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.
Similar content being viewed by others
References
I.M. Sobol, Sensitivity estimates for nonlinear mathematical models. Math. Modell. Comput. Exp. (MMCE), 1, 4.407 (1993)
Rabitz H., Alıcs Ö.: General foundations of high dimensional model representations. J. Math. Chem. 25, 197–233 (1999)
Alıcs Ö., Rabitz H.: Efficient implementation of high dimensional model representations. J. Math. Chem. 29, 127–142 (2001)
Li G., Rosenthal C., Rabitz H.: High dimensional model representations. J. Math. Chem. A 105, 7765–7777 (2001)
Demiralp M.: High dimensional model representation and its application varieties. Math. Res. 9, 146–159 (2003)
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)
Ziehn T., Tomlin A.S.: GUI-HDMR—A software tool for global sensitivity analysis of complex models. Environ. Modell. Softw. 24, 775–785 (2009)
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)
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)
Chowdhury R., Rao B.N.: Hybrid high dimensional model representation for reliability analysis. Comput. Methods Appl. Mech. Eng. 198, 753–765 (2009)
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)
Banerjee I., Ierapetritou M.G.: Design optimization under parameter uncertainty for general black-box models. Ind. Eng. Chem. Res 41, 6687–6697 (2002)
Banerjee I., Ierapetritou M.G.: Parametric process synthesis for general nonlinear models. Comput. Chem. Eng. 27, 1499–1512 (2003)
Banerjee I., Ierapetritou M.G.: Model independent parametric decision making. Ann. Oper. Res. 132, 135–155 (2004)
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)
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)
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)
M. Demiralp, Logarithmic High Dimensional Model Representation, 6th WSEAS International Conference on Mathematics (MATH’06), May 27–29 (İstanbul, Turkey, 2006), pp. 157–161
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. 68–73
M. Demiralp, Convergence issues in the Gaussian weighted multidimensional fluctuation expansion for the univariate numerical Integration. WSEAS Tracsaction Math. 4, 486–492
M. Demiralp, Fluctuationlessness Theorem to Approximate Univariate Functions Matrix Representations (submitted)
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)
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)
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)
Tunga B., Demiralp M.: Constancy maximization based weight optimization in high dimensional model representation. Numer. Algorithms 52(3), 435–459 (2009)
Demiralp M.: Fluctuationlessness theorem to approximate multivariate functions’ matrix representations. WSEAS Trans. Math. 8, 258–297 (2009)
A. Gil, J. Segura, N.M. Temme, Gauss quadrature, Numerical Methods for Special Functions, SIAM (2007)
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)
W. Oevel, F. Postel, S. Wehmeier, J. Gerhard, The MuPAD Tutorial, Springer, 2000
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Tunga, B., Demiralp, M. Fluctuation free multivariate integration based logarithmic HDMR in multivariate function representation. J Math Chem 49, 894–909 (2011). https://doi.org/10.1007/s10910-010-9786-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10910-010-9786-z