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Quasi-interpolation for high-dimensional function approximation

Wenwu Gao, Jiecheng Wang, Zhengjie Sun, Gregory E. Fasshauer

TL;DR

Both theoretical analysis and numerical implementations provide evidence that the proposed quasi-interpolation scheme is robust and is capable of mitigating the curse of dimensionality for approximating high-dimensional functions.

Abstract

The paper proposes a general quasi-interpolation scheme for high-dimensional function approximation. To facilitate error analysis, we view our quasi-interpolation as a two-step procedure. In the first step, we approximate a target function by a purpose-built convolution operator (with an error term referred to as convolution error). In the second step, we discretize the underlying convolution operator using certain quadrature rules at the given sampling data sites (with an error term called discretization error). The final approximation error is obtained as an optimally balanced sum of these two errors, which in turn views our quasi-interpolation as a regularization technique that balances convolution error and discretization error. As a concrete example, we construct a sparse grid quasi-interpolation scheme for high-dimensional function approximation. Both theoretical analysis and numerical implementations provide evidence that our quasi-interpolation scheme is robust and capable of mitigating the curse of dimensionality for approximating high-dimensional functions.

Quasi-interpolation for high-dimensional function approximation

TL;DR

Both theoretical analysis and numerical implementations provide evidence that the proposed quasi-interpolation scheme is robust and is capable of mitigating the curse of dimensionality for approximating high-dimensional functions.

Abstract

The paper proposes a general quasi-interpolation scheme for high-dimensional function approximation. To facilitate error analysis, we view our quasi-interpolation as a two-step procedure. In the first step, we approximate a target function by a purpose-built convolution operator (with an error term referred to as convolution error). In the second step, we discretize the underlying convolution operator using certain quadrature rules at the given sampling data sites (with an error term called discretization error). The final approximation error is obtained as an optimally balanced sum of these two errors, which in turn views our quasi-interpolation as a regularization technique that balances convolution error and discretization error. As a concrete example, we construct a sparse grid quasi-interpolation scheme for high-dimensional function approximation. Both theoretical analysis and numerical implementations provide evidence that our quasi-interpolation scheme is robust and capable of mitigating the curse of dimensionality for approximating high-dimensional functions.
Paper Structure (13 sections, 9 theorems, 91 equations, 3 figures, 6 tables)

This paper contains 13 sections, 9 theorems, 91 equations, 3 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations and definitions
  4. Periodization
  5. MQ trigonometric function
  6. Sparse grids
  7. The main result
  8. Sparse grid quasi-interpolation for approximating periodic functions defined on $\mathbb{T}^d$
  9. Quasi-interpolation defined on $[0,1]^d$
  10. Numerical Simulations
  11. Periodic function approximation over $\mathbb{T}^d$
  12. Non-periodic function approximation over unit hypercube
  13. Conclusions and discussions

Key Result

Lemma 2.1

Let a product transformation $\boldsymbol{\gamma}$, a product weight function $\boldsymbol{\omega}$, and the corresponding transformed function $f$ be defined in Equation periodicfunction. Then for any function $g\in L_2([0,1]^d,\boldsymbol{\omega})\cap C_{\text{mix}}^l([0,1]^d)$, we have $f\in\math for all multi-indices $\boldsymbol{\alpha}$ with $\|\boldsymbol{\alpha}\|_{\infty}\leq l$. Here $C_

Figures (3)

  • Figure 1: Full grid and sparse grid at level 7 on $[0,1]^{2}$, with $129^{2}=16641$ and $1281$ grid points, respectively
  • Figure 2: Numerical results of approximating derivatives of $f_d\in W^{4}_{\infty}([0,1]^{d})$ on sparse grid
  • Figure 3: Numerical results of approximating high-dimensional non-periodic function

Theorems & Definitions (17)

  • Definition 2.1
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Remark 3.1
  • Lemma 3.1
  • proof
  • Theorem 3.1
  • ...and 7 more