Randomized Complexity of Parametric Integration and the Role of Adaption I. Finite Dimensional Case

TL;DR

This paper analyzes the randomized complexity of vector valued mean computation, a discrete analogue of parametric integration, establishing foundational results for Sobolev space analysis and addressing the role of adaptation in linear randomized problems.

Contribution

It provides new bounds for randomized minimal errors in vector mean computation and clarifies the impact of adaptation in linear problem complexity in the randomized setting.

Findings

Established bounds for randomized minimal errors in vector mean computation.

Extended previous complexity results to the finite-dimensional case.

Solved a fundamental problem regarding the power of adaptation in linear randomized problems.

Abstract

We study the randomized $n$-th minimal errors (and hence the complexity) of vector valued mean computation, which is the discrete version of parametric integration. The results of the present paper form the basis for the complexity analysis of parametric integration in Sobolev spaces, which will be presented in Part 2. Altogether this extends previous results of Heinrich and Sindambiwe (J.\ Complexity, 15 (1999), 317--341) and Wiegand (Shaker Verlag, 2006). Moreover, a basic problem of Information-Based Complexity on the power of adaption for linear problems in the randomized setting is solved.