一种构造多个秩1格的确定性算法
A Deterministic Algorithm for Constructing Multiple Rank-1 Lattices of Near-Optimal Size
摘要：
在本文中，我们提出了第一个已知的确定性算法的构造多秩1格的周期函数的多变量的近似。该算法通过将用于某些$ d $维频率集$ I \subset [N]^d $的潜在大的重构单个秩-1格转换成小得多的秩-1格的集合来工作，这些秩-1格允许具有$ I $中的系数的三角多项式的准确且有效的重构（并且因此用于多变量周期函数的近似）。理论上证明了多个秩1格中的采样点总数小于$ \mathcal{O}\left( |I| \log^{ 2 }(N |I|) \right) $，其常数独立于$d$，并且通过在这些点上对具有傅立叶支持的$ I $三角多项式的样本进行一维快速傅立叶变换，我们在少于$ \mathcal{O}\left(d\,|I|\log^4 (N|I|)\right) $的总操作中获得所有傅里叶系数的精确重建。此外，我们提出了第二个多秩-1格的构造算法，该算法构造的网格，甚至更少的采样点的成本，只能重建精确的三角多项式，而不是有额外的理论近似。这两种算法进行了数值测试，并超过了理论界。值得注意的是，我们观察到过采样因子#samples $/|I|$对于第一算法在$ |I| $中似乎仅以几何方式增长，并且在第二算法中似乎以4为界接近最优。
英文摘要：
In this paper we present the first known deterministic algorithm for the construction of multiple rank-1 lattices for the approximation of periodic functions of many variables. The algorithm works by converting a potentially large reconstructing single rank-1 lattice for some $ d $-dimensional frequency set $ I \subset [N]^d $ into a collection of much smaller rank-1 lattices which allow for accurate and efficient reconstruction of trigonometric polynomials with coefficients in $ I $ (and, therefore, for the approximation of multivariate periodic functions). The total number of sampling points in the resulting multiple rank-1 lattices is theoretically shown to be less than $ \mathcal{O}\left( |I| \log^{ 2 }(N |I|) \right) $ with constants independent of $d$, and by performing one-dimensional fast Fourier transforms on samples of trigonometric polynomials with Fourier support in $ I $ at these points, we obtain exact reconstruction of all Fourier coefficients in fewer than $ \mathcal{O}\left(d\,|I|\log^4 (N|I|)\right) $ total operations. Additionally, we present a second multiple rank-1 lattice construction algorithm which constructs lattices with even fewer sampling points at the cost of only being able to reconstruct exact trigonometric polynomials rather than having additional theoretical approximation. Both algorithms are tested numerically and surpass the theoretical bounds. Notably, we observe that the oversampling factors #samples$/|I|$ appear to grow only logarithmically in $ |I| $ for the first algorithm and appear near-optimally bounded by four in the second algorithm.
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