SIAM Journal on Numerical Analysis, Volume 59, Issue 5, Page 2393-2429, January 2021.
We propose an approximation method for high-dimensional 1-periodic functions based on the multivariate ANOVA decomposition. We provide analysis of classical ANOVA decomposition on the torus and prove some important properties, such as the inheritance of smoothness for Sobolev type spaces and the weighted Wiener algebra. We exploit special kinds of sparsity in the ANOVA decomposition with the aim of approximating a function in a scattered data or black-box approximation scenario. This method allows us to simultaneously achieve an importance ranking on dimensions and dimension interactions (referred to as an attribute ranking in some applications). In scattered data approximation we rely on a special algorithm based on the non-equispaced fast Fourier transform (or NFFT) for fast multiplication with arising Fourier matrices. For black-box approximation we choose the well-known rank-1 lattices as sampling schemes and show properties of the arising special lattices.

中文翻译：

---

使用基于傅立叶的方法逼近高维周期函数

SIAM 数值分析杂志，第 59 卷，第 5 期，第 2393-2429 页，2021 年 1 月。
我们提出了一种基于多元方差分析的高维一周期函数的近似方法。我们对圆环上的经典 ANOVA 分解进行了分析，并证明了一些重要的性质，例如 Sobolev 类型空间的平滑继承和加权维纳代数。我们在 ANOVA 分解中利用特殊类型的稀疏性，目的是逼近分散数据或黑盒逼近场景中的函数。这种方法允许我们同时实现维度和维度交互的重要性排序（在某些应用中称为属性排序）。在分散数据近似中，我们依靠一种基于非等距快速傅立叶变换（或 NFFT）的特殊算法来与产生的傅立叶矩阵进行快速乘法。