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Selected aspects of tractability analysis

Peter Kritzer

TL;DR

This paper surveys tractability analysis for multivariate problems, focusing on linear operators between Hilbert spaces and the role of singular values in information-based complexity. It highlights both classical results (polynomial and weak tractability) and modern extensions to exponential and generalized tractability, using Korobov spaces and their weighted variants as primary examples. The work connects eigenvalue decay, tensor-product structure, and average/weighted settings to precise tractability criteria, offering concrete conditions under which problems become tractable and illustrating these with detailed Korobov-space analyses and generalized tractability frameworks. The findings provide a rigorous lens for assessing the feasibility of high-dimensional numerical tasks and guide the design of information strategies (e.g., standard vs. all information) and weighting schemes to achieve favorable tractability properties.

Abstract

We give an overview of certain aspects of tractability analysis of multivariate problems. This paper is not intended to give a complete account of the subject, but provides an insight into how the theory works for particular types of problems. We mainly focus on linear problems on Hilbert spaces, and mostly allow arbitrary linear information. In such cases, tractability analysis is closely linked to an analysis of the singular values of the operator under consideration. We also highlight the more recent developments regarding exponential and generalized tractability. The theoretical results are illustrated by several examples throughout the article.

Selected aspects of tractability analysis

TL;DR

This paper surveys tractability analysis for multivariate problems, focusing on linear operators between Hilbert spaces and the role of singular values in information-based complexity. It highlights both classical results (polynomial and weak tractability) and modern extensions to exponential and generalized tractability, using Korobov spaces and their weighted variants as primary examples. The work connects eigenvalue decay, tensor-product structure, and average/weighted settings to precise tractability criteria, offering concrete conditions under which problems become tractable and illustrating these with detailed Korobov-space analyses and generalized tractability frameworks. The findings provide a rigorous lens for assessing the feasibility of high-dimensional numerical tasks and guide the design of information strategies (e.g., standard vs. all information) and weighting schemes to achieve favorable tractability properties.

Abstract

We give an overview of certain aspects of tractability analysis of multivariate problems. This paper is not intended to give a complete account of the subject, but provides an insight into how the theory works for particular types of problems. We mainly focus on linear problems on Hilbert spaces, and mostly allow arbitrary linear information. In such cases, tractability analysis is closely linked to an analysis of the singular values of the operator under consideration. We also highlight the more recent developments regarding exponential and generalized tractability. The theoretical results are illustrated by several examples throughout the article.
Paper Structure (12 sections, 7 theorems, 84 equations)

This paper contains 12 sections, 7 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Example: problems on Korobov spaces
  3. Tractability for linear problems on Hilbert spaces
  4. Criteria for tractability
  5. Linear tensor product problems
  6. The average case setting
  7. Example: problems on Korobov spaces, Part 2
  8. Exponential tractability
  9. Example: problems on Korobov spaces, Part 3
  10. Generalized tractability
  11. Example: problems on Korobov spaces, Part 4
  12. Conclusion

Key Result

Theorem 1

Let $\{{\mathcal{F}}_d\}_{d\in{\mathbb{N}}}$ and $\{{\mathcal{G}}_d\}_{d\in{\mathbb{N}}}$ be Hilbert spaces, and let $\{S_d\colon{\mathcal{F}}_d\to {\mathcal{G}}_d\}_{d\in{\mathbb{N}}}$ be compact linear operators. Consider information from $\Lambda^{\rm all}$ and the absolute worst case setting on The exponent of SPT is given by

Theorems & Definitions (8)

  • Theorem 1: Woźniakowski
  • proof : Sketch of the proof of Theorem \ref{['thm:ALG_SPT']}
  • Theorem 2: Woźniakowski
  • Theorem 3: Werschulz, Woźniakowski
  • Theorem 4: Woźniakowski
  • Theorem 5: Kritzer, Woźniakowski
  • Theorem 6: Kritzer, Woźniakowski
  • Theorem 7: Emenike, Kritzer, Hickernell