Entropy, Randomization, Derandomization, and Discrepancy

Title:

Entropy, Randomization, Derandomization, and Discrepancy

Author(s):

Gnewuch, Michael

Date:

2011

Type:

Technical reports

Department:

Computer Science

Permanent URL:

http://hdl.handle.net/10022/AC:P:10678

Series:

Columbia University Computer Science Technical Reports

Part Number:

CUCS02011

Publisher:

Department of Computer Science, Columbia University

Publisher Location:

New York

Abstract:

The star discrepancy is a measure of how uniformly distributed a finite point set is in the ddimensional unit cube. It is related to highdimensional numerical integration of certain function classes as expressed by the KoksmaHlawka inequality. A sharp version of this inequality states that the worstcase error of approximating the integral of functions from the unit ball of some Sobolev space by an equalweight cubature is exactly the star discrepancy of the set of sample points. In many applications, as, e.g., in physics, quantum chemistry or finance, it is essential to approximate highdimensional integrals. Thus with regard to the Koksma Hlawka inequality the following three questions are very important: (i) What are good bounds with explicitly given dependence on the dimension d for the smallest possible discrepancy of any npoint set for moderate n? (ii) How can we construct point sets efficiently that satisfy such bounds? (iii) How can we calculate the discrepancy of given point sets efficiently? We want to discuss these questions and survey and explain some approaches to tackle them relying on metric entropy, randomization, and derandomization.

Subject(s):

Computer science
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