Improved Sampling Inequalities for Sparse Grids and High-Dimensional Functions with Effective Low Dimension

2026/07/29 by Christian Rieger, Holger Wendland
Mathematics · Computer Science · #math.NA #cs.NA #msc:65D05 #msc:41A25 #msc:41A63

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arxiv created 2026/07/29 · arxiv updated 2026/07/30

Abstract

The approximation of high-dimensional functions is a challenging task due to the often appearing curse of dimensionality. In this paper, we combine sparse grid with anchored projection techniques to derive sampling inequalities for Sobolev functions of a dominating mixed regularity which are effectively low dimensional. To this end, we derive new sampling inequalities for sparse grids and combine these with recently investigated regression processes of non-matching sampling processes.

Citations

• Constructive Approximation of High-Dimensional Functions with Small Efficient Dimension with Applications in Uncertainty Quantification
• Extending error bounds for radial basis function interpolation to measuring the error in higher order Sobolev norms
• Uncertainty Quantification Using Periodic Random Variables
• Quasi-Monte Carlo Finite Element Methods for a Class of Elliptic Partial Differential Equations with Random Coefficients
• Variance components and generalized Sobol' indices

Related

• Towards optimal sampling for learning sparse approximation in high dimensions
• Sampling on energy-norm based sparse grids for the optimal recovery of Sobolev type functions in H^γ
• Sparse-grid sampling recovery and numerical integration of functions having mixed smoothness
• Weighted sampling recovery of functions with mixed smoothness
• High-dimensional sparse trigonometric approximation in the uniform norm and consequences for sampling recovery