Singular value decomposition versus sparse grids: refined complexity estimates

We compare the cost complexities of two approximation schemes for functions that live on the product domain $\varOmega _1\times \varOmega _2$ of sufficiently smooth domains $\varOmega _1\subset \mathbb{R}^{n_1}$ and $\varOmega _2\subset \mathbb{R}^{n_2}$, namely the singular value / Karhunen–Lòeve decomposition and the sparse grid representation. We assume that appropriate finite element methods with associated orders $r_1$ and $r_2$ of accuracy are given on the domains $\varOmega _1$ and $\varOmega _2$, respectively. This setting reflects practical needs, since often black-box solvers are used in numerical simulation, which restrict the freedom in the choice of the underlying discretization. We compare the cost complexities of the associated singular value decomposition and the associated sparse grid approximation. It turns out that, in this situation, the approximation by the sparse grid is always equal or superior to the approximation by the singular value decomposition. The results in this article improve and generalize those from the study by Griebel & Harbrecht (2014, Approximation of bi-variate functions. Singular value decomposition versus sparse grids. IMA J. Numer. Anal., 34, 28–54). Especially, we consider the approximation of functions from generalized isotropic and anisotropic Sobolev spaces.