+1, for non-empty subset u⊆ \1,…,d\ and width(∅):=1. Strong polynomial tractability is achieved by an explicit quasi-Monte Carlo rule using a multiset union of Korobov's p-sets. We also show that, if we replace width(supp(\boldsymbolk)) with 1 for all \boldsymbolk∈ ℤd in the above definition of norm, multivariate integration is polynomially tractable but not strongly polynomially tractable.