Some Results on the Complexity of Numerical Integration

TL;DR

This survey reviews the historical and recent developments in the complexity of numerical integration, focusing on high-dimensional and oscillatory integrals, highlighting challenges and new insights in the field.

Contribution

It provides a comprehensive overview from Bakhvalov's foundational work to recent advances, including corrections of earlier errors and new results on the curse of dimension and oscillatory integrals.

Findings

Discussion of the curse of dimension in numerical integration

Analysis of the complexity of oscillatory integrals

Corrections of previous errors in the literature

Abstract

This is a survey (21 pages, 124 references) written for the MCQMC 2014 conference in Leuven, April 2014. We start with the seminal paper of Bakhvalov (1959) and end with new results on the curse of dimension and on the complexity of oscillatory integrals. Some small errors of earlier versions are corrected.