Advertisement

Journal of Mathematical Chemistry

, Volume 49, Issue 4, pp 894–909 | Cite as

Fluctuation free multivariate integration based logarithmic HDMR in multivariate function representation

  • Burcu Tunga
  • Metin Demiralp
Original Paper

Abstract

This paper focuses on the Logarithmic High Dimensional Model Representation (Logarithmic HDMR) method which is a divide–and–conquer algorithm developed for multivariate function representation in terms of less-variate functions to reduce both the mathematical and the computational complexities. The main purpose of this work is to bypass the evaluation of N–tuple integrations appearing in Logarithmic HDMR by using the features of a new theorem named as Fluctuationlessness Approximation Theorem. This theorem can be used to evaluate the complicated integral structures of any scientific problem whose values can not be easily obtained analytically and it brings an approximation to the values of these integrals with the help of the matrix representation of functions. The Fluctuation Free Multivariate Integration Based Logarithmic HDMR method gives us the ability of reducing the complexity of the scientific problems of chemistry, physics, mathematics and engineering. A number of numerical implementations are also given at the end of the paper to show the performance of this new method.

Keywords

High dimensional model representation Multivariate functions Approximation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    I.M. Sobol, Sensitivity estimates for nonlinear mathematical models. Math. Modell. Comput. Exp. (MMCE), 1, 4.407 (1993)Google Scholar
  2. 2.
    Rabitz H., Alıcs Ö.: General foundations of high dimensional model representations. J. Math. Chem. 25, 197–233 (1999)CrossRefGoogle Scholar
  3. 3.
    Alıcs Ö., Rabitz H.: Efficient implementation of high dimensional model representations. J. Math. Chem. 29, 127–142 (2001)CrossRefGoogle Scholar
  4. 4.
    Li G., Rosenthal C., Rabitz H.: High dimensional model representations. J. Math. Chem. A 105, 7765–7777 (2001)Google Scholar
  5. 5.
    Demiralp M.: High dimensional model representation and its application varieties. Math. Res. 9, 146–159 (2003)Google Scholar
  6. 6.
    Ziehn T., Tomlin A.S.: 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
  7. 7.
    Ziehn T., Tomlin A.S.: GUI-HDMR—A software tool for global sensitivity analysis of complex models. Environ. Modell. Softw. 24, 775–785 (2009)CrossRefGoogle Scholar
  8. 8.
    Sridharan J., Chen T.: Modeling multiple input switching of CMOS gates in DSM technology using HDMR. Proc. Des. Autom. Test Eur. 1–3, 624–629 (2006)Google Scholar
  9. 9.
    Rao B.N., Chowdhury R.: 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.
    Chowdhury R., Rao B.N.: Hybrid high dimensional model representation for reliability analysis. Comput. Methods Appl. Mech. Eng. 198, 753–765 (2009)CrossRefGoogle Scholar
  11. 11.
    Gomez M.C., Tchijov V., Leon F., Aguilar A.: A tool to improve the execution time of air quality models. Environ. Modell. Softw. 23, 27–34 (2008)CrossRefGoogle Scholar
  12. 12.
    Banerjee I., Ierapetritou M.G.: Design optimization under parameter uncertainty for general black-box models. Ind. Eng. Chem. Res 41, 6687–6697 (2002)CrossRefGoogle Scholar
  13. 13.
    Banerjee I., Ierapetritou M.G.: Parametric process synthesis for general nonlinear models. Comput. Chem. Eng. 27, 1499–1512 (2003)CrossRefGoogle Scholar
  14. 14.
    Banerjee I., Ierapetritou M.G.: Model independent parametric decision making. Ann. Oper. Res. 132, 135–155 (2004)CrossRefGoogle Scholar
  15. 15.
    Shan S., Wang G.G.: Survey of modeling and optimization strategies to solve high-dimensional design problems with computationally-expensive black-box functions. Struct. Multidiscip. Optim. 41, 219–241 (2010)CrossRefGoogle Scholar
  16. 16.
    Tunga M.A., Demiralp M.: A factorized high dimensional model representation on the partitioned random discrete data. Appl. Num. Anal. Comp. Math. 1, 231–241 (2004)CrossRefGoogle Scholar
  17. 17.
    Tunga M.A., Demiralp M.: A factorized high dimensional model representation on the nodes of a finite hyperprismatic regular grid. Appl. Math. Comput. 164, 865–883 (2005)CrossRefGoogle Scholar
  18. 18.
    M. Demiralp, Logarithmic High Dimensional Model Representation, 6th WSEAS International Conference on Mathematics (MATH’06), May 27–29 (İstanbul, Turkey, 2006), pp. 157–161Google Scholar
  19. 19.
    M. Demiralp, A New Fluctuation Expansion Based Method for the Univariate Numerical Integration Under Gaussian Weights, WSEAS-2005 Proceedings, WSEAS 8-th International Conference on Applied Mathematics, 16–18 December (Tenerife, Spain, 2005), pp. 68–73Google Scholar
  20. 20.
    M. Demiralp, Convergence issues in the Gaussian weighted multidimensional fluctuation expansion for the univariate numerical Integration. WSEAS Tracsaction Math. 4, 486–492Google Scholar
  21. 21.
    M. Demiralp, Fluctuationlessness Theorem to Approximate Univariate Functions Matrix Representations (submitted)Google Scholar
  22. 22.
    Altnbaak S.U., Demiralp M.: Solutions to linear matrix ordinary differential equations via minimal, regular, and excessive space extension based universalization: convergence and error estimates for truncation approximants in the homogeneous case with premultiplying polynomial coefficient matrix. J. Math. Chem. 48(2), 266 (2010)CrossRefGoogle Scholar
  23. 23.
    Altnbaak S.U., Demiralp M.: Solutions to linear matrix ordinary differential equations via minimal, regular, and excessive space extension based universalization: perturbative matrix splines, convergence and error estimate issues for polynomial coefficients in the homogeneous case. J. Math. Chem. 48(2), 253 (2010)CrossRefGoogle Scholar
  24. 24.
    B. Tunga, M. Demiralp, The influence of the support functions on the quality of enhanced multivariance product representation, J. Math. Chem. (in press). doi: 10.1007/s10910-010-9714-2 (2010)
  25. 25.
    Tunga B., Demiralp M.: Constancy maximization based weight optimization in high dimensional model representation. Numer. Algorithms 52(3), 435–459 (2009)CrossRefGoogle Scholar
  26. 26.
    Demiralp M.: Fluctuationlessness theorem to approximate multivariate functions’ matrix representations. WSEAS Trans. Math. 8, 258–297 (2009)Google Scholar
  27. 27.
    A. Gil, J. Segura, N.M. Temme, Gauss quadrature, Numerical Methods for Special Functions, SIAM (2007)Google Scholar
  28. 28.
    William H., Flannery B.P., Teukolsky S.A., Vetterling W.T.: Gaussian Quadratures and Orthogonal Polynomials, Numerical Recipes in C. 2nd edn. Cambridge University Press, Cambridge, MA (1988)Google Scholar
  29. 29.
    W. Oevel, F. Postel, S. Wehmeier, J. Gerhard, The MuPAD Tutorial, Springer, 2000Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Informatics Institute, Computational Science and Engineering Program, Group for Science and Methods of Computation Maslakİstanbul Technical UniversityİstanbulTurkey

Personalised recommendations