Advertisement

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
  • 60 Downloads
Part of the following topical collections:
  1. Mathematical Problems in Medical Imaging and Earth Sciences

Abstract

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.

Keywords

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

Mathematics Subject Classification

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

Notes

References

  1. Akram, M., Amna, I., Michel, V.: A study of differential operators for complete orthonormal systems on a 3D ball. Int. J. Pure Appl. Math. 73, 489–506 (2011)MathSciNetzbMATHGoogle Scholar
  2. Amirbekyan, A.: The application of reproducing kernel based spline approximation to seismic surface and body wave tomography: theoretical aspects and numerical results. Ph.D. Thesis, Geomathematics Group, Department of Mathematics, University of Kaiserslautern. http://kluedo.ub.uni-kl.de/volltexte/2007/2103/ (2007). Accessed 13 May 2015
  3. Amirbekyan, A., Michel, V.: Splines on the three-dimensional ball and their application to seismic body wave tomography. Inverse Prob. 24, 015022 (25pp) (2008)MathSciNetCrossRefGoogle Scholar
  4. Amstler, C., Zinterhof, P.: Uniform distribution, discrepancy and reproducing kernel Hilbert spaces. J. Complex. 17, 497–515 (2001)MathSciNetCrossRefGoogle Scholar
  5. Ballani, L., Engels, J., Grafarend, E.W.: Global base functions for the mass density in the interior of a massive body (Earth). Manuscr. Geod. 18, 99–114 (1993)Google Scholar
  6. Beran, R.J.: Testing for uniformity on a compact homogeneous space. J. Appl. Probab. 5, 177–195 (1968)MathSciNetCrossRefGoogle Scholar
  7. Brauchart, J.S., Dick, J., Fang, L.: Spatial low-discrepancy sequences, spherical cone discrepancy, and applications in financial modeling. J. Comput. Appl. Math. 286, 28–53 (2015)MathSciNetCrossRefGoogle Scholar
  8. Choirat, C., Seri, R.: Computational aspects of Cui-Freeden statistics for equidistribution on the sphere. Math. Comput. 82, 2137–2156 (2013a)MathSciNetCrossRefGoogle Scholar
  9. Choirat, C., Seri, R.: Numerical properties of generalized discrepancies on spheres of arbitrary dimensions. J. Complex. 29, 216–235 (2013b)MathSciNetCrossRefGoogle Scholar
  10. Cui, J., Freeden, W.: Equidistribution on the sphere. SIAM J. Sci. Comput. 18, 595–609 (1997)MathSciNetCrossRefGoogle Scholar
  11. Dunkl, C.F., Xu, Y.: Orthogonal Polynomials of Several Variables. Cambridge University Press, Cambridge (2014)CrossRefGoogle Scholar
  12. Freeden, W., Gervens, T., Schreiner, M.: Lectures on Constructive Approximation on the Sphere with Applications to Geomathematics. Oxford University Press, Oxford (1998)zbMATHGoogle Scholar
  13. Gill, P.E., Murray, W.: Quasi-Newton methods for unconstrained optimization. IMA J. Appl. Math. 9, 91–108 (1972)MathSciNetCrossRefGoogle Scholar
  14. Gill, P.E., Golub, G.H., Murray, W., Saunders, M.A.: Methods for modifying matrix factorizations. Math. Comput. 28, 505–535 (1974)MathSciNetCrossRefGoogle Scholar
  15. Gill, P.E., Murray, W., Wright, H.M.: Practical Optimization. Academic Press, London (1981)zbMATHGoogle Scholar
  16. Hesse, K.: A lower bound for the worst-case cubature error on spheres of arbitrary dimensions. Numer. Math. 103, 413–433 (2006)MathSciNetCrossRefGoogle Scholar
  17. Ishtiaq, A.: Grid points and generalized discrepancies on the \(d\)-dimensional ball. Ph.D. Thesis, Geomathematics Group, Department of Mathematics, University of Siegen. http://dokumentix.ub.uni-siegen.de/opus/volltexte/2018/1373/ (2018)
  18. Ishtiaq, A., Michel, V.: Pseudodifferential operators, cubature and equidistribution on the 3D-ball—an approach based on orthonormal basis systems. Numer. Funct. Anal. Optim. 38, 891–910 (2017)MathSciNetCrossRefGoogle Scholar
  19. Leucht, A.: Degenerate U- and V- Statistics under weak dependence: asymptotic theory and Bootstrap consistency. Bernoulli 18, 552–585 (2012)MathSciNetCrossRefGoogle Scholar
  20. Leweke, S., Michel, V., Telschow, R.: On the non-uniqueness of gravitational and magnetic field data inversion (survey article). In: Freeden, W., Nashed, M.Z. (eds.) Handbook of Mathematical Geodesy, pp. 883–919. Birkhäuser, Basel (2018)CrossRefGoogle Scholar
  21. Michel, V.: Lectures on Constructive Approximation-Fourier, Spline and Wavelet Methods on the Real Line, the Sphere and the Ball. Birkhäuser, New York (2013)zbMATHGoogle Scholar
  22. Michel, V., Orzlowski, S.: On the null space of a class of Fredholm integral equations of the first kind. J. Inverse Ill-Posed Probl. 24, 687–710 (2016)MathSciNetCrossRefGoogle Scholar
  23. Müller, C.: Spherical Harmonics. Springer, Berlin (1996)zbMATHGoogle Scholar
  24. Nocedal, J., Wright, J.S.: Numerical Optimization. Springer, New York (2006)zbMATHGoogle Scholar
  25. Novak, E., Woźniakowski, H.: Tractability of Multivariate Problems, Volume II: Standard Information for Functionals. European Mathematical Society, Zürich (2008)Google Scholar
  26. Pycke, J.-R.: A decomposition for invariant tests of uniformity on the sphere. Proc. Am. Math. Soc. 135, 2983–2993 (2007a)MathSciNetCrossRefGoogle Scholar
  27. Pycke, J.-R.: U-statistics based on the Green’s function of the Laplacian on the circle and on the sphere. Stat. Probab. Lett. 77, 863–872 (2007b)MathSciNetCrossRefGoogle Scholar
  28. Serfling, R.J.: Approximation Theorems of Mathematical Statistics. Wiley, New York (1980)CrossRefGoogle Scholar
  29. Shorack, G.R.: Probability for Statistics. Springer, New York (2000)zbMATHGoogle Scholar
  30. Sloan, I.H.: When are quasi-Monte Carlo algorithms efficient for high dimensional integrals? J. Complex. 14, 1–33 (1998)MathSciNetCrossRefGoogle Scholar
  31. Sloan, I.H., Womersley, R.S.: Extremal systems of points and numerical integration on the sphere. Adv. Comput. Math. 21, 107–125 (2004)MathSciNetCrossRefGoogle Scholar
  32. Szegö, G.: Orthogonal Polynomials, vol. XXIII. AMS Colloquium Publications, Providence (1939)zbMATHGoogle Scholar
  33. Tscherning, C.C.: Isotropic reproducing kernels for the inner of a sphere or spherical shell and their use as density covariance functions. Math. Geol. 28, 161–168 (1996)MathSciNetCrossRefGoogle Scholar

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

Personalised recommendations