Constructive Approximation of High-Dimensional Functions with Small Efficient Dimension with Applications in Uncertainty Quantification

Christian Rieger; Holger Wendland

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arXiv · 2411.18128

Constructive Approximation of High-Dimensional Functions with Small Efficient Dimension with Applications in Uncertainty Quantification

Christian Rieger·Holger Wendland

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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 set-up 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.

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Christian Rieger, Holger Wendland. 2024-11-27. Constructive Approximation of High-Dimensional Functions with Small Efficient Dimension with Applications in Uncertainty Quantification. https://arxiv.org/abs/2411.18128

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