Constructive Approximation of High-Dimensional Functions with Small Efficient Dimension with Applications in Uncertainty Quantification
Abstract
Abstract. In this paper, we show that the approximation of high-dimensional functions, which are effectively low dimensional, does not suffer from the curse of dimensionality. This is shown first in a general reproducing kernel Hilbert space setup and then specifically for Sobolev and mixed-regularity Sobolev spaces. Finally, efficient estimates are derived for deciding whether a high-dimensional function is effectively low dimensional by studying error bounds in weighted reproducing kernel Hilbert spaces. The results are applied to parametric partial differential equations, a typical problem from uncertainty quantification.
Fields of Study & Subjects
Probabilistic and Robust Engineering Design
Numerical Methods and Algorithms
Mathematical Approximation and Integration
Publication details
- Journal / source
- SIAM Journal on Numerical Analysis
- Publisher
- Society for Industrial & Applied Mathematics (SIAM)
- Volume
- 64
- Issue
- 1
- Pages
- 1–28
- Publication date
- 6 January 2026
- Language
- EN