A brief history of information-based complexity

TL;DR

This paper reviews the development and foundational concepts of information-based complexity, a field studying the computational complexity of continuous problems using partial information, highlighting its history and key contributions.

Contribution

It provides a historical overview of IBC, including its origins, foundational theories, and development up to the present, emphasizing the work of Traub and Wozniakowski.

Findings

IBC addresses continuous problems like PDEs and high-dimensional integration.

Optimal iteration theory was initiated in the 1960s and developed further.

IBC has established tight complexity bounds using adversary arguments.

Abstract

This paper was presented on the occasion of an honorary doctoral degree for Henryk Wozniakowski at Friedrich Schiller University in Jena, Germany on June 6, 2008. Information-based complexity (IBC) is the study of algorithms and computational complexity of continuous problems. Examples of such problems include partial differential equations (in particular, the Schrodinger equation) very high dimensional integration, approximation, continuous optimization, and path integration. Because the computer has only partial information about the continuous mathematical problem adversary arguments at the information level often lead to tight complexity bounds. This may be contrasted with discrete problems where only conjectures that the complexity hierarchy does not collapse are available. This paper discusses precursors to IBC. It reports on the beginning of optimal iteration theory in the early 60s which was published in Traub's 1964 monograph. In the 70s Traub and Wozniakowski began to formulate the foundations of IBC leading to their 1980 monograph. The paper continues with the development of IBC to the present.