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

Christian Rieger, Holger Wendland

TL;DR

It is shown that the approximation of high-dimensional functions, which are effectively low-dimensional, does not suffer from the curse of dimensionality, first in a general reproducing kernel Hilbert space set-up and then specifically for Sobolev and mixed-regularity Sobolev spaces.

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.

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

TL;DR

It is shown that the approximation of high-dimensional functions, which are effectively low-dimensional, does not suffer from the curse of dimensionality, first in a general reproducing kernel Hilbert space set-up and then specifically for Sobolev and mixed-regularity Sobolev spaces.

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.

Paper Structure

This paper contains 9 sections, 24 theorems, 134 equations.

Table of Contents

  1. Introduction
  2. Anchored Decomposition
  3. Approximation of Mismeasured Functions
  4. Application in Sobolev Spaces
  5. Application in Mixed Regularity Sobolev Spaces
  6. Weighted Norms and Their Connection to the Efficient Dimension
  7. Bounds on the Anchored Terms
  8. Application to Parametric Partial Differential Equations
  9. Bounds for Specific Weights

Key Result

Theorem 1.2

Let $\boldsymbol{c}\in[{\boldsymbol{a}},\boldsymbol{b}]$ be a fixed point, the anchor. Let $H\subseteq C[{\boldsymbol{a}},\boldsymbol{b}]$ a linear subspace of continuous functions. Any function $f\in H$ has an anchored decomposition, i.e. it can be written in the form where the components $f_{{{\mathfrak{u}}};\boldsymbol{c}}$ are functions depending only on the variables with indices in ${{\ma

Theorems & Definitions (49)

  • Definition 1.1
  • Theorem 1.2
  • Definition 2.1
  • Lemma 2.2
  • Proposition 2.3
  • proof
  • Theorem 2.4
  • proof
  • Remark 2.5
  • Corollary 2.6
  • ...and 39 more