Theory of generalized discrepancies on a ball of arbitrary finite dimensions and algorithms for finding low-discrepancy point sets

  • Amna IshtiaqEmail author
  • Volker Michel
  • Hans Peter Scheffler
Original Paper
Part of the following topical collections:
  1. Mathematical Problems in Medical Imaging and Earth Sciences


One of the objectives of this paper is to extend the idea of the generalized discrepancy to the ball of arbitrary finite dimensions and to study its properties. We first construct orthonormal systems in higher dimensions, Sobolev spaces as well as particular differential operators. The discrepancy is then derived from an error estimate for numerical integration on a ball. We also present some new statistical and numerical properties of the generalized discrepancy which have also been unknown in three dimensions before. In addition, this paper focuses on constructing and implementing novel algorithms in order to obtain low-discrepancy point grids on the ball. Such low-discrepancy grids are uniformly distributed and are appropriate for quadrature rules (sometimes also called cubature rules) and as centres of localized trial functions in tomographic inverse problems on the ball. For this purpose, we also consider the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm and examine different BFGS updates and line search methods.


Ball Generalized discrepancy Low-discrepancy point grids Numerical integration Sobolev space 

Mathematics Subject Classification

46E35 47G30 60F05 62E20 65K10 65D30 65D32 



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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Geomathematics Group, Department of MathematicsUniversity of SiegenSiegenGermany
  2. 2.Department of MathematicsUniversity of SiegenSiegenGermany

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