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.

中文翻译：

---

奇异值分解与稀疏网格：改进的复杂度估计

对于存在于产品域上的函数，我们比较两种近似方案的成本复杂度| $ \ varOmega _1 \ times \ varOmega _2 $ | 足够平滑的域| $ \ varOmega _1 \ subset \ mathbb {R} ^ {n_1} $ | 和| $ \ varOmega _2 \ subset \ mathbb {R} ^ {n_2} $ |⁠，即奇异值/Karhunen-Lòeve分解和稀疏网格表示。我们假定具有关联顺序| $ r_1 $ |的适当有限元方法。和| $ r_2 $ | 精度在域上给出| $ \ varOmega _1 $ | 和| $ \ varOmega _2 $ |⁠， 分别。该设置反映了实际需求，因为数值模拟中经常使用黑盒求解器，这限制了基础离散化选择的自由度。我们比较了关联的奇异值分解和关联的稀疏网格近似的成本复杂性。结果表明，在这种情况下，稀疏网格的逼近始终等于或优于奇异值分解的逼近。本文中的结果改进和推广那些由格里贝尔＆Harbrecht研究（2014，的双变量函数逼近。奇异值分解还是稀疏的网格。IMA J. NUMER。元素分析，34，28-54）。特别是，我们考虑了广义各向同性的函数逼近和各向异性的Sobolev空间。