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Tractability of non-homogeneous tensor product problems in the worst case setting

Rong Guo, Heping Wang

TL;DR

The paper develops a unified spectral framework to analyze the tractability of non-homogeneous tensor product problems in the worst-case setting, providing necessary and sufficient conditions for strong polynomial, polynomial, quasi-polynomial, uniformly weak, and standard weak tractability, as well as (s,t)-weak tractability and curse-of-dimensionality scenarios under ABS and NOR error criteria. Central to the approach is Property (P), which ties tractability to the decay and distribution of univariate eigenvalues through the sequences h_k, A_*, and B, enabling precise characterizations via Theorem 2.1. The results are then specialized to kernel-based approximation problems—Euler, Wiener, Korobov, Gaussian, and analytic Korobov—yielding explicit conditions and exponents for SPT, PT, QPT, and related notions, along with practical corollaries agreeing with and extending prior work. This work advances understanding of how multidimensional approximation complexity scales with dimension and accuracy, informing algorithm design and highlighting when guarantees degrade or remain tractable in high dimensions.

Abstract

We study multivariate linear tensor product problems with some special properties in the worst case setting. We consider algorithms that use finitely many continuous linear functionals. We use a unified method to investigate tractability of the above multivariate problems, and obtain necessary and sufficient conditions for strong polynomial tractability, polynomial tractability, quasi-polynomial tractability, uniformly weak tractability, $(s,t)$-weak tractability, and weak tractability. Our results can apply to multivariate approximation problems with kernels corresponding to Euler kernels, Wiener kernels, Korobov kernels, Gaussian kernels, and analytic Korobov kernels.

Tractability of non-homogeneous tensor product problems in the worst case setting

TL;DR

The paper develops a unified spectral framework to analyze the tractability of non-homogeneous tensor product problems in the worst-case setting, providing necessary and sufficient conditions for strong polynomial, polynomial, quasi-polynomial, uniformly weak, and standard weak tractability, as well as (s,t)-weak tractability and curse-of-dimensionality scenarios under ABS and NOR error criteria. Central to the approach is Property (P), which ties tractability to the decay and distribution of univariate eigenvalues through the sequences h_k, A_*, and B, enabling precise characterizations via Theorem 2.1. The results are then specialized to kernel-based approximation problems—Euler, Wiener, Korobov, Gaussian, and analytic Korobov—yielding explicit conditions and exponents for SPT, PT, QPT, and related notions, along with practical corollaries agreeing with and extending prior work. This work advances understanding of how multidimensional approximation complexity scales with dimension and accuracy, informing algorithm design and highlighting when guarantees degrade or remain tractable in high dimensions.

Abstract

We study multivariate linear tensor product problems with some special properties in the worst case setting. We consider algorithms that use finitely many continuous linear functionals. We use a unified method to investigate tractability of the above multivariate problems, and obtain necessary and sufficient conditions for strong polynomial tractability, polynomial tractability, quasi-polynomial tractability, uniformly weak tractability, -weak tractability, and weak tractability. Our results can apply to multivariate approximation problems with kernels corresponding to Euler kernels, Wiener kernels, Korobov kernels, Gaussian kernels, and analytic Korobov kernels.

Paper Structure

This paper contains 9 sections, 9 theorems, 178 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and main results
  3. Proof of Theorem \ref{['thm2.1']}
  4. Applications of Theorem \ref{['thm2.1']}
  5. Function approximation with Euler kernels
  6. Function approximation with Wiener kernels
  7. Function approximation with Korobov kernels
  8. Function approximation with Gaussian kernels
  9. Function approximation with analytic Korobov kernels

Key Result

Theorem 2.1

Let $S=\{S_d\}_{d\in\Bbb N}$ be a non-homogeneous tensor product problem with Property (P) in the worst case setting. Then for the absolute error criterion or the normalized error criterion, we have (i) $S$ is strongly polynomially tractable iff and the exponent of SPT is (ii) $S$ is strongly polynomially tractable iff it is polynomially tractable. (iii) $S$ is quasi-polynomially tractable iff

Theorems & Definitions (17)

  • Theorem 2.1
  • Remark 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • Remark 3.1
  • Corollary 4.1
  • Remark 4.2
  • Corollary 4.3
  • ...and 7 more