s -numbers in information-based complexity

Abstract We shall study maximal errors of approximating linear problems. As possible classes of information operators the classes of arbitrary, continuous (nonlinear), or continuous linear information operators are considered. Algorithms also may be arbitrary, continuous (nonlinear), or linear. We focus our interest on two natural questions: (a) For what problems (the dependence on the underlying Banach spaces turns out to be crucial) do different classes of information or algorithms, respectively, yield the same quality of approximation? (b) What are the maximal differences in the errors of different classes? Both questions are treated in both the worst-case and average-case settings. Therefore the paper is divided into Parts A and B. For the study of the worst-case setting the notion of s-scales turns out to be powerful. An appropriate approach is also suggested for the average-case setting. Using the ideas of s-scales and function analytic methods we reprove some known results and obtain some new ones, thus answering questions posed in several papers on this subject.

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