In a recent paper by the same authors, we provided a theoretical foundation for the component-by-component (CBC) construction of lattice algorithms for multivariate $L_2$ approximation in the worst case setting, for functions in a periodic space with general weight parameters. The construction led to an error bound that achieves the best possible rate of convergence for lattice algorithms. Previously available literature covered only weights of a simple form commonly known as product weights. In this paper we address the computational aspect of the construction. We develop fast CBC construction of lattice algorithms for special forms of weight parameters, including the so-called POD weights and SPOD weights which arise from PDE applications, making the lattice algorithms truly applicable in practice. With $d$ denoting the dimension and $n$ the number of lattice points, we show that the construction cost is $\mathcal{O}(d\,n\log(n) + d^2\log(d)\,n)$ for POD weights, and $\mathcal{O}(d\,n\log(n) + d^3\sigma^2\,n)$ for SPOD weights of degree $\sigma\ge 2$. The resulting lattice generating vectors can be used in other lattice-based approximation algorithms, including kernel methods or splines.

中文翻译：

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用于具有 POD 和 SPOD 权重的多元近似的格算法的快速逐个组件构建

在同一作者最近的一篇论文中，我们为在最坏情况下的多元 $L_2$ 近似的格算法的逐个组件 (CBC) 构造提供了理论基础，对于具有一般权重参数的周期空间中的函数. 该构造导致了一个误差界限，它实现了格算法的最佳收敛速度。先前可用的文献仅涵盖通常称为产品权重的简单形式的权重。在本文中，我们讨论了构造的计算方面。我们为特殊形式的权重参数开发了快速 CBC 构造点阵算法，包括从 PDE 应用程序中产生的所谓 POD 权重和 SPOD 权重，使点阵算法真正适用于实践。$d$ 表示维度，$n$ 表示格点数，我们表明构建成本为 $\mathcal{O}(d\,n\log(n) + d^2\log(d)\ ,n)$ 表示 POD 权重，$\mathcal{O}(d\,n\log(n) + d^3\sigma^2\,n)$ 表示度数为 $\sigma\ge 2$ 的 SPOD 权重. 生成的点阵生成向量可用于其他基于点阵的近似算法，包括核方法或样条。