Variable transformations in combination with wavelets and ANOVA for high-dimensional approximation (Q6561375)

Let \(\Omega \in \{{\mathbb T}^d,\,{\mathbb R}^d,\, [0,\,1]^d\}\) or tensor products of these domains. The authors consider the approximation of a high-dimensional function \(f:\,\Omega \to \mathbb C\) from discrete sample points, which are distributed to an arbitrary density. They show that it is possible to transform the good approximation results for periodic functions on the torus \({\mathbb T}^d\) (see [\textit{L. Lippert} et al., Numer. Math. 154, No. 1--2, 155--207 (2023; Zbl 1528.41030)]) to the domain \(\Omega\). This method combines the least squares approximation on \({\mathbb T}^d\) and the truncation of the ANOVA (analysis of variance) decomposition with a variable transformation and a density estimation. The error decay rates and fast algorithms are transferred from the torus \({\mathbb T}^d\) to the domain \(\Omega\). A new extension method, which benefits from the Chui-Wang wavelets, allows the approximation of non-periodic functions too. Numerical experiments illustrate the performance of these procedures.