Tractability of multivariate analytic problems

TL;DR

This paper surveys and extends the study of the computational complexity of multivariate analytic problems, focusing on tractability in terms of the number of variables and logarithmic error bounds, with new results and open questions.

Contribution

It provides a comprehensive survey, introduces new tractability results for analytic problems over general spaces, and discusses future research directions.

Findings

Tractability results for multivariate analytic problems over general spaces.

Extension of existing results from Korobov spaces to broader settings.

Identification of open questions in the field.

Abstract

In the theory of tractability of multivariate problems one usually studies problems with finite smoothness. Then we want to know which $s$-variate problems can be approximated to within $\varepsilon$ by using, say, polynomially many in $s$ and $\varepsilon^{-1}$ function values or arbitrary linear functionals. There is a recent stream of work for multivariate analytic problems for which we want to answer the usual tractability questions with $\varepsilon^{-1}$ replaced by $1+\log \varepsilon^{-1}$. In this vein of research, multivariate integration and approximation have been studied over Korobov spaces with exponentially fast decaying Fourier coefficients. This is work of J. Dick, G. Larcher, and the authors. There is a natural need to analyze more general analytic problems defined over more general spaces and obtain tractability results in terms of $s$ and $1+\log \varepsilon^{-1}$. The goal of this paper is to survey the existing results, present some new results, and propose further questions for the study of tractability of multivariate analytic questions.