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Breaking the curse of dimensionality

Volume 505 / 2015

Markus Weimar Dissertationes Mathematicae 505 (2015), 1-112 MSC: Primary 68Q25, 41A63; Secondary 41A65, 47N40, 65J10, 65Y20. DOI: 10.4064/dm505-0-1

Abstract

In modern science, efficient numerical treatment of high-dimensional problems becomes more and more important. A fundamental insight of the theory of information-based complexity (IBC for short) is that the computational hardness of a problem cannot be described properly only by the rate of convergence. There exist problems for which an exponential number of information operations is needed in order to reduce the initial error, although there are algorithms which provide an arbitrarily large rate of convergence. Problems that yield this exponential dependence are said to suffer from the curse of dimensionality. While analyzing numerical problems it turns out that we can often vanquish this curse by exploiting additional structural properties. The aim of this paper is to present several approaches of this type. Moreover, a detailed introduction to the field of IBC is given.

Authors

  • Markus WeimarWorkgroup Numerics and Optimization
    Faculty of Mathematics and Computer Science
    Philipps University Marburg
    Hans-Meerwein-Straße
    Lahnberge
    35032 Marburg, Germany
    e-mail

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