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Journal of Mathematical Chemistry

, Volume 51, Issue 7, pp 1784–1801 | Cite as

A novel piecewise multivariate function approximation method via universal matrix representation

  • Süha Tuna
  • Burcu Tunga
Original Paper

Abstract

Multivariance in science and engineering causes problematic situations even for continous and discrete cases. One way to overcome this situation is to decrease the multivariance level of the problem by using a divide—and—conquer based method. In this sense, Enhanced Multivariance Product Representation (EMPR) plays a part in the considered scenario and acts successfully. This method brings up a finite expansion to represent a multivariate function in terms of less-variate functions with the assistance of univariate support functions. This work aims to propose a new EMPR based algorithm which has two new features that improves the determination process of each expansion component through Fluctuation Free Integration method, which is an efficient method in evaluating multiple integrals through a universal matrix representation, and increases the approximation quality through inserting a piecewise structure into the standard EMPR algorithm. This new method is called Fluctuation Free Integration based piecewise EMPR. Some numerical implementations are also given to examine the performance of this proposed method.

Keywords

EMPR Piecewise functions Multidimensional problems  Approximation Matrix representation Numerical integration 

Mathematics Subject Classification (2000)

41A63 41A10 65F99 65D30 

Notes

Acknowledgments

Both authors are grateful to Prof. Demiralp for his contributions and valuable comments.

References

  1. 1.
    B. Tunga, M. Demiralp, The influence of the support functions on the quality of Enhanced Multivariance Product Representation. J. Math. Chem. 48, 827–840 (2010)CrossRefGoogle Scholar
  2. 2.
    M. Demiralp, Dominating the constancy in Enhanced Multivariance Product Representation (EMPR) via support function selection. In Proceedings of the 2nd International Conference on Applied Informatics and Computing Theory (AICT’11), 26–28 September, Keynote Speech, Prague, Czech Republic, pp. 11–12 (2011)Google Scholar
  3. 3.
    B. Kalay, M. Demiralp, Affine transformational Enhanced Multivariance Product Representation (ATEMPR) and its relation to rational approximants. In Proceedings for the 1st IEEEAM Conference on Applied Computer Science (ACS), pp. 336–340 (2010)Google Scholar
  4. 4.
    I.M. Sobol, Sensitivity estimates for nonlinear mathematical models. Math. Model. Comput. Exp. (MMCE) 1(4), 407–414 (1993)Google Scholar
  5. 5.
    M. Demiralp, High dimensional model representation and its application varieties. Math. Res. 9, 146–159 (2003)Google Scholar
  6. 6.
    G. Li, H. Rabitz, General formulation of HDMR component functions with independent and correlated variables. J. Math. Chem. 50, 99–130 (2012)CrossRefGoogle Scholar
  7. 7.
    T. Ziehn, A.S. Tomlin, A global sensitivity study of sulfur chemistry in a premixed methane flame model using HDMR. Int. J. Chem. Kinet. 40, 742–753 (2008)CrossRefGoogle Scholar
  8. 8.
    J. Sridharan, T. Chen, Modeling multiple input switching of CMOS gates in DSM technology using HDMR. Proc. Des. Autom. Test Eur. 1, 624–629 (2006)Google Scholar
  9. 9.
    B.N. Rao, R. Chowdhury, Probabilistic analysis using high dimensional model representation and fast fourier transform. Int. J. Comput. Methods Eng. Sci. Mech. 9, 342–357 (2008)CrossRefGoogle Scholar
  10. 10.
    A. Okan, N.A. Baykara, M. Demiralp, Weight optimization in Enhanced Multivariance Product Representation (EMPR) method. International conference of numerical analysis and applied mathematics. AIP Conf. Proc. 1281, 1935–1938 (2010)CrossRefGoogle Scholar
  11. 11.
    M. Demiralp, A new fluctuation expansion based method for the univariate numerical integration under Gaussian weights. In WSEAS 8-th International Conference on Applied Mathematics, Tenerife, Spain, pp. 68–73 (2005)Google Scholar
  12. 12.
    M. Demiralp, Convergence issues in the Gaussian weighted multidimensional fluctuation expansion for the univariate numerical integration. WSEAS Trans. Math. 4, 486–492 (2005)Google Scholar
  13. 13.
    M. Demiralp, Fluctuationlessness theorem to approximate univariate functions matrix representations. WSEAS Trans. Math. 8, 258–267 (2009)Google Scholar
  14. 14.
    M. Demiralp, Data production for a multivariate function on an orthogonal hyperprismatic grid via fuctuation free matrix representation: completely filled grid case. IJEECE 1, 61–76 (2010)Google Scholar
  15. 15.
    A. Langer, S. Osher, C. Schnlieb, Bregmanized domain decomposition for image restoration. J. Sci. Comput. 54, 549–576 (2013)Google Scholar
  16. 16.
    J. Geer, N.S. Banerjee, Exponentially accurate approximations to piece-wise smooth periodic functions. J. Sci. Comput. 12(3), 253–287 (1997)CrossRefGoogle Scholar
  17. 17.
    M.A. Tunga, M. Demiralp, Multivariate data modelling through piecewise generalized HDMR method. J. Math. Chem. 5, 1711–1726 (2012)CrossRefGoogle Scholar
  18. 18.
    B. Tunga, M. Demiralp, Support function influences on the univariance of the enhanced multivariance product representation. In Seventh International Conference of Computational Methods in Sciences and Engineering, Rhodes, Greece, 2009Google Scholar
  19. 19.
    B. Tunga, M. Demiralp, Fluctuation free multivariate integration based logarithmic HDMR in multivariate function representation. J. Math. Chem. 49, 894–909 (2011)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Computational Science and Engineering Program, Informatics Instituteİstanbul Technical UniversityMaslak, IstanbulTurkey
  2. 2.Department of Mathematics Engineering, Faculty of Science and Lettersİstanbul Technical UniversityMaslak, IstanbulTurkey

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